/*****************************************************************************/ /* */ /* 888888888 ,o, / 888 */ /* 888 88o88o " o8888o 88o8888o o88888o 888 o88888o */ /* 888 888 888 88b 888 888 888 888 888 d888 88b */ /* 888 888 888 o88^o888 888 888 "88888" 888 8888oo888 */ /* 888 888 888 C888 888 888 888 / 888 q888 */ /* 888 888 888 "88o^888 888 888 Cb 888 "88oooo" */ /* "8oo8D */ /* */ /* A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator. */ /* (triangle.c) */ /* */ /* Version 1.3 */ /* July 19, 1996 */ /* */ /* Copyright 1996 */ /* Jonathan Richard Shewchuk */ /* School of Computer Science */ /* Carnegie Mellon University */ /* 5000 Forbes Avenue */ /* Pittsburgh, Pennsylvania 15213-3891 */ /* jrs@cs.cmu.edu */ /* */ /* This program may be freely redistributed under the condition that the */ /* copyright notices (including this entire header and the copyright */ /* notice printed when the `-h' switch is selected) are not removed, and */ /* no compensation is received. Private, research, and institutional */ /* use is free. You may distribute modified versions of this code UNDER */ /* THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS MADE TO IT IN THE */ /* SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL AUTHOR, BOTH SOURCE */ /* AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT CHARGE, AND CLEAR */ /* NOTICE IS GIVEN OF THE MODIFICATIONS. Distribution of this code as */ /* part of a commercial system is permissible ONLY BY DIRECT ARRANGEMENT */ /* WITH THE AUTHOR. (If you are not directly supplying this code to a */ /* customer, and you are instead telling them how they can obtain it for */ /* free, then you are not required to make any arrangement with me.) */ /* */ /* Hypertext instructions for Triangle are available on the Web at */ /* */ /* http://www.cs.cmu.edu/~quake/triangle.html */ /* */ /* Some of the references listed below are marked [*]. These are available */ /* for downloading from the Web page */ /* */ /* http://www.cs.cmu.edu/~quake/triangle.research.html */ /* */ /* A paper discussing some aspects of Triangle is available. See Jonathan */ /* Richard Shewchuk, "Triangle: Engineering a 2D Quality Mesh Generator */ /* and Delaunay Triangulator," First Workshop on Applied Computational */ /* Geometry, ACM, May 1996. [*] */ /* */ /* Triangle was created as part of the Archimedes project in the School of */ /* Computer Science at Carnegie Mellon University. Archimedes is a */ /* system for compiling parallel finite element solvers. For further */ /* information, see Anja Feldmann, Omar Ghattas, John R. Gilbert, Gary L. */ /* Miller, David R. O'Hallaron, Eric J. Schwabe, Jonathan R. Shewchuk, */ /* and Shang-Hua Teng, "Automated Parallel Solution of Unstructured PDE */ /* Problems." To appear in Communications of the ACM, we hope. */ /* */ /* The quality mesh generation algorithm is due to Jim Ruppert, "A */ /* Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh */ /* Generation," Journal of Algorithms 18(3):548-585, May 1995. [*] */ /* */ /* My implementation of the divide-and-conquer and incremental Delaunay */ /* triangulation algorithms follows closely the presentation of Guibas */ /* and Stolfi, even though I use a triangle-based data structure instead */ /* of their quad-edge data structure. (In fact, I originally implemented */ /* Triangle using the quad-edge data structure, but switching to a */ /* triangle-based data structure sped Triangle by a factor of two.) The */ /* mesh manipulation primitives and the two aforementioned Delaunay */ /* triangulation algorithms are described by Leonidas J. Guibas and Jorge */ /* Stolfi, "Primitives for the Manipulation of General Subdivisions and */ /* the Computation of Voronoi Diagrams," ACM Transactions on Graphics */ /* 4(2):74-123, April 1985. */ /* */ /* Their O(n log n) divide-and-conquer algorithm is adapted from Der-Tsai */ /* Lee and Bruce J. Schachter, "Two Algorithms for Constructing the */ /* Delaunay Triangulation," International Journal of Computer and */ /* Information Science 9(3):219-242, 1980. The idea to improve the */ /* divide-and-conquer algorithm by alternating between vertical and */ /* horizontal cuts was introduced by Rex A. Dwyer, "A Faster Divide-and- */ /* Conquer Algorithm for Constructing Delaunay Triangulations," */ /* Algorithmica 2(2):137-151, 1987. */ /* */ /* The incremental insertion algorithm was first proposed by C. L. Lawson, */ /* "Software for C1 Surface Interpolation," in Mathematical Software III, */ /* John R. Rice, editor, Academic Press, New York, pp. 161-194, 1977. */ /* For point location, I use the algorithm of Ernst P. Mucke, Isaac */ /* Saias, and Binhai Zhu, "Fast Randomized Point Location Without */ /* Preprocessing in Two- and Three-dimensional Delaunay Triangulations," */ /* Proceedings of the Twelfth Annual Symposium on Computational Geometry, */ /* ACM, May 1996. [*] If I were to randomize the order of point */ /* insertion (I currently don't bother), their result combined with the */ /* result of Leonidas J. Guibas, Donald E. Knuth, and Micha Sharir, */ /* "Randomized Incremental Construction of Delaunay and Voronoi */ /* Diagrams," Algorithmica 7(4):381-413, 1992, would yield an expected */ /* O(n^{4/3}) bound on running time. */ /* */ /* The O(n log n) sweepline Delaunay triangulation algorithm is taken from */ /* Steven Fortune, "A Sweepline Algorithm for Voronoi Diagrams", */ /* Algorithmica 2(2):153-174, 1987. A random sample of edges on the */ /* boundary of the triangulation are maintained in a splay tree for the */ /* purpose of point location. Splay trees are described by Daniel */ /* Dominic Sleator and Robert Endre Tarjan, "Self-Adjusting Binary Search */ /* Trees," Journal of the ACM 32(3):652-686, July 1985. */ /* */ /* The algorithms for exact computation of the signs of determinants are */ /* described in Jonathan Richard Shewchuk, "Adaptive Precision Floating- */ /* Point Arithmetic and Fast Robust Geometric Predicates," Technical */ /* Report CMU-CS-96-140, School of Computer Science, Carnegie Mellon */ /* University, Pittsburgh, Pennsylvania, May 1996. [*] (Submitted to */ /* Discrete & Computational Geometry.) An abbreviated version appears as */ /* Jonathan Richard Shewchuk, "Robust Adaptive Floating-Point Geometric */ /* Predicates," Proceedings of the Twelfth Annual Symposium on Computa- */ /* tional Geometry, ACM, May 1996. [*] Many of the ideas for my exact */ /* arithmetic routines originate with Douglas M. Priest, "Algorithms for */ /* Arbitrary Precision Floating Point Arithmetic," Tenth Symposium on */ /* Computer Arithmetic, 132-143, IEEE Computer Society Press, 1991. [*] */ /* Many of the ideas for the correct evaluation of the signs of */ /* determinants are taken from Steven Fortune and Christopher J. Van Wyk, */ /* "Efficient Exact Arithmetic for Computational Geometry," Proceedings */ /* of the Ninth Annual Symposium on Computational Geometry, ACM, */ /* pp. 163-172, May 1993, and from Steven Fortune, "Numerical Stability */ /* of Algorithms for 2D Delaunay Triangulations," International Journal */ /* of Computational Geometry & Applications 5(1-2):193-213, March-June */ /* 1995. */ /* */ /* For definitions of and results involving Delaunay triangulations, */ /* constrained and conforming versions thereof, and other aspects of */ /* triangular mesh generation, see the excellent survey by Marshall Bern */ /* and David Eppstein, "Mesh Generation and Optimal Triangulation," in */ /* Computing and Euclidean Geometry, Ding-Zhu Du and Frank Hwang, */ /* editors, World Scientific, Singapore, pp. 23-90, 1992. */ /* */ /* The time for incrementally adding PSLG (planar straight line graph) */ /* segments to create a constrained Delaunay triangulation is probably */ /* O(n^2) per segment in the worst case and O(n) per edge in the common */ /* case, where n is the number of triangles that intersect the segment */ /* before it is inserted. This doesn't count point location, which can */ /* be much more expensive. (This note does not apply to conforming */ /* Delaunay triangulations, for which a different method is used to */ /* insert segments.) */ /* */ /* The time for adding segments to a conforming Delaunay triangulation is */ /* not clear, but does not depend upon n alone. In some cases, very */ /* small features (like a point lying next to a segment) can cause a */ /* single segment to be split an arbitrary number of times. Of course, */ /* floating-point precision is a practical barrier to how much this can */ /* happen. */ /* */ /* The time for deleting a point from a Delaunay triangulation is O(n^2) in */ /* the worst case and O(n) in the common case, where n is the degree of */ /* the point being deleted. I could improve this to expected O(n) time */ /* by "inserting" the neighboring vertices in random order, but n is */ /* usually quite small, so it's not worth the bother. (The O(n) time */ /* for random insertion follows from L. Paul Chew, "Building Voronoi */ /* Diagrams for Convex Polygons in Linear Expected Time," Technical */ /* Report PCS-TR90-147, Department of Mathematics and Computer Science, */ /* Dartmouth College, 1990. */ /* */ /* Ruppert's Delaunay refinement algorithm typically generates triangles */ /* at a linear rate (constant time per triangle) after the initial */ /* triangulation is formed. There may be pathological cases where more */ /* time is required, but these never arise in practice. */ /* */ /* The segment intersection formulae are straightforward. If you want to */ /* see them derived, see Franklin Antonio. "Faster Line Segment */ /* Intersection." In Graphics Gems III (David Kirk, editor), pp. 199- */ /* 202. Academic Press, Boston, 1992. */ /* */ /* If you make any improvements to this code, please please please let me */ /* know, so that I may obtain the improvements. Even if you don't change */ /* the code, I'd still love to hear what it's being used for. */ /* */ /* Disclaimer: Neither I nor Carnegie Mellon warrant this code in any way */ /* whatsoever. This code is provided "as-is". Use at your own risk. */ /* */ /*****************************************************************************/ /* For single precision (which will save some memory and reduce paging), */ /* define the symbol SINGLE by using the -DSINGLE compiler switch or by */ /* writing "#define SINGLE" below. */ /* */ /* For double precision (which will allow you to refine meshes to a smaller */ /* edge length), leave SINGLE undefined. */ /* */ /* Double precision uses more memory, but improves the resolution of the */ /* meshes you can generate with Triangle. It also reduces the likelihood */ /* of a floating exception due to overflow. Finally, it is much faster */ /* than single precision on 64-bit architectures like the DEC Alpha. I */ /* recommend double precision unless you want to generate a mesh for which */ /* you do not have enough memory. */ /* #define SINGLE */ #ifdef SINGLE #define REAL float #else /* not SINGLE */ #define REAL double #endif /* not SINGLE */ /* If yours is not a Unix system, define the NO_TIMER compiler switch to */ /* remove the Unix-specific timing code. */ /* #define NO_TIMER */ /* To insert lots of self-checks for internal errors, define the SELF_CHECK */ /* symbol. This will slow down the program significantly. It is best to */ /* define the symbol using the -DSELF_CHECK compiler switch, but you could */ /* write "#define SELF_CHECK" below. If you are modifying this code, I */ /* recommend you turn self-checks on. */ /* #define SELF_CHECK */ /* To compile Triangle as a callable object library (triangle.o), define the */ /* TRILIBRARY symbol. Read the file triangle.h for details on how to call */ /* the procedure triangulate() that results. */ /* #define TRILIBRARY */ /* It is possible to generate a smaller version of Triangle using one or */ /* both of the following symbols. Define the REDUCED symbol to eliminate */ /* all features that are primarily of research interest; specifically, the */ /* -i, -F, -s, and -C switches. Define the CDT_ONLY symbol to eliminate */ /* all meshing algorithms above and beyond constrained Delaunay */ /* triangulation; specifically, the -r, -q, -a, -S, and -s switches. */ /* These reductions are most likely to be useful when generating an object */ /* library (triangle.o) by defining the TRILIBRARY symbol. */ /* #define REDUCED */ /* #define CDT_ONLY */ /* On some machines, the exact arithmetic routines might be defeated by the */ /* use of internal extended precision floating-point registers. Sometimes */ /* this problem can be fixed by defining certain values to be volatile, */ /* thus forcing them to be stored to memory and rounded off. This isn't */ /* a great solution, though, as it slows Triangle down. */ /* */ /* To try this out, write "#define INEXACT volatile" below. Normally, */ /* however, INEXACT should be defined to be nothing. ("#define INEXACT".) */ #define INEXACT /* Nothing */ /* #define INEXACT volatile */ /* Maximum number of characters in a file name (including the null). */ #define FILENAMESIZE 512 /* Maximum number of characters in a line read from a file (including the */ /* null). */ #define INPUTLINESIZE 512 /* For efficiency, a variety of data structures are allocated in bulk. The */ /* following constants determine how many of each structure is allocated */ /* at once. */ #define TRIPERBLOCK 4092 /* Number of triangles allocated at once. */ #define SHELLEPERBLOCK 508 /* Number of shell edges allocated at once. */ #define POINTPERBLOCK 4092 /* Number of points allocated at once. */ #define VIRUSPERBLOCK 1020 /* Number of virus triangles allocated at once. */ /* Number of encroached segments allocated at once. */ #define BADSEGMENTPERBLOCK 252 /* Number of skinny triangles allocated at once. */ #define BADTRIPERBLOCK 4092 /* Number of splay tree nodes allocated at once. */ #define SPLAYNODEPERBLOCK 508 /* The point marker DEADPOINT is an arbitrary number chosen large enough to */ /* (hopefully) not conflict with user boundary markers. Make sure that it */ /* is small enough to fit into your machine's integer size. */ #define DEADPOINT -1073741824 /* The next line is used to outsmart some very stupid compilers. If your */ /* compiler is smarter, feel free to replace the "int" with "void". */ /* Not that it matters. */ #define VOID int /* Two constants for algorithms based on random sampling. Both constants */ /* have been chosen empirically to optimize their respective algorithms. */ /* Used for the point location scheme of Mucke, Saias, and Zhu, to decide */ /* how large a random sample of triangles to inspect. */ #define SAMPLEFACTOR 11 /* Used in Fortune's sweepline Delaunay algorithm to determine what fraction */ /* of boundary edges should be maintained in the splay tree for point */ /* location on the front. */ #define SAMPLERATE 10 /* A number that speaks for itself, every kissable digit. */ #define PI 3.141592653589793238462643383279502884197169399375105820974944592308 /* Another fave. */ #define SQUAREROOTTWO 1.4142135623730950488016887242096980785696718753769480732 /* And here's one for those of you who are intimidated by math. */ #define ONETHIRD 0.333333333333333333333333333333333333333333333333333333333333 #include #include #include #ifndef NO_TIMER #include #endif /* NO_TIMER */ #ifdef TRILIBRARY #include "triangle.h" #endif /* TRILIBRARY */ /* The following obscenity seems to be necessary to ensure that this program */ /* will port to Dec Alphas running OSF/1, because their stdio.h file commits */ /* the unpardonable sin of including stdlib.h. Hence, malloc(), free(), and */ /* exit() may or may not already be defined at this point. I declare these */ /* functions explicitly because some non-ANSI C compilers lack stdlib.h. */ #ifndef _STDLIB_H_ extern void *malloc(); extern void free(); extern void exit(); extern double strtod(); extern long strtol(); #endif /* _STDLIB_H_ */ /* A few forward declarations. */ void poolrestart(); #ifndef TRILIBRARY char *readline(); char *findfield(); #endif /* not TRILIBRARY */ /* Labels that signify whether a record consists primarily of pointers or of */ /* floating-point words. Used to make decisions about data alignment. */ enum wordtype {POINTER, FLOATINGPOINT}; /* Labels that signify the result of point location. The result of a */ /* search indicates that the point falls in the interior of a triangle, on */ /* an edge, on a vertex, or outside the mesh. */ enum locateresult {INTRIANGLE, ONEDGE, ONVERTEX, OUTSIDE}; /* Labels that signify the result of site insertion. The result indicates */ /* that the point was inserted with complete success, was inserted but */ /* encroaches on a segment, was not inserted because it lies on a segment, */ /* or was not inserted because another point occupies the same location. */ enum insertsiteresult {SUCCESSFULPOINT, ENCROACHINGPOINT, VIOLATINGPOINT, DUPLICATEPOINT}; /* Labels that signify the result of direction finding. The result */ /* indicates that a segment connecting the two query points falls within */ /* the direction triangle, along the left edge of the direction triangle, */ /* or along the right edge of the direction triangle. */ enum finddirectionresult {WITHIN, LEFTCOLLINEAR, RIGHTCOLLINEAR}; /* Labels that signify the result of the circumcenter computation routine. */ /* The return value indicates which edge of the triangle is shortest. */ enum circumcenterresult {OPPOSITEORG, OPPOSITEDEST, OPPOSITEAPEX}; /*****************************************************************************/ /* */ /* The basic mesh data structures */ /* */ /* There are three: points, triangles, and shell edges (abbreviated */ /* `shelle'). These three data structures, linked by pointers, comprise */ /* the mesh. A point simply represents a point in space and its properties.*/ /* A triangle is a triangle. A shell edge is a special data structure used */ /* to represent impenetrable segments in the mesh (including the outer */ /* boundary, boundaries of holes, and internal boundaries separating two */ /* triangulated regions). Shell edges represent boundaries defined by the */ /* user that triangles may not lie across. */ /* */ /* A triangle consists of a list of three vertices, a list of three */ /* adjoining triangles, a list of three adjoining shell edges (when shell */ /* edges are used), an arbitrary number of optional user-defined floating- */ /* point attributes, and an optional area constraint. The latter is an */ /* upper bound on the permissible area of each triangle in a region, used */ /* for mesh refinement. */ /* */ /* For a triangle on a boundary of the mesh, some or all of the neighboring */ /* triangles may not be present. For a triangle in the interior of the */ /* mesh, often no neighboring shell edges are present. Such absent */ /* triangles and shell edges are never represented by NULL pointers; they */ /* are represented by two special records: `dummytri', the triangle that */ /* fills "outer space", and `dummysh', the omnipresent shell edge. */ /* `dummytri' and `dummysh' are used for several reasons; for instance, */ /* they can be dereferenced and their contents examined without causing the */ /* memory protection exception that would occur if NULL were dereferenced. */ /* */ /* However, it is important to understand that a triangle includes other */ /* information as well. The pointers to adjoining vertices, triangles, and */ /* shell edges are ordered in a way that indicates their geometric relation */ /* to each other. Furthermore, each of these pointers contains orientation */ /* information. Each pointer to an adjoining triangle indicates which face */ /* of that triangle is contacted. Similarly, each pointer to an adjoining */ /* shell edge indicates which side of that shell edge is contacted, and how */ /* the shell edge is oriented relative to the triangle. */ /* */ /* Shell edges are found abutting edges of triangles; either sandwiched */ /* between two triangles, or resting against one triangle on an exterior */ /* boundary or hole boundary. */ /* */ /* A shell edge consists of a list of two vertices, a list of two */ /* adjoining shell edges, and a list of two adjoining triangles. One of */ /* the two adjoining triangles may not be present (though there should */ /* always be one), and neighboring shell edges might not be present. */ /* Shell edges also store a user-defined integer "boundary marker". */ /* Typically, this integer is used to indicate what sort of boundary */ /* conditions are to be applied at that location in a finite element */ /* simulation. */ /* */ /* Like triangles, shell edges maintain information about the relative */ /* orientation of neighboring objects. */ /* */ /* Points are relatively simple. A point is a list of floating point */ /* numbers, starting with the x, and y coordinates, followed by an */ /* arbitrary number of optional user-defined floating-point attributes, */ /* followed by an integer boundary marker. During the segment insertion */ /* phase, there is also a pointer from each point to a triangle that may */ /* contain it. Each pointer is not always correct, but when one is, it */ /* speeds up segment insertion. These pointers are assigned values once */ /* at the beginning of the segment insertion phase, and are not used or */ /* updated at any other time. Edge swapping during segment insertion will */ /* render some of them incorrect. Hence, don't rely upon them for */ /* anything. For the most part, points do not have any information about */ /* what triangles or shell edges they are linked to. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* Handles */ /* */ /* The oriented triangle (`triedge') and oriented shell edge (`edge') data */ /* structures defined below do not themselves store any part of the mesh. */ /* The mesh itself is made of `triangle's, `shelle's, and `point's. */ /* */ /* Oriented triangles and oriented shell edges will usually be referred to */ /* as "handles". A handle is essentially a pointer into the mesh; it */ /* allows you to "hold" one particular part of the mesh. Handles are used */ /* to specify the regions in which one is traversing and modifying the mesh.*/ /* A single `triangle' may be held by many handles, or none at all. (The */ /* latter case is not a memory leak, because the triangle is still */ /* connected to other triangles in the mesh.) */ /* */ /* A `triedge' is a handle that holds a triangle. It holds a specific side */ /* of the triangle. An `edge' is a handle that holds a shell edge. It */ /* holds either the left or right side of the edge. */ /* */ /* Navigation about the mesh is accomplished through a set of mesh */ /* manipulation primitives, further below. Many of these primitives take */ /* a handle and produce a new handle that holds the mesh near the first */ /* handle. Other primitives take two handles and glue the corresponding */ /* parts of the mesh together. The exact position of the handles is */ /* important. For instance, when two triangles are glued together by the */ /* bond() primitive, they are glued by the sides on which the handles lie. */ /* */ /* Because points have no information about which triangles they are */ /* attached to, I commonly represent a point by use of a handle whose */ /* origin is the point. A single handle can simultaneously represent a */ /* triangle, an edge, and a point. */ /* */ /*****************************************************************************/ /* The triangle data structure. Each triangle contains three pointers to */ /* adjoining triangles, plus three pointers to vertex points, plus three */ /* pointers to shell edges (defined below; these pointers are usually */ /* `dummysh'). It may or may not also contain user-defined attributes */ /* and/or a floating-point "area constraint". It may also contain extra */ /* pointers for nodes, when the user asks for high-order elements. */ /* Because the size and structure of a `triangle' is not decided until */ /* runtime, I haven't simply defined the type `triangle' to be a struct. */ typedef REAL **triangle; /* Really: typedef triangle *triangle */ /* An oriented triangle: includes a pointer to a triangle and orientation. */ /* The orientation denotes an edge of the triangle. Hence, there are */ /* three possible orientations. By convention, each edge is always */ /* directed to point counterclockwise about the corresponding triangle. */ struct triedge { triangle *tri; int orient; /* Ranges from 0 to 2. */ }; /* The shell data structure. Each shell edge contains two pointers to */ /* adjoining shell edges, plus two pointers to vertex points, plus two */ /* pointers to adjoining triangles, plus one shell marker. */ typedef REAL **shelle; /* Really: typedef shelle *shelle */ /* An oriented shell edge: includes a pointer to a shell edge and an */ /* orientation. The orientation denotes a side of the edge. Hence, there */ /* are two possible orientations. By convention, the edge is always */ /* directed so that the "side" denoted is the right side of the edge. */ struct edge { shelle *sh; int shorient; /* Ranges from 0 to 1. */ }; /* The point data structure. Each point is actually an array of REALs. */ /* The number of REALs is unknown until runtime. An integer boundary */ /* marker, and sometimes a pointer to a triangle, is appended after the */ /* REALs. */ typedef REAL *point; /* A queue used to store encroached segments. Each segment's vertices are */ /* stored so that one can check whether a segment is still the same. */ struct badsegment { struct edge encsegment; /* An encroached segment. */ point segorg, segdest; /* The two vertices. */ struct badsegment *nextsegment; /* Pointer to next encroached segment. */ }; /* A queue used to store bad triangles. The key is the square of the cosine */ /* of the smallest angle of the triangle. Each triangle's vertices are */ /* stored so that one can check whether a triangle is still the same. */ struct badface { struct triedge badfacetri; /* A bad triangle. */ REAL key; /* cos^2 of smallest (apical) angle. */ point faceorg, facedest, faceapex; /* The three vertices. */ struct badface *nextface; /* Pointer to next bad triangle. */ }; /* A node in a heap used to store events for the sweepline Delaunay */ /* algorithm. Nodes do not point directly to their parents or children in */ /* the heap. Instead, each node knows its position in the heap, and can */ /* look up its parent and children in a separate array. The `eventptr' */ /* points either to a `point' or to a triangle (in encoded format, so that */ /* an orientation is included). In the latter case, the origin of the */ /* oriented triangle is the apex of a "circle event" of the sweepline */ /* algorithm. To distinguish site events from circle events, all circle */ /* events are given an invalid (smaller than `xmin') x-coordinate `xkey'. */ struct event { REAL xkey, ykey; /* Coordinates of the event. */ VOID *eventptr; /* Can be a point or the location of a circle event. */ int heapposition; /* Marks this event's position in the heap. */ }; /* A node in the splay tree. Each node holds an oriented ghost triangle */ /* that represents a boundary edge of the growing triangulation. When a */ /* circle event covers two boundary edges with a triangle, so that they */ /* are no longer boundary edges, those edges are not immediately deleted */ /* from the tree; rather, they are lazily deleted when they are next */ /* encountered. (Since only a random sample of boundary edges are kept */ /* in the tree, lazy deletion is faster.) `keydest' is used to verify */ /* that a triangle is still the same as when it entered the splay tree; if */ /* it has been rotated (due to a circle event), it no longer represents a */ /* boundary edge and should be deleted. */ struct splaynode { struct triedge keyedge; /* Lprev of an edge on the front. */ point keydest; /* Used to verify that splay node is still live. */ struct splaynode *lchild, *rchild; /* Children in splay tree. */ }; /* A type used to allocate memory. firstblock is the first block of items. */ /* nowblock is the block from which items are currently being allocated. */ /* nextitem points to the next slab of free memory for an item. */ /* deaditemstack is the head of a linked list (stack) of deallocated items */ /* that can be recycled. unallocateditems is the number of items that */ /* remain to be allocated from nowblock. */ /* */ /* Traversal is the process of walking through the entire list of items, and */ /* is separate from allocation. Note that a traversal will visit items on */ /* the "deaditemstack" stack as well as live items. pathblock points to */ /* the block currently being traversed. pathitem points to the next item */ /* to be traversed. pathitemsleft is the number of items that remain to */ /* be traversed in pathblock. */ /* */ /* itemwordtype is set to POINTER or FLOATINGPOINT, and is used to suggest */ /* what sort of word the record is primarily made up of. alignbytes */ /* determines how new records should be aligned in memory. itembytes and */ /* itemwords are the length of a record in bytes (after rounding up) and */ /* words. itemsperblock is the number of items allocated at once in a */ /* single block. items is the number of currently allocated items. */ /* maxitems is the maximum number of items that have been allocated at */ /* once; it is the current number of items plus the number of records kept */ /* on deaditemstack. */ struct memorypool { VOID **firstblock, **nowblock; VOID *nextitem; VOID *deaditemstack; VOID **pathblock; VOID *pathitem; enum wordtype itemwordtype; int alignbytes; int itembytes, itemwords; int itemsperblock; long items, maxitems; int unallocateditems; int pathitemsleft; }; /* Variables used to allocate memory for triangles, shell edges, points, */ /* viri (triangles being eaten), bad (encroached) segments, bad (skinny */ /* or too large) triangles, and splay tree nodes. */ struct memorypool triangles; struct memorypool shelles; struct memorypool points; struct memorypool viri; struct memorypool badsegments; struct memorypool badtriangles; struct memorypool splaynodes; /* Variables that maintain the bad triangle queues. The tails are pointers */ /* to the pointers that have to be filled in to enqueue an item. */ struct badface *queuefront[64]; struct badface **queuetail[64]; REAL xmin, xmax, ymin, ymax; /* x and y bounds. */ REAL xminextreme; /* Nonexistent x value used as a flag in sweepline. */ int inpoints; /* Number of input points. */ int inelements; /* Number of input triangles. */ int insegments; /* Number of input segments. */ int holes; /* Number of input holes. */ int regions; /* Number of input regions. */ long edges; /* Number of output edges. */ int mesh_dim; /* Dimension (ought to be 2). */ int nextras; /* Number of attributes per point. */ int eextras; /* Number of attributes per triangle. */ long hullsize; /* Number of edges of convex hull. */ int triwords; /* Total words per triangle. */ int shwords; /* Total words per shell edge. */ int pointmarkindex; /* Index to find boundary marker of a point. */ int point2triindex; /* Index to find a triangle adjacent to a point. */ int highorderindex; /* Index to find extra nodes for high-order elements. */ int elemattribindex; /* Index to find attributes of a triangle. */ int areaboundindex; /* Index to find area bound of a triangle. */ int checksegments; /* Are there segments in the triangulation yet? */ int readnodefile; /* Has a .node file been read? */ long samples; /* Number of random samples for point location. */ unsigned long randomseed; /* Current random number seed. */ REAL splitter; /* Used to split REAL factors for exact multiplication. */ REAL epsilon; /* Floating-point machine epsilon. */ REAL resulterrbound; REAL ccwerrboundA, ccwerrboundB, ccwerrboundC; REAL iccerrboundA, iccerrboundB, iccerrboundC; long incirclecount; /* Number of incircle tests performed. */ long counterclockcount; /* Number of counterclockwise tests performed. */ long hyperbolacount; /* Number of right-of-hyperbola tests performed. */ long circumcentercount; /* Number of circumcenter calculations performed. */ long circletopcount; /* Number of circle top calculations performed. */ /* Switches for the triangulator. */ /* poly: -p switch. refine: -r switch. */ /* quality: -q switch. */ /* minangle: minimum angle bound, specified after -q switch. */ /* goodangle: cosine squared of minangle. */ /* vararea: -a switch without number. */ /* fixedarea: -a switch with number. */ /* maxarea: maximum area bound, specified after -a switch. */ /* regionattrib: -A switch. convex: -c switch. */ /* firstnumber: inverse of -z switch. All items are numbered starting */ /* from firstnumber. */ /* edgesout: -e switch. voronoi: -v switch. */ /* neighbors: -n switch. geomview: -g switch. */ /* nobound: -B switch. nopolywritten: -P switch. */ /* nonodewritten: -N switch. noelewritten: -E switch. */ /* noiterationnum: -I switch. noholes: -O switch. */ /* noexact: -X switch. */ /* order: element order, specified after -o switch. */ /* nobisect: count of how often -Y switch is selected. */ /* steiner: maximum number of Steiner points, specified after -S switch. */ /* steinerleft: number of Steiner points not yet used. */ /* incremental: -i switch. sweepline: -F switch. */ /* dwyer: inverse of -l switch. */ /* splitseg: -s switch. */ /* docheck: -C switch. */ /* quiet: -Q switch. verbose: count of how often -V switch is selected. */ /* useshelles: -p, -r, -q, or -c switch; determines whether shell edges */ /* are used at all. */ /* */ /* Read the instructions to find out the meaning of these switches. */ int poly, refine, quality, vararea, fixedarea, regionattrib, convex; int firstnumber; int edgesout, voronoi, neighbors, geomview; int nobound, nopolywritten, nonodewritten, noelewritten, noiterationnum; int noholes, noexact; int incremental, sweepline, dwyer; int splitseg; int docheck; int quiet, verbose; int useshelles; int order; int nobisect; int steiner, steinerleft; REAL minangle, goodangle; REAL maxarea; /* Variables for file names. */ #ifndef TRILIBRARY char innodefilename[FILENAMESIZE]; char inelefilename[FILENAMESIZE]; char inpolyfilename[FILENAMESIZE]; char areafilename[FILENAMESIZE]; char outnodefilename[FILENAMESIZE]; char outelefilename[FILENAMESIZE]; char outpolyfilename[FILENAMESIZE]; char edgefilename[FILENAMESIZE]; char vnodefilename[FILENAMESIZE]; char vedgefilename[FILENAMESIZE]; char neighborfilename[FILENAMESIZE]; char offfilename[FILENAMESIZE]; #endif /* not TRILIBRARY */ /* Triangular bounding box points. */ point infpoint1, infpoint2, infpoint3; /* Pointer to the `triangle' that occupies all of "outer space". */ triangle *dummytri; triangle *dummytribase; /* Keep base address so we can free() it later. */ /* Pointer to the omnipresent shell edge. Referenced by any triangle or */ /* shell edge that isn't really connected to a shell edge at that */ /* location. */ shelle *dummysh; shelle *dummyshbase; /* Keep base address so we can free() it later. */ /* Pointer to a recently visited triangle. Improves point location if */ /* proximate points are inserted sequentially. */ struct triedge recenttri; /*****************************************************************************/ /* */ /* Mesh manipulation primitives. Each triangle contains three pointers to */ /* other triangles, with orientations. Each pointer points not to the */ /* first byte of a triangle, but to one of the first three bytes of a */ /* triangle. It is necessary to extract both the triangle itself and the */ /* orientation. To save memory, I keep both pieces of information in one */ /* pointer. To make this possible, I assume that all triangles are aligned */ /* to four-byte boundaries. The `decode' routine below decodes a pointer, */ /* extracting an orientation (in the range 0 to 2) and a pointer to the */ /* beginning of a triangle. The `encode' routine compresses a pointer to a */ /* triangle and an orientation into a single pointer. My assumptions that */ /* triangles are four-byte-aligned and that the `unsigned long' type is */ /* long enough to hold a pointer are two of the few kludges in this program.*/ /* */ /* Shell edges are manipulated similarly. A pointer to a shell edge */ /* carries both an address and an orientation in the range 0 to 1. */ /* */ /* The other primitives take an oriented triangle or oriented shell edge, */ /* and return an oriented triangle or oriented shell edge or point; or they */ /* change the connections in the data structure. */ /* */ /*****************************************************************************/ /********* Mesh manipulation primitives begin here *********/ /** **/ /** **/ /* Fast lookup arrays to speed some of the mesh manipulation primitives. */ int plus1mod3[3] = {1, 2, 0}; int minus1mod3[3] = {2, 0, 1}; /********* Primitives for triangles *********/ /* */ /* */ /* decode() converts a pointer to an oriented triangle. The orientation is */ /* extracted from the two least significant bits of the pointer. */ #define decode(ptr, triedge) \ (triedge).orient = (int) ((unsigned long) (ptr) & (unsigned long) 3l); \ (triedge).tri = (triangle *) \ ((unsigned long) (ptr) ^ (unsigned long) (triedge).orient) /* encode() compresses an oriented triangle into a single pointer. It */ /* relies on the assumption that all triangles are aligned to four-byte */ /* boundaries, so the two least significant bits of (triedge).tri are zero.*/ #define encode(triedge) \ (triangle) ((unsigned long) (triedge).tri | (unsigned long) (triedge).orient) /* The following edge manipulation primitives are all described by Guibas */ /* and Stolfi. However, they use an edge-based data structure, whereas I */ /* am using a triangle-based data structure. */ /* sym() finds the abutting triangle, on the same edge. Note that the */ /* edge direction is necessarily reversed, because triangle/edge handles */ /* are always directed counterclockwise around the triangle. */ #define sym(triedge1, triedge2) \ ptr = (triedge1).tri[(triedge1).orient]; \ decode(ptr, triedge2); #define symself(triedge) \ ptr = (triedge).tri[(triedge).orient]; \ decode(ptr, triedge); /* lnext() finds the next edge (counterclockwise) of a triangle. */ #define lnext(triedge1, triedge2) \ (triedge2).tri = (triedge1).tri; \ (triedge2).orient = plus1mod3[(triedge1).orient] #define lnextself(triedge) \ (triedge).orient = plus1mod3[(triedge).orient] /* lprev() finds the previous edge (clockwise) of a triangle. */ #define lprev(triedge1, triedge2) \ (triedge2).tri = (triedge1).tri; \ (triedge2).orient = minus1mod3[(triedge1).orient] #define lprevself(triedge) \ (triedge).orient = minus1mod3[(triedge).orient] /* onext() spins counterclockwise around a point; that is, it finds the next */ /* edge with the same origin in the counterclockwise direction. This edge */ /* will be part of a different triangle. */ #define onext(triedge1, triedge2) \ lprev(triedge1, triedge2); \ symself(triedge2); #define onextself(triedge) \ lprevself(triedge); \ symself(triedge); /* oprev() spins clockwise around a point; that is, it finds the next edge */ /* with the same origin in the clockwise direction. This edge will be */ /* part of a different triangle. */ #define oprev(triedge1, triedge2) \ sym(triedge1, triedge2); \ lnextself(triedge2); #define oprevself(triedge) \ symself(triedge); \ lnextself(triedge); /* dnext() spins counterclockwise around a point; that is, it finds the next */ /* edge with the same destination in the counterclockwise direction. This */ /* edge will be part of a different triangle. */ #define dnext(triedge1, triedge2) \ sym(triedge1, triedge2); \ lprevself(triedge2); #define dnextself(triedge) \ symself(triedge); \ lprevself(triedge); /* dprev() spins clockwise around a point; that is, it finds the next edge */ /* with the same destination in the clockwise direction. This edge will */ /* be part of a different triangle. */ #define dprev(triedge1, triedge2) \ lnext(triedge1, triedge2); \ symself(triedge2); #define dprevself(triedge) \ lnextself(triedge); \ symself(triedge); /* rnext() moves one edge counterclockwise about the adjacent triangle. */ /* (It's best understood by reading Guibas and Stolfi. It involves */ /* changing triangles twice.) */ #define rnext(triedge1, triedge2) \ sym(triedge1, triedge2); \ lnextself(triedge2); \ symself(triedge2); #define rnextself(triedge) \ symself(triedge); \ lnextself(triedge); \ symself(triedge); /* rnext() moves one edge clockwise about the adjacent triangle. */ /* (It's best understood by reading Guibas and Stolfi. It involves */ /* changing triangles twice.) */ #define rprev(triedge1, triedge2) \ sym(triedge1, triedge2); \ lprevself(triedge2); \ symself(triedge2); #define rprevself(triedge) \ symself(triedge); \ lprevself(triedge); \ symself(triedge); /* These primitives determine or set the origin, destination, or apex of a */ /* triangle. */ #define org(triedge, pointptr) \ pointptr = (point) (triedge).tri[plus1mod3[(triedge).orient] + 3] #define dest(triedge, pointptr) \ pointptr = (point) (triedge).tri[minus1mod3[(triedge).orient] + 3] #define apex(triedge, pointptr) \ pointptr = (point) (triedge).tri[(triedge).orient + 3] #define setorg(triedge, pointptr) \ (triedge).tri[plus1mod3[(triedge).orient] + 3] = (triangle) pointptr #define setdest(triedge, pointptr) \ (triedge).tri[minus1mod3[(triedge).orient] + 3] = (triangle) pointptr #define setapex(triedge, pointptr) \ (triedge).tri[(triedge).orient + 3] = (triangle) pointptr #define setvertices2null(triedge) \ (triedge).tri[3] = (triangle) NULL; \ (triedge).tri[4] = (triangle) NULL; \ (triedge).tri[5] = (triangle) NULL; /* Bond two triangles together. */ #define bond(triedge1, triedge2) \ (triedge1).tri[(triedge1).orient] = encode(triedge2); \ (triedge2).tri[(triedge2).orient] = encode(triedge1) /* Dissolve a bond (from one side). Note that the other triangle will still */ /* think it's connected to this triangle. Usually, however, the other */ /* triangle is being deleted entirely, or bonded to another triangle, so */ /* it doesn't matter. */ #define dissolve(triedge) \ (triedge).tri[(triedge).orient] = (triangle) dummytri /* Copy a triangle/edge handle. */ #define triedgecopy(triedge1, triedge2) \ (triedge2).tri = (triedge1).tri; \ (triedge2).orient = (triedge1).orient /* Test for equality of triangle/edge handles. */ #define triedgeequal(triedge1, triedge2) \ (((triedge1).tri == (triedge2).tri) && \ ((triedge1).orient == (triedge2).orient)) /* Primitives to infect or cure a triangle with the virus. These rely on */ /* the assumption that all shell edges are aligned to four-byte boundaries.*/ #define infect(triedge) \ (triedge).tri[6] = (triangle) \ ((unsigned long) (triedge).tri[6] | (unsigned long) 2l) #define uninfect(triedge) \ (triedge).tri[6] = (triangle) \ ((unsigned long) (triedge).tri[6] & ~ (unsigned long) 2l) /* Test a triangle for viral infection. */ #define infected(triedge) \ (((unsigned long) (triedge).tri[6] & (unsigned long) 2l) != 0) /* Check or set a triangle's attributes. */ #define elemattribute(triedge, attnum) \ ((REAL *) (triedge).tri)[elemattribindex + (attnum)] #define setelemattribute(triedge, attnum, value) \ ((REAL *) (triedge).tri)[elemattribindex + (attnum)] = value /* Check or set a triangle's maximum area bound. */ #define areabound(triedge) ((REAL *) (triedge).tri)[areaboundindex] #define setareabound(triedge, value) \ ((REAL *) (triedge).tri)[areaboundindex] = value /********* Primitives for shell edges *********/ /* */ /* */ /* sdecode() converts a pointer to an oriented shell edge. The orientation */ /* is extracted from the least significant bit of the pointer. The two */ /* least significant bits (one for orientation, one for viral infection) */ /* are masked out to produce the real pointer. */ #define sdecode(sptr, edge) \ (edge).shorient = (int) ((unsigned long) (sptr) & (unsigned long) 1l); \ (edge).sh = (shelle *) \ ((unsigned long) (sptr) & ~ (unsigned long) 3l) /* sencode() compresses an oriented shell edge into a single pointer. It */ /* relies on the assumption that all shell edges are aligned to two-byte */ /* boundaries, so the least significant bit of (edge).sh is zero. */ #define sencode(edge) \ (shelle) ((unsigned long) (edge).sh | (unsigned long) (edge).shorient) /* ssym() toggles the orientation of a shell edge. */ #define ssym(edge1, edge2) \ (edge2).sh = (edge1).sh; \ (edge2).shorient = 1 - (edge1).shorient #define ssymself(edge) \ (edge).shorient = 1 - (edge).shorient /* spivot() finds the other shell edge (from the same segment) that shares */ /* the same origin. */ #define spivot(edge1, edge2) \ sptr = (edge1).sh[(edge1).shorient]; \ sdecode(sptr, edge2) #define spivotself(edge) \ sptr = (edge).sh[(edge).shorient]; \ sdecode(sptr, edge) /* snext() finds the next shell edge (from the same segment) in sequence; */ /* one whose origin is the input shell edge's destination. */ #define snext(edge1, edge2) \ sptr = (edge1).sh[1 - (edge1).shorient]; \ sdecode(sptr, edge2) #define snextself(edge) \ sptr = (edge).sh[1 - (edge).shorient]; \ sdecode(sptr, edge) /* These primitives determine or set the origin or destination of a shell */ /* edge. */ #define sorg(edge, pointptr) \ pointptr = (point) (edge).sh[2 + (edge).shorient] #define sdest(edge, pointptr) \ pointptr = (point) (edge).sh[3 - (edge).shorient] #define setsorg(edge, pointptr) \ (edge).sh[2 + (edge).shorient] = (shelle) pointptr #define setsdest(edge, pointptr) \ (edge).sh[3 - (edge).shorient] = (shelle) pointptr /* These primitives read or set a shell marker. Shell markers are used to */ /* hold user boundary information. */ #define mark(edge) (* (int *) ((edge).sh + 6)) #define setmark(edge, value) \ * (int *) ((edge).sh + 6) = value /* Bond two shell edges together. */ #define sbond(edge1, edge2) \ (edge1).sh[(edge1).shorient] = sencode(edge2); \ (edge2).sh[(edge2).shorient] = sencode(edge1) /* Dissolve a shell edge bond (from one side). Note that the other shell */ /* edge will still think it's connected to this shell edge. */ #define sdissolve(edge) \ (edge).sh[(edge).shorient] = (shelle) dummysh /* Copy a shell edge. */ #define shellecopy(edge1, edge2) \ (edge2).sh = (edge1).sh; \ (edge2).shorient = (edge1).shorient /* Test for equality of shell edges. */ #define shelleequal(edge1, edge2) \ (((edge1).sh == (edge2).sh) && \ ((edge1).shorient == (edge2).shorient)) /********* Primitives for interacting triangles and shell edges *********/ /* */ /* */ /* tspivot() finds a shell edge abutting a triangle. */ #define tspivot(triedge, edge) \ sptr = (shelle) (triedge).tri[6 + (triedge).orient]; \ sdecode(sptr, edge) /* stpivot() finds a triangle abutting a shell edge. It requires that the */ /* variable `ptr' of type `triangle' be defined. */ #define stpivot(edge, triedge) \ ptr = (triangle) (edge).sh[4 + (edge).shorient]; \ decode(ptr, triedge) /* Bond a triangle to a shell edge. */ #define tsbond(triedge, edge) \ (triedge).tri[6 + (triedge).orient] = (triangle) sencode(edge); \ (edge).sh[4 + (edge).shorient] = (shelle) encode(triedge) /* Dissolve a bond (from the triangle side). */ #define tsdissolve(triedge) \ (triedge).tri[6 + (triedge).orient] = (triangle) dummysh /* Dissolve a bond (from the shell edge side). */ #define stdissolve(edge) \ (edge).sh[4 + (edge).shorient] = (shelle) dummytri /********* Primitives for points *********/ /* */ /* */ #define pointmark(pt) ((int *) (pt))[pointmarkindex] #define setpointmark(pt, value) \ ((int *) (pt))[pointmarkindex] = value #define point2tri(pt) ((triangle *) (pt))[point2triindex] #define setpoint2tri(pt, value) \ ((triangle *) (pt))[point2triindex] = value /** **/ /** **/ /********* Mesh manipulation primitives end here *********/ /********* User interaction routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* syntax() Print list of command line switches. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void syntax() { #ifdef CDT_ONLY #ifdef REDUCED printf("triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n"); #else /* not REDUCED */ printf("triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n"); #endif /* not REDUCED */ #else /* not CDT_ONLY */ #ifdef REDUCED printf("triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n"); #else /* not REDUCED */ printf("triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n"); #endif /* not REDUCED */ #endif /* not CDT_ONLY */ printf(" -p Triangulates a Planar Straight Line Graph (.poly file).\n"); #ifndef CDT_ONLY printf(" -r Refines a previously generated mesh.\n"); printf( " -q Quality mesh generation. A minimum angle may be specified.\n"); printf(" -a Applies a maximum triangle area constraint.\n"); #endif /* not CDT_ONLY */ printf( " -A Applies attributes to identify elements in certain regions.\n"); printf(" -c Encloses the convex hull with segments.\n"); printf(" -e Generates an edge list.\n"); printf(" -v Generates a Voronoi diagram.\n"); printf(" -n Generates a list of triangle neighbors.\n"); printf(" -g Generates an .off file for Geomview.\n"); printf(" -B Suppresses output of boundary information.\n"); printf(" -P Suppresses output of .poly file.\n"); printf(" -N Suppresses output of .node file.\n"); printf(" -E Suppresses output of .ele file.\n"); printf(" -I Suppresses mesh iteration numbers.\n"); printf(" -O Ignores holes in .poly file.\n"); printf(" -X Suppresses use of exact arithmetic.\n"); printf(" -z Numbers all items starting from zero (rather than one).\n"); printf(" -o2 Generates second-order subparametric elements.\n"); #ifndef CDT_ONLY printf(" -Y Suppresses boundary segment splitting.\n"); printf(" -S Specifies maximum number of added Steiner points.\n"); #endif /* not CDT_ONLY */ #ifndef REDUCED printf(" -i Uses incremental method, rather than divide-and-conquer.\n"); printf(" -F Uses Fortune's sweepline algorithm, rather than d-and-c.\n"); #endif /* not REDUCED */ printf(" -l Uses vertical cuts only, rather than alternating cuts.\n"); #ifndef REDUCED #ifndef CDT_ONLY printf( " -s Force segments into mesh by splitting (instead of using CDT).\n"); #endif /* not CDT_ONLY */ printf(" -C Check consistency of final mesh.\n"); #endif /* not REDUCED */ printf(" -Q Quiet: No terminal output except errors.\n"); printf(" -V Verbose: Detailed information on what I'm doing.\n"); printf(" -h Help: Detailed instructions for Triangle.\n"); exit(0); } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* info() Print out complete instructions. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void info() { printf("Triangle\n"); printf( "A Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.\n"); printf("Version 1.3\n\n"); printf( "Copyright 1996 Jonathan Richard Shewchuk (bugs/comments to jrs@cs.cmu.edu)\n" ); printf("School of Computer Science / Carnegie Mellon University\n"); printf("5000 Forbes Avenue / Pittsburgh, Pennsylvania 15213-3891\n"); printf( "Created as part of the Archimedes project (tools for parallel FEM).\n"); printf( "Supported in part by NSF Grant CMS-9318163 and an NSERC 1967 Scholarship.\n"); printf("There is no warranty whatsoever. Use at your own risk.\n"); #ifdef SINGLE printf("This executable is compiled for single precision arithmetic.\n\n\n"); #else /* not SINGLE */ printf("This executable is compiled for double precision arithmetic.\n\n\n"); #endif /* not SINGLE */ printf( "Triangle generates exact Delaunay triangulations, constrained Delaunay\n"); printf( "triangulations, and quality conforming Delaunay triangulations. The latter\n" ); printf( "can be generated with no small angles, and are thus suitable for finite\n"); printf( "element analysis. If no command line switches are specified, your .node\n"); printf( "input file will be read, and the Delaunay triangulation will be returned in\n" ); printf(".node and .ele output files. The command syntax is:\n\n"); #ifdef CDT_ONLY #ifdef REDUCED printf("triangle [-pAcevngBPNEIOXzo_lQVh] input_file\n\n"); #else /* not REDUCED */ printf("triangle [-pAcevngBPNEIOXzo_iFlCQVh] input_file\n\n"); #endif /* not REDUCED */ #else /* not CDT_ONLY */ #ifdef REDUCED printf("triangle [-prq__a__AcevngBPNEIOXzo_YS__lQVh] input_file\n\n"); #else /* not REDUCED */ printf("triangle [-prq__a__AcevngBPNEIOXzo_YS__iFlsCQVh] input_file\n\n"); #endif /* not REDUCED */ #endif /* not CDT_ONLY */ printf( "Underscores indicate that numbers may optionally follow certain switches;\n"); printf( "do not leave any space between a switch and its numeric parameter.\n"); printf( "input_file must be a file with extension .node, or extension .poly if the\n"); printf( "-p switch is used. If -r is used, you must supply .node and .ele files,\n"); printf( "and possibly a .poly file and .area file as well. The formats of these\n"); printf("files are described below.\n\n"); printf("Command Line Switches:\n\n"); printf( " -p Reads a Planar Straight Line Graph (.poly file), which can specify\n" ); printf( " points, segments, holes, and regional attributes and area\n"); printf( " constraints. Will generate a constrained Delaunay triangulation\n"); printf( " fitting the input; or, if -s, -q, or -a is used, a conforming\n"); printf( " Delaunay triangulation. If -p is not used, Triangle reads a .node\n" ); printf(" file by default.\n"); printf( " -r Refines a previously generated mesh. The mesh is read from a .node\n" ); printf( " file and an .ele file. If -p is also used, a .poly file is read\n"); printf( " and used to constrain edges in the mesh. Further details on\n"); printf(" refinement are given below.\n"); printf( " -q Quality mesh generation by Jim Ruppert's Delaunay refinement\n"); printf( " algorithm. Adds points to the mesh to ensure that no angles\n"); printf( " smaller than 20 degrees occur. An alternative minimum angle may be\n" ); printf( " specified after the `q'. If the minimum angle is 20.7 degrees or\n"); printf( " smaller, the triangulation algorithm is theoretically guaranteed to\n" ); printf( " terminate (assuming infinite precision arithmetic - Triangle may\n"); printf( " fail to terminate if you run out of precision). In practice, the\n"); printf( " algorithm often succeeds for minimum angles up to 33.8 degrees.\n"); printf( " For highly refined meshes, however, it may be necessary to reduce\n"); printf( " the minimum angle to well below 20 to avoid problems associated\n"); printf( " with insufficient floating-point precision. The specified angle\n"); printf(" may include a decimal point.\n"); printf( " -a Imposes a maximum triangle area. If a number follows the `a', no\n"); printf( " triangle will be generated whose area is larger than that number.\n"); printf( " If no number is specified, an .area file (if -r is used) or .poly\n"); printf( " file (if -r is not used) specifies a number of maximum area\n"); printf( " constraints. An .area file contains a separate area constraint for\n" ); printf( " each triangle, and is useful for refining a finite element mesh\n"); printf( " based on a posteriori error estimates. A .poly file can optionally\n" ); printf( " contain an area constraint for each segment-bounded region, thereby\n" ); printf( " enforcing triangle densities in a first triangulation. You can\n"); printf( " impose both a fixed area constraint and a varying area constraint\n"); printf( " by invoking the -a switch twice, once with and once without a\n"); printf( " number following. Each area specified may include a decimal point.\n" ); printf( " -A Assigns an additional attribute to each triangle that identifies\n"); printf( " what segment-bounded region each triangle belongs to. Attributes\n"); printf( " are assigned to regions by the .poly file. If a region is not\n"); printf( " explicitly marked by the .poly file, triangles in that region are\n"); printf( " assigned an attribute of zero. The -A switch has an effect only\n"); printf(" when the -p switch is used and the -r switch is not.\n"); printf( " -c Creates segments on the convex hull of the triangulation. If you\n"); printf( " are triangulating a point set, this switch causes a .poly file to\n"); printf( " be written, containing all edges in the convex hull. (By default,\n" ); printf( " a .poly file is written only if a .poly file is read.) If you are\n" ); printf( " triangulating a PSLG, this switch specifies that the interior of\n"); printf( " the convex hull of the PSLG should be triangulated. If you do not\n" ); printf( " use this switch when triangulating a PSLG, it is assumed that you\n"); printf( " have identified the region to be triangulated by surrounding it\n"); printf( " with segments of the input PSLG. Beware: if you are not careful,\n" ); printf( " this switch can cause the introduction of an extremely thin angle\n"); printf( " between a PSLG segment and a convex hull segment, which can cause\n"); printf( " overrefinement or failure if Triangle runs out of precision. If\n"); printf( " you are refining a mesh, the -c switch works differently; it\n"); printf( " generates the set of boundary edges of the mesh, rather than the\n"); printf(" convex hull.\n"); printf( " -e Outputs (to an .edge file) a list of edges of the triangulation.\n"); printf( " -v Outputs the Voronoi diagram associated with the triangulation.\n"); printf(" Does not attempt to detect degeneracies.\n"); printf( " -n Outputs (to a .neigh file) a list of triangles neighboring each\n"); printf(" triangle.\n"); printf( " -g Outputs the mesh to an Object File Format (.off) file, suitable for\n" ); printf(" viewing with the Geometry Center's Geomview package.\n"); printf( " -B No boundary markers in the output .node, .poly, and .edge output\n"); printf( " files. See the detailed discussion of boundary markers below.\n"); printf( " -P No output .poly file. Saves disk space, but you lose the ability\n"); printf( " to impose segment constraints on later refinements of the mesh.\n"); printf(" -N No output .node file.\n"); printf(" -E No output .ele file.\n"); printf( " -I No iteration numbers. Suppresses the output of .node and .poly\n"); printf( " files, so your input files won't be overwritten. (If your input is\n" ); printf( " a .poly file only, a .node file will be written.) Cannot be used\n"); printf( " with the -r switch, because that would overwrite your input .ele\n"); printf( " file. Shouldn't be used with the -s, -q, or -a switch if you are\n"); printf( " using a .node file for input, because no .node file will be\n"); printf(" written, so there will be no record of any added points.\n"); printf(" -O No holes. Ignores the holes in the .poly file.\n"); printf( " -X No exact arithmetic. Normally, Triangle uses exact floating-point\n" ); printf( " arithmetic for certain tests if it thinks the inexact tests are not\n" ); printf( " accurate enough. Exact arithmetic ensures the robustness of the\n"); printf( " triangulation algorithms, despite floating-point roundoff error.\n"); printf( " Disabling exact arithmetic with the -X switch will cause a small\n"); printf( " improvement in speed and create the possibility (albeit small) that\n" ); printf( " Triangle will fail to produce a valid mesh. Not recommended.\n"); printf( " -z Numbers all items starting from zero (rather than one). Note that\n" ); printf( " this switch is normally overrided by the value used to number the\n"); printf( " first point of the input .node or .poly file. However, this switch\n" ); printf(" is useful when calling Triangle from another program.\n"); printf( " -o2 Generates second-order subparametric elements with six nodes each.\n" ); printf( " -Y No new points on the boundary. This switch is useful when the mesh\n" ); printf( " boundary must be preserved so that it conforms to some adjacent\n"); printf( " mesh. Be forewarned that you will probably sacrifice some of the\n"); printf( " quality of the mesh; Triangle will try, but the resulting mesh may\n" ); printf( " contain triangles of poor aspect ratio. Works well if all the\n"); printf( " boundary points are closely spaced. Specify this switch twice\n"); printf( " (`-YY') to prevent all segment splitting, including internal\n"); printf(" boundaries.\n"); printf( " -S Specifies the maximum number of Steiner points (points that are not\n" ); printf( " in the input, but are added to meet the constraints of minimum\n"); printf( " angle and maximum area). The default is to allow an unlimited\n"); printf( " number. If you specify this switch with no number after it,\n"); printf( " the limit is set to zero. Triangle always adds points at segment\n"); printf( " intersections, even if it needs to use more points than the limit\n"); printf( " you set. When Triangle inserts segments by splitting (-s), it\n"); printf( " always adds enough points to ensure that all the segments appear in\n" ); printf( " the triangulation, again ignoring the limit. Be forewarned that\n"); printf( " the -S switch may result in a conforming triangulation that is not\n" ); printf( " truly Delaunay, because Triangle may be forced to stop adding\n"); printf( " points when the mesh is in a state where a segment is non-Delaunay\n" ); printf( " and needs to be split. If so, Triangle will print a warning.\n"); printf( " -i Uses an incremental rather than divide-and-conquer algorithm to\n"); printf( " form a Delaunay triangulation. Try it if the divide-and-conquer\n"); printf(" algorithm fails.\n"); printf( " -F Uses Steven Fortune's sweepline algorithm to form a Delaunay\n"); printf( " triangulation. Warning: does not use exact arithmetic for all\n"); printf(" calculations. An exact result is not guaranteed.\n"); printf( " -l Uses only vertical cuts in the divide-and-conquer algorithm. By\n"); printf( " default, Triangle uses alternating vertical and horizontal cuts,\n"); printf( " which usually improve the speed except with point sets that are\n"); printf( " small or short and wide. This switch is primarily of theoretical\n"); printf(" interest.\n"); printf( " -s Specifies that segments should be forced into the triangulation by\n" ); printf( " recursively splitting them at their midpoints, rather than by\n"); printf( " generating a constrained Delaunay triangulation. Segment splitting\n" ); printf( " is true to Ruppert's original algorithm, but can create needlessly\n" ); printf(" small triangles near external small features.\n"); printf( " -C Check the consistency of the final mesh. Uses exact arithmetic for\n" ); printf( " checking, even if the -X switch is used. Useful if you suspect\n"); printf(" Triangle is buggy.\n"); printf( " -Q Quiet: Suppresses all explanation of what Triangle is doing, unless\n" ); printf(" an error occurs.\n"); printf( " -V Verbose: Gives detailed information about what Triangle is doing.\n"); printf( " Add more `V's for increasing amount of detail. `-V' gives\n"); printf( " information on algorithmic progress and more detailed statistics.\n"); printf( " `-VV' gives point-by-point details, and will print so much that\n"); printf( " Triangle will run much more slowly. `-VVV' gives information only\n" ); printf(" a debugger could love.\n"); printf(" -h Help: Displays these instructions.\n"); printf("\n"); printf("Definitions:\n"); printf("\n"); printf( " A Delaunay triangulation of a point set is a triangulation whose vertices\n" ); printf( " are the point set, having the property that no point in the point set\n"); printf( " falls in the interior of the circumcircle (circle that passes through all\n" ); printf(" three vertices) of any triangle in the triangulation.\n\n"); printf( " A Voronoi diagram of a point set is a subdivision of the plane into\n"); printf( " polygonal regions (some of which may be infinite), where each region is\n"); printf( " the set of points in the plane that are closer to some input point than\n"); printf( " to any other input point. (The Voronoi diagram is the geometric dual of\n" ); printf(" the Delaunay triangulation.)\n\n"); printf( " A Planar Straight Line Graph (PSLG) is a collection of points and\n"); printf( " segments. Segments are simply edges, whose endpoints are points in the\n"); printf( " PSLG. The file format for PSLGs (.poly files) is described below.\n"); printf("\n"); printf( " A constrained Delaunay triangulation of a PSLG is similar to a Delaunay\n"); printf( " triangulation, but each PSLG segment is present as a single edge in the\n"); printf( " triangulation. (A constrained Delaunay triangulation is not truly a\n"); printf(" Delaunay triangulation.)\n\n"); printf( " A conforming Delaunay triangulation of a PSLG is a true Delaunay\n"); printf( " triangulation in which each PSLG segment may have been subdivided into\n"); printf( " several edges by the insertion of additional points. These inserted\n"); printf( " points are necessary to allow the segments to exist in the mesh while\n"); printf(" maintaining the Delaunay property.\n\n"); printf("File Formats:\n\n"); printf( " All files may contain comments prefixed by the character '#'. Points,\n"); printf( " triangles, edges, holes, and maximum area constraints must be numbered\n"); printf( " consecutively, starting from either 1 or 0. Whichever you choose, all\n"); printf( " input files must be consistent; if the nodes are numbered from 1, so must\n" ); printf( " be all other objects. Triangle automatically detects your choice while\n"); printf( " reading the .node (or .poly) file. (When calling Triangle from another\n"); printf( " program, use the -z switch if you wish to number objects from zero.)\n"); printf(" Examples of these file formats are given below.\n\n"); printf(" .node files:\n"); printf( " First line: <# of points> <# of attributes>\n"); printf( " <# of boundary markers (0 or 1)>\n" ); printf( " Remaining lines: [attributes] [boundary marker]\n"); printf("\n"); printf( " The attributes, which are typically floating-point values of physical\n"); printf( " quantities (such as mass or conductivity) associated with the nodes of\n" ); printf( " a finite element mesh, are copied unchanged to the output mesh. If -s,\n" ); printf( " -q, or -a is selected, each new Steiner point added to the mesh will\n"); printf(" have attributes assigned to it by linear interpolation.\n\n"); printf( " If the fourth entry of the first line is `1', the last column of the\n"); printf( " remainder of the file is assumed to contain boundary markers. Boundary\n" ); printf( " markers are used to identify boundary points and points resting on PSLG\n" ); printf( " segments; a complete description appears in a section below. The .node\n" ); printf( " file produced by Triangle will contain boundary markers in the last\n"); printf(" column unless they are suppressed by the -B switch.\n\n"); printf(" .ele files:\n"); printf( " First line: <# of triangles> <# of attributes>\n"); printf( " Remaining lines: ... [attributes]\n" ); printf("\n"); printf( " Points are indices into the corresponding .node file. The first three\n" ); printf( " points are the corners, and are listed in counterclockwise order around\n" ); printf( " each triangle. (The remaining points, if any, depend on the type of\n"); printf( " finite element used.) The attributes are just like those of .node\n"); printf( " files. Because there is no simple mapping from input to output\n"); printf( " triangles, an attempt is made to interpolate attributes, which may\n"); printf( " result in a good deal of diffusion of attributes among nearby triangles\n" ); printf( " as the triangulation is refined. Diffusion does not occur across\n"); printf( " segments, so attributes used to identify segment-bounded regions remain\n" ); printf( " intact. In output .ele files, all triangles have three points each\n"); printf( " unless the -o2 switch is used, in which case they have six, and the\n"); printf( " fourth, fifth, and sixth points lie on the midpoints of the edges\n"); printf(" opposite the first, second, and third corners.\n\n"); printf(" .poly files:\n"); printf( " First line: <# of points> <# of attributes>\n"); printf( " <# of boundary markers (0 or 1)>\n" ); printf( " Following lines: [attributes] [boundary marker]\n"); printf(" One line: <# of segments> <# of boundary markers (0 or 1)>\n"); printf( " Following lines: [boundary marker]\n"); printf(" One line: <# of holes>\n"); printf(" Following lines: \n"); printf( " Optional line: <# of regional attributes and/or area constraints>\n"); printf( " Optional following lines: \n"); printf("\n"); printf( " A .poly file represents a PSLG, as well as some additional information.\n" ); printf( " The first section lists all the points, and is identical to the format\n" ); printf( " of .node files. <# of points> may be set to zero to indicate that the\n" ); printf( " points are listed in a separate .node file; .poly files produced by\n"); printf( " Triangle always have this format. This has the advantage that a point\n" ); printf( " set may easily be triangulated with or without segments. (The same\n"); printf( " effect can be achieved, albeit using more disk space, by making a copy\n" ); printf( " of the .poly file with the extension .node; all sections of the file\n"); printf(" but the first are ignored.)\n\n"); printf( " The second section lists the segments. Segments are edges whose\n"); printf( " presence in the triangulation is enforced. Each segment is specified\n"); printf( " by listing the indices of its two endpoints. This means that you must\n" ); printf( " include its endpoints in the point list. If -s, -q, and -a are not\n"); printf( " selected, Triangle will produce a constrained Delaunay triangulation,\n"); printf( " in which each segment appears as a single edge in the triangulation.\n"); printf( " If -q or -a is selected, Triangle will produce a conforming Delaunay\n"); printf( " triangulation, in which segments may be subdivided into smaller edges.\n" ); printf(" Each segment, like each point, may have a boundary marker.\n\n"); printf( " The third section lists holes (and concavities, if -c is selected) in\n"); printf( " the triangulation. Holes are specified by identifying a point inside\n"); printf( " each hole. After the triangulation is formed, Triangle creates holes\n"); printf( " by eating triangles, spreading out from each hole point until its\n"); printf( " progress is blocked by PSLG segments; you must be careful to enclose\n"); printf( " each hole in segments, or your whole triangulation may be eaten away.\n"); printf( " If the two triangles abutting a segment are eaten, the segment itself\n"); printf( " is also eaten. Do not place a hole directly on a segment; if you do,\n"); printf(" Triangle will choose one side of the segment arbitrarily.\n\n"); printf( " The optional fourth section lists regional attributes (to be assigned\n"); printf( " to all triangles in a region) and regional constraints on the maximum\n"); printf( " triangle area. Triangle will read this section only if the -A switch\n"); printf( " is used or the -a switch is used without a number following it, and the\n" ); printf( " -r switch is not used. Regional attributes and area constraints are\n"); printf( " propagated in the same manner as holes; you specify a point for each\n"); printf( " attribute and/or constraint, and the attribute and/or constraint will\n"); printf( " affect the whole region (bounded by segments) containing the point. If\n" ); printf( " two values are written on a line after the x and y coordinate, the\n"); printf( " former is assumed to be a regional attribute (but will only be applied\n" ); printf( " if the -A switch is selected), and the latter is assumed to be a\n"); printf( " regional area constraint (but will only be applied if the -a switch is\n" ); printf( " selected). You may also specify just one value after the coordinates,\n" ); printf( " which can serve as both an attribute and an area constraint, depending\n" ); printf( " on the choice of switches. If you are using the -A and -a switches\n"); printf( " simultaneously and wish to assign an attribute to some region without\n"); printf(" imposing an area constraint, use a negative maximum area.\n\n"); printf( " When a triangulation is created from a .poly file, you must either\n"); printf( " enclose the entire region to be triangulated in PSLG segments, or\n"); printf( " use the -c switch, which encloses the convex hull of the input point\n"); printf( " set. If you do not use the -c switch, Triangle will eat all triangles\n" ); printf( " on the outer boundary that are not protected by segments; if you are\n"); printf( " not careful, your whole triangulation may be eaten away. If you do\n"); printf( " use the -c switch, you can still produce concavities by appropriate\n"); printf(" placement of holes just inside the convex hull.\n\n"); printf( " An ideal PSLG has no intersecting segments, nor any points that lie\n"); printf( " upon segments (except, of course, the endpoints of each segment.) You\n" ); printf( " aren't required to make your .poly files ideal, but you should be aware\n" ); printf( " of what can go wrong. Segment intersections are relatively safe -\n"); printf( " Triangle will calculate the intersection points for you and add them to\n" ); printf( " the triangulation - as long as your machine's floating-point precision\n" ); printf( " doesn't become a problem. You are tempting the fates if you have three\n" ); printf( " segments that cross at the same location, and expect Triangle to figure\n" ); printf( " out where the intersection point is. Thanks to floating-point roundoff\n" ); printf( " error, Triangle will probably decide that the three segments intersect\n" ); printf( " at three different points, and you will find a minuscule triangle in\n"); printf( " your output - unless Triangle tries to refine the tiny triangle, uses\n"); printf( " up the last bit of machine precision, and fails to terminate at all.\n"); printf( " You're better off putting the intersection point in the input files,\n"); printf( " and manually breaking up each segment into two. Similarly, if you\n"); printf( " place a point at the middle of a segment, and hope that Triangle will\n"); printf( " break up the segment at that point, you might get lucky. On the other\n" ); printf( " hand, Triangle might decide that the point doesn't lie precisely on the\n" ); printf( " line, and you'll have a needle-sharp triangle in your output - or a lot\n" ); printf(" of tiny triangles if you're generating a quality mesh.\n\n"); printf( " When Triangle reads a .poly file, it also writes a .poly file, which\n"); printf( " includes all edges that are part of input segments. If the -c switch\n"); printf( " is used, the output .poly file will also include all of the edges on\n"); printf( " the convex hull. Hence, the output .poly file is useful for finding\n"); printf( " edges associated with input segments and setting boundary conditions in\n" ); printf( " finite element simulations. More importantly, you will need it if you\n" ); printf( " plan to refine the output mesh, and don't want segments to be missing\n"); printf(" in later triangulations.\n\n"); printf(" .area files:\n"); printf(" First line: <# of triangles>\n"); printf(" Following lines: \n\n"); printf( " An .area file associates with each triangle a maximum area that is used\n" ); printf( " for mesh refinement. As with other file formats, every triangle must\n"); printf( " be represented, and they must be numbered consecutively. A triangle\n"); printf( " may be left unconstrained by assigning it a negative maximum area.\n"); printf("\n"); printf(" .edge files:\n"); printf(" First line: <# of edges> <# of boundary markers (0 or 1)>\n"); printf( " Following lines: [boundary marker]\n"); printf("\n"); printf( " Endpoints are indices into the corresponding .node file. Triangle can\n" ); printf( " produce .edge files (use the -e switch), but cannot read them. The\n"); printf( " optional column of boundary markers is suppressed by the -B switch.\n"); printf("\n"); printf( " In Voronoi diagrams, one also finds a special kind of edge that is an\n"); printf( " infinite ray with only one endpoint. For these edges, a different\n"); printf(" format is used:\n\n"); printf(" -1 \n\n"); printf( " The `direction' is a floating-point vector that indicates the direction\n" ); printf(" of the infinite ray.\n\n"); printf(" .neigh files:\n"); printf( " First line: <# of triangles> <# of neighbors per triangle (always 3)>\n" ); printf( " Following lines: \n"); printf("\n"); printf( " Neighbors are indices into the corresponding .ele file. An index of -1\n" ); printf( " indicates a mesh boundary, and therefore no neighbor. Triangle can\n"); printf( " produce .neigh files (use the -n switch), but cannot read them.\n"); printf("\n"); printf( " The first neighbor of triangle i is opposite the first corner of\n"); printf(" triangle i, and so on.\n\n"); printf("Boundary Markers:\n\n"); printf( " Boundary markers are tags used mainly to identify which output points and\n" ); printf( " edges are associated with which PSLG segment, and to identify which\n"); printf( " points and edges occur on a boundary of the triangulation. A common use\n" ); printf( " is to determine where boundary conditions should be applied to a finite\n"); printf( " element mesh. You can prevent boundary markers from being written into\n"); printf(" files produced by Triangle by using the -B switch.\n\n"); printf( " The boundary marker associated with each segment in an output .poly file\n" ); printf(" or edge in an output .edge file is chosen as follows:\n"); printf( " - If an output edge is part or all of a PSLG segment with a nonzero\n"); printf( " boundary marker, then the edge is assigned the same marker.\n"); printf( " - Otherwise, if the edge occurs on a boundary of the triangulation\n"); printf( " (including boundaries of holes), then the edge is assigned the marker\n" ); printf(" one (1).\n"); printf(" - Otherwise, the edge is assigned the marker zero (0).\n"); printf( " The boundary marker associated with each point in an output .node file is\n" ); printf(" chosen as follows:\n"); printf( " - If a point is assigned a nonzero boundary marker in the input file,\n"); printf( " then it is assigned the same marker in the output .node file.\n"); printf( " - Otherwise, if the point lies on a PSLG segment (including the\n"); printf( " segment's endpoints) with a nonzero boundary marker, then the point\n"); printf( " is assigned the same marker. If the point lies on several such\n"); printf(" segments, one of the markers is chosen arbitrarily.\n"); printf( " - Otherwise, if the point occurs on a boundary of the triangulation,\n"); printf(" then the point is assigned the marker one (1).\n"); printf(" - Otherwise, the point is assigned the marker zero (0).\n"); printf("\n"); printf( " If you want Triangle to determine for you which points and edges are on\n"); printf( " the boundary, assign them the boundary marker zero (or use no markers at\n" ); printf( " all) in your input files. Alternatively, you can mark some of them and\n"); printf(" leave others marked zero, allowing Triangle to label them.\n\n"); printf("Triangulation Iteration Numbers:\n\n"); printf( " Because Triangle can read and refine its own triangulations, input\n"); printf( " and output files have iteration numbers. For instance, Triangle might\n"); printf( " read the files mesh.3.node, mesh.3.ele, and mesh.3.poly, refine the\n"); printf( " triangulation, and output the files mesh.4.node, mesh.4.ele, and\n"); printf(" mesh.4.poly. Files with no iteration number are treated as if\n"); printf( " their iteration number is zero; hence, Triangle might read the file\n"); printf( " points.node, triangulate it, and produce the files points.1.node and\n"); printf(" points.1.ele.\n\n"); printf( " Iteration numbers allow you to create a sequence of successively finer\n"); printf( " meshes suitable for multigrid methods. They also allow you to produce a\n" ); printf( " sequence of meshes using error estimate-driven mesh refinement.\n"); printf("\n"); printf( " If you're not using refinement or quality meshing, and you don't like\n"); printf( " iteration numbers, use the -I switch to disable them. This switch will\n"); printf( " also disable output of .node and .poly files to prevent your input files\n" ); printf( " from being overwritten. (If the input is a .poly file that contains its\n" ); printf(" own points, a .node file will be written.)\n\n"); printf("Examples of How to Use Triangle:\n\n"); printf( " `triangle dots' will read points from dots.node, and write their Delaunay\n" ); printf( " triangulation to dots.1.node and dots.1.ele. (dots.1.node will be\n"); printf( " identical to dots.node.) `triangle -I dots' writes the triangulation to\n" ); printf( " dots.ele instead. (No additional .node file is needed, so none is\n"); printf(" written.)\n\n"); printf( " `triangle -pe object.1' will read a PSLG from object.1.poly (and possibly\n" ); printf( " object.1.node, if the points are omitted from object.1.poly) and write\n"); printf(" their constrained Delaunay triangulation to object.2.node and\n"); printf( " object.2.ele. The segments will be copied to object.2.poly, and all\n"); printf(" edges will be written to object.2.edge.\n\n"); printf( " `triangle -pq31.5a.1 object' will read a PSLG from object.poly (and\n"); printf( " possibly object.node), generate a mesh whose angles are all greater than\n" ); printf( " 31.5 degrees and whose triangles all have area smaller than 0.1, and\n"); printf( " write the mesh to object.1.node and object.1.ele. Each segment may have\n" ); printf( " been broken up into multiple edges; the resulting constrained edges are\n"); printf(" written to object.1.poly.\n\n"); printf( " Here is a sample file `box.poly' describing a square with a square hole:\n" ); printf("\n"); printf( " # A box with eight points in 2D, no attributes, one boundary marker.\n"); printf(" 8 2 0 1\n"); printf(" # Outer box has these vertices:\n"); printf(" 1 0 0 0\n"); printf(" 2 0 3 0\n"); printf(" 3 3 0 0\n"); printf(" 4 3 3 33 # A special marker for this point.\n"); printf(" # Inner square has these vertices:\n"); printf(" 5 1 1 0\n"); printf(" 6 1 2 0\n"); printf(" 7 2 1 0\n"); printf(" 8 2 2 0\n"); printf(" # Five segments with boundary markers.\n"); printf(" 5 1\n"); printf(" 1 1 2 5 # Left side of outer box.\n"); printf(" 2 5 7 0 # Segments 2 through 5 enclose the hole.\n"); printf(" 3 7 8 0\n"); printf(" 4 8 6 10\n"); printf(" 5 6 5 0\n"); printf(" # One hole in the middle of the inner square.\n"); printf(" 1\n"); printf(" 1 1.5 1.5\n\n"); printf( " Note that some segments are missing from the outer square, so one must\n"); printf( " use the `-c' switch. After `triangle -pqc box.poly', here is the output\n" ); printf( " file `box.1.node', with twelve points. The last four points were added\n"); printf( " to meet the angle constraint. Points 1, 2, and 9 have markers from\n"); printf( " segment 1. Points 6 and 8 have markers from segment 4. All the other\n"); printf( " points but 4 have been marked to indicate that they lie on a boundary.\n"); printf("\n"); printf(" 12 2 0 1\n"); printf(" 1 0 0 5\n"); printf(" 2 0 3 5\n"); printf(" 3 3 0 1\n"); printf(" 4 3 3 33\n"); printf(" 5 1 1 1\n"); printf(" 6 1 2 10\n"); printf(" 7 2 1 1\n"); printf(" 8 2 2 10\n"); printf(" 9 0 1.5 5\n"); printf(" 10 1.5 0 1\n"); printf(" 11 3 1.5 1\n"); printf(" 12 1.5 3 1\n"); printf(" # Generated by triangle -pqc box.poly\n\n"); printf(" Here is the output file `box.1.ele', with twelve triangles.\n\n"); printf(" 12 3 0\n"); printf(" 1 5 6 9\n"); printf(" 2 10 3 7\n"); printf(" 3 6 8 12\n"); printf(" 4 9 1 5\n"); printf(" 5 6 2 9\n"); printf(" 6 7 3 11\n"); printf(" 7 11 4 8\n"); printf(" 8 7 5 10\n"); printf(" 9 12 2 6\n"); printf(" 10 8 7 11\n"); printf(" 11 5 1 10\n"); printf(" 12 8 4 12\n"); printf(" # Generated by triangle -pqc box.poly\n\n"); printf( " Here is the output file `box.1.poly'. Note that segments have been added\n" ); printf( " to represent the convex hull, and some segments have been split by newly\n" ); printf( " added points. Note also that <# of points> is set to zero to indicate\n"); printf(" that the points should be read from the .node file.\n\n"); printf(" 0 2 0 1\n"); printf(" 12 1\n"); printf(" 1 1 9 5\n"); printf(" 2 5 7 1\n"); printf(" 3 8 7 1\n"); printf(" 4 6 8 10\n"); printf(" 5 5 6 1\n"); printf(" 6 3 10 1\n"); printf(" 7 4 11 1\n"); printf(" 8 2 12 1\n"); printf(" 9 9 2 5\n"); printf(" 10 10 1 1\n"); printf(" 11 11 3 1\n"); printf(" 12 12 4 1\n"); printf(" 1\n"); printf(" 1 1.5 1.5\n"); printf(" # Generated by triangle -pqc box.poly\n\n"); printf("Refinement and Area Constraints:\n\n"); printf( " The -r switch causes a mesh (.node and .ele files) to be read and\n"); printf( " refined. If the -p switch is also used, a .poly file is read and used to\n" ); printf( " specify edges that are constrained and cannot be eliminated (although\n"); printf( " they can be divided into smaller edges) by the refinement process.\n"); printf("\n"); printf( " When you refine a mesh, you generally want to impose tighter quality\n"); printf( " constraints. One way to accomplish this is to use -q with a larger\n"); printf( " angle, or -a followed by a smaller area than you used to generate the\n"); printf( " mesh you are refining. Another way to do this is to create an .area\n"); printf( " file, which specifies a maximum area for each triangle, and use the -a\n"); printf( " switch (without a number following). Each triangle's area constraint is\n" ); printf( " applied to that triangle. Area constraints tend to diffuse as the mesh\n"); printf( " is refined, so if there are large variations in area constraint between\n"); printf(" adjacent triangles, you may not get the results you want.\n\n"); printf( " If you are refining a mesh composed of linear (three-node) elements, the\n" ); printf( " output mesh will contain all the nodes present in the input mesh, in the\n" ); printf( " same order, with new nodes added at the end of the .node file. However,\n" ); printf( " there is no guarantee that each output element is contained in a single\n"); printf( " input element. Often, output elements will overlap two input elements,\n"); printf( " and input edges are not present in the output mesh. Hence, a sequence of\n" ); printf( " refined meshes will form a hierarchy of nodes, but not a hierarchy of\n"); printf( " elements. If you a refining a mesh of higher-order elements, the\n"); printf( " hierarchical property applies only to the nodes at the corners of an\n"); printf(" element; other nodes may not be present in the refined mesh.\n\n"); printf( " It is important to understand that maximum area constraints in .poly\n"); printf( " files are handled differently from those in .area files. A maximum area\n" ); printf( " in a .poly file applies to the whole (segment-bounded) region in which a\n" ); printf( " point falls, whereas a maximum area in an .area file applies to only one\n" ); printf( " triangle. Area constraints in .poly files are used only when a mesh is\n"); printf( " first generated, whereas area constraints in .area files are used only to\n" ); printf( " refine an existing mesh, and are typically based on a posteriori error\n"); printf( " estimates resulting from a finite element simulation on that mesh.\n"); printf("\n"); printf( " `triangle -rq25 object.1' will read object.1.node and object.1.ele, then\n" ); printf( " refine the triangulation to enforce a 25 degree minimum angle, and then\n"); printf( " write the refined triangulation to object.2.node and object.2.ele.\n"); printf("\n"); printf( " `triangle -rpaa6.2 z.3' will read z.3.node, z.3.ele, z.3.poly, and\n"); printf( " z.3.area. After reconstructing the mesh and its segments, Triangle will\n" ); printf( " refine the mesh so that no triangle has area greater than 6.2, and\n"); printf( " furthermore the triangles satisfy the maximum area constraints in\n"); printf( " z.3.area. The output is written to z.4.node, z.4.ele, and z.4.poly.\n"); printf("\n"); printf( " The sequence `triangle -qa1 x', `triangle -rqa.3 x.1', `triangle -rqa.1\n"); printf( " x.2' creates a sequence of successively finer meshes x.1, x.2, and x.3,\n"); printf(" suitable for multigrid.\n\n"); printf("Convex Hulls and Mesh Boundaries:\n\n"); printf( " If the input is a point set (rather than a PSLG), Triangle produces its\n"); printf( " convex hull as a by-product in the output .poly file if you use the -c\n"); printf( " switch. There are faster algorithms for finding a two-dimensional convex\n" ); printf( " hull than triangulation, of course, but this one comes for free. If the\n" ); printf( " input is an unconstrained mesh (you are using the -r switch but not the\n"); printf( " -p switch), Triangle produces a list of its boundary edges (including\n"); printf(" hole boundaries) as a by-product if you use the -c switch.\n\n"); printf("Voronoi Diagrams:\n\n"); printf( " The -v switch produces a Voronoi diagram, in files suffixed .v.node and\n"); printf( " .v.edge. For example, `triangle -v points' will read points.node,\n"); printf( " produce its Delaunay triangulation in points.1.node and points.1.ele,\n"); printf( " and produce its Voronoi diagram in points.1.v.node and points.1.v.edge.\n"); printf( " The .v.node file contains a list of all Voronoi vertices, and the .v.edge\n" ); printf( " file contains a list of all Voronoi edges, some of which may be infinite\n" ); printf( " rays. (The choice of filenames makes it easy to run the set of Voronoi\n"); printf(" vertices through Triangle, if so desired.)\n\n"); printf( " This implementation does not use exact arithmetic to compute the Voronoi\n" ); printf( " vertices, and does not check whether neighboring vertices are identical.\n" ); printf( " Be forewarned that if the Delaunay triangulation is degenerate or\n"); printf( " near-degenerate, the Voronoi diagram may have duplicate points, crossing\n" ); printf( " edges, or infinite rays whose direction vector is zero. Also, if you\n"); printf( " generate a constrained (as opposed to conforming) Delaunay triangulation,\n" ); printf( " or if the triangulation has holes, the corresponding Voronoi diagram is\n"); printf(" likely to have crossing edges and unlikely to make sense.\n\n"); printf("Mesh Topology:\n\n"); printf( " You may wish to know which triangles are adjacent to a certain Delaunay\n"); printf( " edge in an .edge file, which Voronoi regions are adjacent to a certain\n"); printf( " Voronoi edge in a .v.edge file, or which Voronoi regions are adjacent to\n" ); printf( " each other. All of this information can be found by cross-referencing\n"); printf( " output files with the recollection that the Delaunay triangulation and\n"); printf(" the Voronoi diagrams are planar duals.\n\n"); printf( " Specifically, edge i of an .edge file is the dual of Voronoi edge i of\n"); printf( " the corresponding .v.edge file, and is rotated 90 degrees counterclock-\n"); printf( " wise from the Voronoi edge. Triangle j of an .ele file is the dual of\n"); printf( " vertex j of the corresponding .v.node file; and Voronoi region k is the\n"); printf(" dual of point k of the corresponding .node file.\n\n"); printf( " Hence, to find the triangles adjacent to a Delaunay edge, look at the\n"); printf( " vertices of the corresponding Voronoi edge; their dual triangles are on\n"); printf( " the left and right of the Delaunay edge, respectively. To find the\n"); printf( " Voronoi regions adjacent to a Voronoi edge, look at the endpoints of the\n" ); printf( " corresponding Delaunay edge; their dual regions are on the right and left\n" ); printf( " of the Voronoi edge, respectively. To find which Voronoi regions are\n"); printf(" adjacent to each other, just read the list of Delaunay edges.\n"); printf("\n"); printf("Statistics:\n"); printf("\n"); printf( " After generating a mesh, Triangle prints a count of the number of points,\n" ); printf( " triangles, edges, boundary edges, and segments in the output mesh. If\n"); printf( " you've forgotten the statistics for an existing mesh, the -rNEP switches\n" ); printf( " (or -rpNEP if you've got a .poly file for the existing mesh) will\n"); printf(" regenerate these statistics without writing any output.\n\n"); printf( " The -V switch produces extended statistics, including a rough estimate\n"); printf( " of memory use and a histogram of triangle aspect ratios and angles in the\n" ); printf(" mesh.\n\n"); printf("Exact Arithmetic:\n\n"); printf( " Triangle uses adaptive exact arithmetic to perform what computational\n"); printf( " geometers call the `orientation' and `incircle' tests. If the floating-\n" ); printf( " point arithmetic of your machine conforms to the IEEE 754 standard (as\n"); printf( " most workstations do), and does not use extended precision internal\n"); printf( " registers, then your output is guaranteed to be an absolutely true\n"); printf(" Delaunay or conforming Delaunay triangulation, roundoff error\n"); printf( " notwithstanding. The word `adaptive' implies that these arithmetic\n"); printf( " routines compute the result only to the precision necessary to guarantee\n" ); printf( " correctness, so they are usually nearly as fast as their approximate\n"); printf( " counterparts. The exact tests can be disabled with the -X switch. On\n"); printf( " most inputs, this switch will reduce the computation time by about eight\n" ); printf( " percent - it's not worth the risk. There are rare difficult inputs\n"); printf( " (having many collinear and cocircular points), however, for which the\n"); printf( " difference could be a factor of two. These are precisely the inputs most\n" ); printf(" likely to cause errors if you use the -X switch.\n\n"); printf( " Unfortunately, these routines don't solve every numerical problem. Exact\n" ); printf( " arithmetic is not used to compute the positions of points, because the\n"); printf( " bit complexity of point coordinates would grow without bound. Hence,\n"); printf( " segment intersections aren't computed exactly; in very unusual cases,\n"); printf( " roundoff error in computing an intersection point might actually lead to\n" ); printf( " an inverted triangle and an invalid triangulation. (This is one reason\n"); printf( " to compute your own intersection points in your .poly files.) Similarly,\n" ); printf( " exact arithmetic is not used to compute the vertices of the Voronoi\n"); printf(" diagram.\n\n"); printf( " Underflow and overflow can also cause difficulties; the exact arithmetic\n" ); printf( " routines do not ameliorate out-of-bounds exponents, which can arise\n"); printf( " during the orientation and incircle tests. As a rule of thumb, you\n"); printf( " should ensure that your input values are within a range such that their\n"); printf( " third powers can be taken without underflow or overflow. Underflow can\n"); printf( " silently prevent the tests from being performed exactly, while overflow\n"); printf(" will typically cause a floating exception.\n\n"); printf("Calling Triangle from Another Program:\n\n"); printf(" Read the file triangle.h for details.\n\n"); printf("Troubleshooting:\n\n"); printf(" Please read this section before mailing me bugs.\n\n"); printf(" `My output mesh has no triangles!'\n\n"); printf( " If you're using a PSLG, you've probably failed to specify a proper set\n" ); printf( " of bounding segments, or forgotten to use the -c switch. Or you may\n"); printf( " have placed a hole badly. To test these possibilities, try again with\n" ); printf( " the -c and -O switches. Alternatively, all your input points may be\n"); printf( " collinear, in which case you can hardly expect to triangulate them.\n"); printf("\n"); printf(" `Triangle doesn't terminate, or just crashes.'\n"); printf("\n"); printf( " Bad things can happen when triangles get so small that the distance\n"); printf( " between their vertices isn't much larger than the precision of your\n"); printf( " machine's arithmetic. If you've compiled Triangle for single-precision\n" ); printf( " arithmetic, you might do better by recompiling it for double-precision.\n" ); printf( " Then again, you might just have to settle for more lenient constraints\n" ); printf( " on the minimum angle and the maximum area than you had planned.\n"); printf("\n"); printf( " You can minimize precision problems by ensuring that the origin lies\n"); printf( " inside your point set, or even inside the densest part of your\n"); printf( " mesh. On the other hand, if you're triangulating an object whose x\n"); printf( " coordinates all fall between 6247133 and 6247134, you're not leaving\n"); printf(" much floating-point precision for Triangle to work with.\n\n"); printf( " Precision problems can occur covertly if the input PSLG contains two\n"); printf( " segments that meet (or intersect) at a very small angle, or if such an\n" ); printf( " angle is introduced by the -c switch, which may occur if a point lies\n"); printf( " ever-so-slightly inside the convex hull, and is connected by a PSLG\n"); printf( " segment to a point on the convex hull. If you don't realize that a\n"); printf( " small angle is being formed, you might never discover why Triangle is\n"); printf( " crashing. To check for this possibility, use the -S switch (with an\n"); printf( " appropriate limit on the number of Steiner points, found by trial-and-\n" ); printf( " error) to stop Triangle early, and view the output .poly file with\n"); printf( " Show Me (described below). Look carefully for small angles between\n"); printf( " segments; zoom in closely, as such segments might look like a single\n"); printf(" segment from a distance.\n\n"); printf( " If some of the input values are too large, Triangle may suffer a\n"); printf( " floating exception due to overflow when attempting to perform an\n"); printf( " orientation or incircle test. (Read the section on exact arithmetic\n"); printf( " above.) Again, I recommend compiling Triangle for double (rather\n"); printf(" than single) precision arithmetic.\n\n"); printf( " `The numbering of the output points doesn't match the input points.'\n"); printf("\n"); printf( " You may have eaten some of your input points with a hole, or by placing\n" ); printf(" them outside the area enclosed by segments.\n\n"); printf( " `Triangle executes without incident, but when I look at the resulting\n"); printf( " mesh, it has overlapping triangles or other geometric inconsistencies.'\n"); printf("\n"); printf( " If you select the -X switch, Triangle's divide-and-conquer Delaunay\n"); printf( " triangulation algorithm occasionally makes mistakes due to floating-\n"); printf( " point roundoff error. Although these errors are rare, don't use the -X\n" ); printf(" switch. If you still have problems, please report the bug.\n"); printf("\n"); printf( " Strange things can happen if you've taken liberties with your PSLG. Do\n"); printf( " you have a point lying in the middle of a segment? Triangle sometimes\n"); printf( " copes poorly with that sort of thing. Do you want to lay out a collinear\n" ); printf( " row of evenly spaced, segment-connected points? Have you simply defined\n" ); printf( " one long segment connecting the leftmost point to the rightmost point,\n"); printf( " and a bunch of points lying along it? This method occasionally works,\n"); printf( " especially with horizontal and vertical lines, but often it doesn't, and\n" ); printf( " you'll have to connect each adjacent pair of points with a separate\n"); printf(" segment. If you don't like it, tough.\n\n"); printf( " Furthermore, if you have segments that intersect other than at their\n"); printf( " endpoints, try not to let the intersections fall extremely close to PSLG\n" ); printf(" points or each other.\n\n"); printf( " If you have problems refining a triangulation not produced by Triangle:\n"); printf( " Are you sure the triangulation is geometrically valid? Is it formatted\n"); printf( " correctly for Triangle? Are the triangles all listed so the first three\n" ); printf(" points are their corners in counterclockwise order?\n\n"); printf("Show Me:\n\n"); printf( " Triangle comes with a separate program named `Show Me', whose primary\n"); printf( " purpose is to draw meshes on your screen or in PostScript. Its secondary\n" ); printf( " purpose is to check the validity of your input files, and do so more\n"); printf( " thoroughly than Triangle does. Show Me requires that you have the X\n"); printf( " Windows system. If you didn't receive Show Me with Triangle, complain to\n" ); printf(" whomever you obtained Triangle from, then send me mail.\n\n"); printf("Triangle on the Web:\n\n"); printf( " To see an illustrated, updated version of these instructions, check out\n"); printf("\n"); printf(" http://www.cs.cmu.edu/~quake/triangle.html\n"); printf("\n"); printf("A Brief Plea:\n"); printf("\n"); printf( " If you use Triangle, and especially if you use it to accomplish real\n"); printf( " work, I would like very much to hear from you. A short letter or email\n"); printf( " (to jrs@cs.cmu.edu) describing how you use Triangle will mean a lot to\n"); printf( " me. The more people I know are using this program, the more easily I can\n" ); printf( " justify spending time on improvements and on the three-dimensional\n"); printf( " successor to Triangle, which in turn will benefit you. Also, I can put\n"); printf( " you on a list to receive email whenever a new version of Triangle is\n"); printf(" available.\n\n"); printf( " If you use a mesh generated by Triangle in a publication, please include\n" ); printf(" an acknowledgment as well.\n\n"); printf("Research credit:\n\n"); printf( " Of course, I can take credit for only a fraction of the ideas that made\n"); printf( " this mesh generator possible. Triangle owes its existence to the efforts\n" ); printf( " of many fine computational geometers and other researchers, including\n"); printf( " Marshall Bern, L. Paul Chew, Boris Delaunay, Rex A. Dwyer, David\n"); printf( " Eppstein, Steven Fortune, Leonidas J. Guibas, Donald E. Knuth, C. L.\n"); printf( " Lawson, Der-Tsai Lee, Ernst P. Mucke, Douglas M. Priest, Jim Ruppert,\n"); printf( " Isaac Saias, Bruce J. Schachter, Micha Sharir, Jorge Stolfi, Christopher\n" ); printf( " J. Van Wyk, David F. Watson, and Binhai Zhu. See the comments at the\n"); printf(" beginning of the source code for references.\n\n"); exit(0); } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* internalerror() Ask the user to send me the defective product. Exit. */ /* */ /*****************************************************************************/ void internalerror() { printf(" Please report this bug to jrs@cs.cmu.edu\n"); printf(" Include the message above, your input data set, and the exact\n"); printf(" command line you used to run Triangle.\n"); exit(1); } /*****************************************************************************/ /* */ /* parsecommandline() Read the command line, identify switches, and set */ /* up options and file names. */ /* */ /* The effects of this routine are felt entirely through global variables. */ /* */ /*****************************************************************************/ void parsecommandline(argc, argv) int argc; char **argv; { #ifdef TRILIBRARY #define STARTINDEX 0 #else /* not TRILIBRARY */ #define STARTINDEX 1 int increment; int meshnumber; #endif /* not TRILIBRARY */ int i, j, k; char workstring[FILENAMESIZE]; poly = refine = quality = vararea = fixedarea = regionattrib = convex = 0; firstnumber = 1; edgesout = voronoi = neighbors = geomview = 0; nobound = nopolywritten = nonodewritten = noelewritten = noiterationnum = 0; noholes = noexact = 0; incremental = sweepline = 0; dwyer = 1; splitseg = 0; docheck = 0; nobisect = 0; steiner = -1; order = 1; minangle = 0.0; maxarea = -1.0; quiet = verbose = 0; #ifndef TRILIBRARY innodefilename[0] = '\0'; #endif /* not TRILIBRARY */ for (i = STARTINDEX; i < argc; i++) { #ifndef TRILIBRARY if (argv[i][0] == '-') { #endif /* not TRILIBRARY */ for (j = STARTINDEX; argv[i][j] != '\0'; j++) { if (argv[i][j] == 'p') { poly = 1; } #ifndef CDT_ONLY if (argv[i][j] == 'r') { refine = 1; } if (argv[i][j] == 'q') { quality = 1; if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) || (argv[i][j + 1] == '.')) { k = 0; while (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) || (argv[i][j + 1] == '.')) { j++; workstring[k] = argv[i][j]; k++; } workstring[k] = '\0'; minangle = (REAL) strtod(workstring, (char **) NULL); } else { minangle = 20.0; } } if (argv[i][j] == 'a') { quality = 1; if (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) || (argv[i][j + 1] == '.')) { fixedarea = 1; k = 0; while (((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) || (argv[i][j + 1] == '.')) { j++; workstring[k] = argv[i][j]; k++; } workstring[k] = '\0'; maxarea = (REAL) strtod(workstring, (char **) NULL); if (maxarea <= 0.0) { printf("Error: Maximum area must be greater than zero.\n"); exit(1); } } else { vararea = 1; } } #endif /* not CDT_ONLY */ if (argv[i][j] == 'A') { regionattrib = 1; } if (argv[i][j] == 'c') { convex = 1; } if (argv[i][j] == 'z') { firstnumber = 0; } if (argv[i][j] == 'e') { edgesout = 1; } if (argv[i][j] == 'v') { voronoi = 1; } if (argv[i][j] == 'n') { neighbors = 1; } if (argv[i][j] == 'g') { geomview = 1; } if (argv[i][j] == 'B') { nobound = 1; } if (argv[i][j] == 'P') { nopolywritten = 1; } if (argv[i][j] == 'N') { nonodewritten = 1; } if (argv[i][j] == 'E') { noelewritten = 1; } #ifndef TRILIBRARY if (argv[i][j] == 'I') { noiterationnum = 1; } #endif /* not TRILIBRARY */ if (argv[i][j] == 'O') { noholes = 1; } if (argv[i][j] == 'X') { noexact = 1; } if (argv[i][j] == 'o') { if (argv[i][j + 1] == '2') { j++; order = 2; } } #ifndef CDT_ONLY if (argv[i][j] == 'Y') { nobisect++; } if (argv[i][j] == 'S') { steiner = 0; while ((argv[i][j + 1] >= '0') && (argv[i][j + 1] <= '9')) { j++; steiner = steiner * 10 + (int) (argv[i][j] - '0'); } } #endif /* not CDT_ONLY */ #ifndef REDUCED if (argv[i][j] == 'i') { incremental = 1; } if (argv[i][j] == 'F') { sweepline = 1; } #endif /* not REDUCED */ if (argv[i][j] == 'l') { dwyer = 0; } #ifndef REDUCED #ifndef CDT_ONLY if (argv[i][j] == 's') { splitseg = 1; } #endif /* not CDT_ONLY */ if (argv[i][j] == 'C') { docheck = 1; } #endif /* not REDUCED */ if (argv[i][j] == 'Q') { quiet = 1; } if (argv[i][j] == 'V') { verbose++; } #ifndef TRILIBRARY if ((argv[i][j] == 'h') || (argv[i][j] == 'H') || (argv[i][j] == '?')) { info(); } #endif /* not TRILIBRARY */ } #ifndef TRILIBRARY } else { strncpy(innodefilename, argv[i], FILENAMESIZE - 1); innodefilename[FILENAMESIZE - 1] = '\0'; } #endif /* not TRILIBRARY */ } #ifndef TRILIBRARY if (innodefilename[0] == '\0') { syntax(); } if (!strcmp(&innodefilename[strlen(innodefilename) - 5], ".node")) { innodefilename[strlen(innodefilename) - 5] = '\0'; } if (!strcmp(&innodefilename[strlen(innodefilename) - 5], ".poly")) { innodefilename[strlen(innodefilename) - 5] = '\0'; poly = 1; } #ifndef CDT_ONLY if (!strcmp(&innodefilename[strlen(innodefilename) - 4], ".ele")) { innodefilename[strlen(innodefilename) - 4] = '\0'; refine = 1; } if (!strcmp(&innodefilename[strlen(innodefilename) - 5], ".area")) { innodefilename[strlen(innodefilename) - 5] = '\0'; refine = 1; quality = 1; vararea = 1; } #endif /* not CDT_ONLY */ #endif /* not TRILIBRARY */ steinerleft = steiner; useshelles = poly || refine || quality || convex; goodangle = cos(minangle * PI / 180.0); goodangle *= goodangle; if (refine && noiterationnum) { printf( "Error: You cannot use the -I switch when refining a triangulation.\n"); exit(1); } /* Be careful not to allocate space for element area constraints that */ /* will never be assigned any value (other than the default -1.0). */ if (!refine && !poly) { vararea = 0; } /* Be careful not to add an extra attribute to each element unless the */ /* input supports it (PSLG in, but not refining a preexisting mesh). */ if (refine || !poly) { regionattrib = 0; } #ifndef TRILIBRARY strcpy(inpolyfilename, innodefilename); strcpy(inelefilename, innodefilename); strcpy(areafilename, innodefilename); increment = 0; strcpy(workstring, innodefilename); j = 1; while (workstring[j] != '\0') { if ((workstring[j] == '.') && (workstring[j + 1] != '\0')) { increment = j + 1; } j++; } meshnumber = 0; if (increment > 0) { j = increment; do { if ((workstring[j] >= '0') && (workstring[j] <= '9')) { meshnumber = meshnumber * 10 + (int) (workstring[j] - '0'); } else { increment = 0; } j++; } while (workstring[j] != '\0'); } if (noiterationnum) { strcpy(outnodefilename, innodefilename); strcpy(outelefilename, innodefilename); strcpy(edgefilename, innodefilename); strcpy(vnodefilename, innodefilename); strcpy(vedgefilename, innodefilename); strcpy(neighborfilename, innodefilename); strcpy(offfilename, innodefilename); strcat(outnodefilename, ".node"); strcat(outelefilename, ".ele"); strcat(edgefilename, ".edge"); strcat(vnodefilename, ".v.node"); strcat(vedgefilename, ".v.edge"); strcat(neighborfilename, ".neigh"); strcat(offfilename, ".off"); } else if (increment == 0) { strcpy(outnodefilename, innodefilename); strcpy(outpolyfilename, innodefilename); strcpy(outelefilename, innodefilename); strcpy(edgefilename, innodefilename); strcpy(vnodefilename, innodefilename); strcpy(vedgefilename, innodefilename); strcpy(neighborfilename, innodefilename); strcpy(offfilename, innodefilename); strcat(outnodefilename, ".1.node"); strcat(outpolyfilename, ".1.poly"); strcat(outelefilename, ".1.ele"); strcat(edgefilename, ".1.edge"); strcat(vnodefilename, ".1.v.node"); strcat(vedgefilename, ".1.v.edge"); strcat(neighborfilename, ".1.neigh"); strcat(offfilename, ".1.off"); } else { workstring[increment] = '%'; workstring[increment + 1] = 'd'; workstring[increment + 2] = '\0'; sprintf(outnodefilename, workstring, meshnumber + 1); strcpy(outpolyfilename, outnodefilename); strcpy(outelefilename, outnodefilename); strcpy(edgefilename, outnodefilename); strcpy(vnodefilename, outnodefilename); strcpy(vedgefilename, outnodefilename); strcpy(neighborfilename, outnodefilename); strcpy(offfilename, outnodefilename); strcat(outnodefilename, ".node"); strcat(outpolyfilename, ".poly"); strcat(outelefilename, ".ele"); strcat(edgefilename, ".edge"); strcat(vnodefilename, ".v.node"); strcat(vedgefilename, ".v.edge"); strcat(neighborfilename, ".neigh"); strcat(offfilename, ".off"); } strcat(innodefilename, ".node"); strcat(inpolyfilename, ".poly"); strcat(inelefilename, ".ele"); strcat(areafilename, ".area"); #endif /* not TRILIBRARY */ } /** **/ /** **/ /********* User interaction routines begin here *********/ /********* Debugging routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* printtriangle() Print out the details of a triangle/edge handle. */ /* */ /* I originally wrote this procedure to simplify debugging; it can be */ /* called directly from the debugger, and presents information about a */ /* triangle/edge handle in digestible form. It's also used when the */ /* highest level of verbosity (`-VVV') is specified. */ /* */ /*****************************************************************************/ void printtriangle(t) struct triedge *t; { struct triedge printtri; struct edge printsh; point printpoint; printf("triangle x%lx with orientation %d:\n", (unsigned long) t->tri, t->orient); decode(t->tri[0], printtri); if (printtri.tri == dummytri) { printf(" [0] = Outer space\n"); } else { printf(" [0] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient); } decode(t->tri[1], printtri); if (printtri.tri == dummytri) { printf(" [1] = Outer space\n"); } else { printf(" [1] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient); } decode(t->tri[2], printtri); if (printtri.tri == dummytri) { printf(" [2] = Outer space\n"); } else { printf(" [2] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient); } org(*t, printpoint); if (printpoint == (point) NULL) printf(" Origin[%d] = NULL\n", (t->orient + 1) % 3 + 3); else printf(" Origin[%d] = x%lx (%.12g, %.12g)\n", (t->orient + 1) % 3 + 3, (unsigned long) printpoint, printpoint[0], printpoint[1]); dest(*t, printpoint); if (printpoint == (point) NULL) printf(" Dest [%d] = NULL\n", (t->orient + 2) % 3 + 3); else printf(" Dest [%d] = x%lx (%.12g, %.12g)\n", (t->orient + 2) % 3 + 3, (unsigned long) printpoint, printpoint[0], printpoint[1]); apex(*t, printpoint); if (printpoint == (point) NULL) printf(" Apex [%d] = NULL\n", t->orient + 3); else printf(" Apex [%d] = x%lx (%.12g, %.12g)\n", t->orient + 3, (unsigned long) printpoint, printpoint[0], printpoint[1]); if (useshelles) { sdecode(t->tri[6], printsh); if (printsh.sh != dummysh) { printf(" [6] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient); } sdecode(t->tri[7], printsh); if (printsh.sh != dummysh) { printf(" [7] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient); } sdecode(t->tri[8], printsh); if (printsh.sh != dummysh) { printf(" [8] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient); } } if (vararea) { printf(" Area constraint: %.4g\n", areabound(*t)); } } /*****************************************************************************/ /* */ /* printshelle() Print out the details of a shell edge handle. */ /* */ /* I originally wrote this procedure to simplify debugging; it can be */ /* called directly from the debugger, and presents information about a */ /* shell edge handle in digestible form. It's also used when the highest */ /* level of verbosity (`-VVV') is specified. */ /* */ /*****************************************************************************/ void printshelle(s) struct edge *s; { struct edge printsh; struct triedge printtri; point printpoint; printf("shell edge x%lx with orientation %d and mark %d:\n", (unsigned long) s->sh, s->shorient, mark(*s)); sdecode(s->sh[0], printsh); if (printsh.sh == dummysh) { printf(" [0] = No shell\n"); } else { printf(" [0] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient); } sdecode(s->sh[1], printsh); if (printsh.sh == dummysh) { printf(" [1] = No shell\n"); } else { printf(" [1] = x%lx %d\n", (unsigned long) printsh.sh, printsh.shorient); } sorg(*s, printpoint); if (printpoint == (point) NULL) printf(" Origin[%d] = NULL\n", 2 + s->shorient); else printf(" Origin[%d] = x%lx (%.12g, %.12g)\n", 2 + s->shorient, (unsigned long) printpoint, printpoint[0], printpoint[1]); sdest(*s, printpoint); if (printpoint == (point) NULL) printf(" Dest [%d] = NULL\n", 3 - s->shorient); else printf(" Dest [%d] = x%lx (%.12g, %.12g)\n", 3 - s->shorient, (unsigned long) printpoint, printpoint[0], printpoint[1]); decode(s->sh[4], printtri); if (printtri.tri == dummytri) { printf(" [4] = Outer space\n"); } else { printf(" [4] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient); } decode(s->sh[5], printtri); if (printtri.tri == dummytri) { printf(" [5] = Outer space\n"); } else { printf(" [5] = x%lx %d\n", (unsigned long) printtri.tri, printtri.orient); } } /** **/ /** **/ /********* Debugging routines end here *********/ /********* Memory management routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* poolinit() Initialize a pool of memory for allocation of items. */ /* */ /* This routine initializes the machinery for allocating items. A `pool' */ /* is created whose records have size at least `bytecount'. Items will be */ /* allocated in `itemcount'-item blocks. Each item is assumed to be a */ /* collection of words, and either pointers or floating-point values are */ /* assumed to be the "primary" word type. (The "primary" word type is used */ /* to determine alignment of items.) If `alignment' isn't zero, all items */ /* will be `alignment'-byte aligned in memory. `alignment' must be either */ /* a multiple or a factor of the primary word size; powers of two are safe. */ /* `alignment' is normally used to create a few unused bits at the bottom */ /* of each item's pointer, in which information may be stored. */ /* */ /* Don't change this routine unless you understand it. */ /* */ /*****************************************************************************/ void poolinit(pool, bytecount, itemcount, wtype, alignment) struct memorypool *pool; int bytecount; int itemcount; enum wordtype wtype; int alignment; { int wordsize; /* Initialize values in the pool. */ pool->itemwordtype = wtype; wordsize = (pool->itemwordtype == POINTER) ? sizeof(VOID *) : sizeof(REAL); /* Find the proper alignment, which must be at least as large as: */ /* - The parameter `alignment'. */ /* - The primary word type, to avoid unaligned accesses. */ /* - sizeof(VOID *), so the stack of dead items can be maintained */ /* without unaligned accesses. */ if (alignment > wordsize) { pool->alignbytes = alignment; } else { pool->alignbytes = wordsize; } if (sizeof(VOID *) > pool->alignbytes) { pool->alignbytes = sizeof(VOID *); } pool->itemwords = ((bytecount + pool->alignbytes - 1) / pool->alignbytes) * (pool->alignbytes / wordsize); pool->itembytes = pool->itemwords * wordsize; pool->itemsperblock = itemcount; /* Allocate a block of items. Space for `itemsperblock' items and one */ /* pointer (to point to the next block) are allocated, as well as space */ /* to ensure alignment of the items. */ pool->firstblock = (VOID **) malloc(pool->itemsperblock * pool->itembytes + sizeof(VOID *) + pool->alignbytes); if (pool->firstblock == (VOID **) NULL) { printf("Error: Out of memory.\n"); exit(1); } /* Set the next block pointer to NULL. */ *(pool->firstblock) = (VOID *) NULL; poolrestart(pool); } /*****************************************************************************/ /* */ /* poolrestart() Deallocate all items in a pool. */ /* */ /* The pool is returned to its starting state, except that no memory is */ /* freed to the operating system. Rather, the previously allocated blocks */ /* are ready to be reused. */ /* */ /*****************************************************************************/ void poolrestart(pool) struct memorypool *pool; { unsigned long alignptr; pool->items = 0; pool->maxitems = 0; /* Set the currently active block. */ pool->nowblock = pool->firstblock; /* Find the first item in the pool. Increment by the size of (VOID *). */ alignptr = (unsigned long) (pool->nowblock + 1); /* Align the item on an `alignbytes'-byte boundary. */ pool->nextitem = (VOID *) (alignptr + (unsigned long) pool->alignbytes - (alignptr % (unsigned long) pool->alignbytes)); /* There are lots of unallocated items left in this block. */ pool->unallocateditems = pool->itemsperblock; /* The stack of deallocated items is empty. */ pool->deaditemstack = (VOID *) NULL; } /*****************************************************************************/ /* */ /* pooldeinit() Free to the operating system all memory taken by a pool. */ /* */ /*****************************************************************************/ void pooldeinit(pool) struct memorypool *pool; { while (pool->firstblock != (VOID **) NULL) { pool->nowblock = (VOID **) *(pool->firstblock); free(pool->firstblock); pool->firstblock = pool->nowblock; } } /*****************************************************************************/ /* */ /* poolalloc() Allocate space for an item. */ /* */ /*****************************************************************************/ VOID *poolalloc(pool) struct memorypool *pool; { VOID *newitem; VOID **newblock; unsigned long alignptr; /* First check the linked list of dead items. If the list is not */ /* empty, allocate an item from the list rather than a fresh one. */ if (pool->deaditemstack != (VOID *) NULL) { newitem = pool->deaditemstack; /* Take first item in list. */ pool->deaditemstack = * (VOID **) pool->deaditemstack; } else { /* Check if there are any free items left in the current block. */ if (pool->unallocateditems == 0) { /* Check if another block must be allocated. */ if (*(pool->nowblock) == (VOID *) NULL) { /* Allocate a new block of items, pointed to by the previous block. */ newblock = (VOID **) malloc(pool->itemsperblock * pool->itembytes + sizeof(VOID *) + pool->alignbytes); if (newblock == (VOID **) NULL) { printf("Error: Out of memory.\n"); exit(1); } *(pool->nowblock) = (VOID *) newblock; /* The next block pointer is NULL. */ *newblock = (VOID *) NULL; } /* Move to the new block. */ pool->nowblock = (VOID **) *(pool->nowblock); /* Find the first item in the block. */ /* Increment by the size of (VOID *). */ alignptr = (unsigned long) (pool->nowblock + 1); /* Align the item on an `alignbytes'-byte boundary. */ pool->nextitem = (VOID *) (alignptr + (unsigned long) pool->alignbytes - (alignptr % (unsigned long) pool->alignbytes)); /* There are lots of unallocated items left in this block. */ pool->unallocateditems = pool->itemsperblock; } /* Allocate a new item. */ newitem = pool->nextitem; /* Advance `nextitem' pointer to next free item in block. */ if (pool->itemwordtype == POINTER) { pool->nextitem = (VOID *) ((VOID **) pool->nextitem + pool->itemwords); } else { pool->nextitem = (VOID *) ((REAL *) pool->nextitem + pool->itemwords); } pool->unallocateditems--; pool->maxitems++; } pool->items++; return newitem; } /*****************************************************************************/ /* */ /* pooldealloc() Deallocate space for an item. */ /* */ /* The deallocated space is stored in a queue for later reuse. */ /* */ /*****************************************************************************/ void pooldealloc(pool, dyingitem) struct memorypool *pool; VOID *dyingitem; { /* Push freshly killed item onto stack. */ *((VOID **) dyingitem) = pool->deaditemstack; pool->deaditemstack = dyingitem; pool->items--; } /*****************************************************************************/ /* */ /* traversalinit() Prepare to traverse the entire list of items. */ /* */ /* This routine is used in conjunction with traverse(). */ /* */ /*****************************************************************************/ void traversalinit(pool) struct memorypool *pool; { unsigned long alignptr; /* Begin the traversal in the first block. */ pool->pathblock = pool->firstblock; /* Find the first item in the block. Increment by the size of (VOID *). */ alignptr = (unsigned long) (pool->pathblock + 1); /* Align with item on an `alignbytes'-byte boundary. */ pool->pathitem = (VOID *) (alignptr + (unsigned long) pool->alignbytes - (alignptr % (unsigned long) pool->alignbytes)); /* Set the number of items left in the current block. */ pool->pathitemsleft = pool->itemsperblock; } /*****************************************************************************/ /* */ /* traverse() Find the next item in the list. */ /* */ /* This routine is used in conjunction with traversalinit(). Be forewarned */ /* that this routine successively returns all items in the list, including */ /* deallocated ones on the deaditemqueue. It's up to you to figure out */ /* which ones are actually dead. Why? I don't want to allocate extra */ /* space just to demarcate dead items. It can usually be done more */ /* space-efficiently by a routine that knows something about the structure */ /* of the item. */ /* */ /*****************************************************************************/ VOID *traverse(pool) struct memorypool *pool; { VOID *newitem; unsigned long alignptr; /* Stop upon exhausting the list of items. */ if (pool->pathitem == pool->nextitem) { return (VOID *) NULL; } /* Check whether any untraversed items remain in the current block. */ if (pool->pathitemsleft == 0) { /* Find the next block. */ pool->pathblock = (VOID **) *(pool->pathblock); /* Find the first item in the block. Increment by the size of (VOID *). */ alignptr = (unsigned long) (pool->pathblock + 1); /* Align with item on an `alignbytes'-byte boundary. */ pool->pathitem = (VOID *) (alignptr + (unsigned long) pool->alignbytes - (alignptr % (unsigned long) pool->alignbytes)); /* Set the number of items left in the current block. */ pool->pathitemsleft = pool->itemsperblock; } newitem = pool->pathitem; /* Find the next item in the block. */ if (pool->itemwordtype == POINTER) { pool->pathitem = (VOID *) ((VOID **) pool->pathitem + pool->itemwords); } else { pool->pathitem = (VOID *) ((REAL *) pool->pathitem + pool->itemwords); } pool->pathitemsleft--; return newitem; } /*****************************************************************************/ /* */ /* dummyinit() Initialize the triangle that fills "outer space" and the */ /* omnipresent shell edge. */ /* */ /* The triangle that fills "outer space", called `dummytri', is pointed to */ /* by every triangle and shell edge on a boundary (be it outer or inner) of */ /* the triangulation. Also, `dummytri' points to one of the triangles on */ /* the convex hull (until the holes and concavities are carved), making it */ /* possible to find a starting triangle for point location. */ /* */ /* The omnipresent shell edge, `dummysh', is pointed to by every triangle */ /* or shell edge that doesn't have a full complement of real shell edges */ /* to point to. */ /* */ /*****************************************************************************/ void dummyinit(trianglewords, shellewords) int trianglewords; int shellewords; { unsigned long alignptr; /* `triwords' and `shwords' are used by the mesh manipulation primitives */ /* to extract orientations of triangles and shell edges from pointers. */ triwords = trianglewords; /* Initialize `triwords' once and for all. */ shwords = shellewords; /* Initialize `shwords' once and for all. */ /* Set up `dummytri', the `triangle' that occupies "outer space". */ dummytribase = (triangle *) malloc(triwords * sizeof(triangle) + triangles.alignbytes); if (dummytribase == (triangle *) NULL) { printf("Error: Out of memory.\n"); exit(1); } /* Align `dummytri' on a `triangles.alignbytes'-byte boundary. */ alignptr = (unsigned long) dummytribase; dummytri = (triangle *) (alignptr + (unsigned long) triangles.alignbytes - (alignptr % (unsigned long) triangles.alignbytes)); /* Initialize the three adjoining triangles to be "outer space". These */ /* will eventually be changed by various bonding operations, but their */ /* values don't really matter, as long as they can legally be */ /* dereferenced. */ dummytri[0] = (triangle) dummytri; dummytri[1] = (triangle) dummytri; dummytri[2] = (triangle) dummytri; /* Three NULL vertex points. */ dummytri[3] = (triangle) NULL; dummytri[4] = (triangle) NULL; dummytri[5] = (triangle) NULL; if (useshelles) { /* Set up `dummysh', the omnipresent "shell edge" pointed to by any */ /* triangle side or shell edge end that isn't attached to a real shell */ /* edge. */ dummyshbase = (shelle *) malloc(shwords * sizeof(shelle) + shelles.alignbytes); if (dummyshbase == (shelle *) NULL) { printf("Error: Out of memory.\n"); exit(1); } /* Align `dummysh' on a `shelles.alignbytes'-byte boundary. */ alignptr = (unsigned long) dummyshbase; dummysh = (shelle *) (alignptr + (unsigned long) shelles.alignbytes - (alignptr % (unsigned long) shelles.alignbytes)); /* Initialize the two adjoining shell edges to be the omnipresent shell */ /* edge. These will eventually be changed by various bonding */ /* operations, but their values don't really matter, as long as they */ /* can legally be dereferenced. */ dummysh[0] = (shelle) dummysh; dummysh[1] = (shelle) dummysh; /* Two NULL vertex points. */ dummysh[2] = (shelle) NULL; dummysh[3] = (shelle) NULL; /* Initialize the two adjoining triangles to be "outer space". */ dummysh[4] = (shelle) dummytri; dummysh[5] = (shelle) dummytri; /* Set the boundary marker to zero. */ * (int *) (dummysh + 6) = 0; /* Initialize the three adjoining shell edges of `dummytri' to be */ /* the omnipresent shell edge. */ dummytri[6] = (triangle) dummysh; dummytri[7] = (triangle) dummysh; dummytri[8] = (triangle) dummysh; } } /*****************************************************************************/ /* */ /* initializepointpool() Calculate the size of the point data structure */ /* and initialize its memory pool. */ /* */ /* This routine also computes the `pointmarkindex' and `point2triindex' */ /* indices used to find values within each point. */ /* */ /*****************************************************************************/ void initializepointpool() { int pointsize; /* The index within each point at which the boundary marker is found. */ /* Ensure the point marker is aligned to a sizeof(int)-byte address. */ pointmarkindex = ((mesh_dim + nextras) * sizeof(REAL) + sizeof(int) - 1) / sizeof(int); pointsize = (pointmarkindex + 1) * sizeof(int); if (poly) { /* The index within each point at which a triangle pointer is found. */ /* Ensure the pointer is aligned to a sizeof(triangle)-byte address. */ point2triindex = (pointsize + sizeof(triangle) - 1) / sizeof(triangle); pointsize = (point2triindex + 1) * sizeof(triangle); } /* Initialize the pool of points. */ poolinit(&points, pointsize, POINTPERBLOCK, (sizeof(REAL) >= sizeof(triangle)) ? FLOATINGPOINT : POINTER, 0); } /*****************************************************************************/ /* */ /* initializetrisegpools() Calculate the sizes of the triangle and shell */ /* edge data structures and initialize their */ /* memory pools. */ /* */ /* This routine also computes the `highorderindex', `elemattribindex', and */ /* `areaboundindex' indices used to find values within each triangle. */ /* */ /*****************************************************************************/ void initializetrisegpools() { int trisize; /* The index within each triangle at which the extra nodes (above three) */ /* associated with high order elements are found. There are three */ /* pointers to other triangles, three pointers to corners, and possibly */ /* three pointers to shell edges before the extra nodes. */ highorderindex = 6 + (useshelles * 3); /* The number of bytes occupied by a triangle. */ trisize = ((order + 1) * (order + 2) / 2 + (highorderindex - 3)) * sizeof(triangle); /* The index within each triangle at which its attributes are found, */ /* where the index is measured in REALs. */ elemattribindex = (trisize + sizeof(REAL) - 1) / sizeof(REAL); /* The index within each triangle at which the maximum area constraint */ /* is found, where the index is measured in REALs. Note that if the */ /* `regionattrib' flag is set, an additional attribute will be added. */ areaboundindex = elemattribindex + eextras + regionattrib; /* If triangle attributes or an area bound are needed, increase the number */ /* of bytes occupied by a triangle. */ if (vararea) { trisize = (areaboundindex + 1) * sizeof(REAL); } else if (eextras + regionattrib > 0) { trisize = areaboundindex * sizeof(REAL); } /* If a Voronoi diagram or triangle neighbor graph is requested, make */ /* sure there's room to store an integer index in each triangle. This */ /* integer index can occupy the same space as the shell edges or */ /* attributes or area constraint or extra nodes. */ if ((voronoi || neighbors) && (trisize < 6 * sizeof(triangle) + sizeof(int))) { trisize = 6 * sizeof(triangle) + sizeof(int); } /* Having determined the memory size of a triangle, initialize the pool. */ poolinit(&triangles, trisize, TRIPERBLOCK, POINTER, 4); if (useshelles) { /* Initialize the pool of shell edges. */ poolinit(&shelles, 6 * sizeof(triangle) + sizeof(int), SHELLEPERBLOCK, POINTER, 4); /* Initialize the "outer space" triangle and omnipresent shell edge. */ dummyinit(triangles.itemwords, shelles.itemwords); } else { /* Initialize the "outer space" triangle. */ dummyinit(triangles.itemwords, 0); } } /*****************************************************************************/ /* */ /* triangledealloc() Deallocate space for a triangle, marking it dead. */ /* */ /*****************************************************************************/ void triangledealloc(dyingtriangle) triangle *dyingtriangle; { /* Set triangle's vertices to NULL. This makes it possible to */ /* detect dead triangles when traversing the list of all triangles. */ dyingtriangle[3] = (triangle) NULL; dyingtriangle[4] = (triangle) NULL; dyingtriangle[5] = (triangle) NULL; pooldealloc(&triangles, (VOID *) dyingtriangle); } /*****************************************************************************/ /* */ /* triangletraverse() Traverse the triangles, skipping dead ones. */ /* */ /*****************************************************************************/ triangle *triangletraverse() { triangle *newtriangle; do { newtriangle = (triangle *) traverse(&triangles); if (newtriangle == (triangle *) NULL) { return (triangle *) NULL; } } while (newtriangle[3] == (triangle) NULL); /* Skip dead ones. */ return newtriangle; } /*****************************************************************************/ /* */ /* shelledealloc() Deallocate space for a shell edge, marking it dead. */ /* */ /*****************************************************************************/ void shelledealloc(dyingshelle) shelle *dyingshelle; { /* Set shell edge's vertices to NULL. This makes it possible to */ /* detect dead shells when traversing the list of all shells. */ dyingshelle[2] = (shelle) NULL; dyingshelle[3] = (shelle) NULL; pooldealloc(&shelles, (VOID *) dyingshelle); } /*****************************************************************************/ /* */ /* shelletraverse() Traverse the shell edges, skipping dead ones. */ /* */ /*****************************************************************************/ shelle *shelletraverse() { shelle *newshelle; do { newshelle = (shelle *) traverse(&shelles); if (newshelle == (shelle *) NULL) { return (shelle *) NULL; } } while (newshelle[2] == (shelle) NULL); /* Skip dead ones. */ return newshelle; } /*****************************************************************************/ /* */ /* pointdealloc() Deallocate space for a point, marking it dead. */ /* */ /*****************************************************************************/ void pointdealloc(dyingpoint) point dyingpoint; { /* Mark the point as dead. This makes it possible to detect dead points */ /* when traversing the list of all points. */ setpointmark(dyingpoint, DEADPOINT); pooldealloc(&points, (VOID *) dyingpoint); } /*****************************************************************************/ /* */ /* pointtraverse() Traverse the points, skipping dead ones. */ /* */ /*****************************************************************************/ point pointtraverse() { point newpoint; do { newpoint = (point) traverse(&points); if (newpoint == (point) NULL) { return (point) NULL; } } while (pointmark(newpoint) == DEADPOINT); /* Skip dead ones. */ return newpoint; } /*****************************************************************************/ /* */ /* badsegmentdealloc() Deallocate space for a bad segment, marking it */ /* dead. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void badsegmentdealloc(dyingseg) struct edge *dyingseg; { /* Set segment's orientation to -1. This makes it possible to */ /* detect dead segments when traversing the list of all segments. */ dyingseg->shorient = -1; pooldealloc(&badsegments, (VOID *) dyingseg); } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* badsegmenttraverse() Traverse the bad segments, skipping dead ones. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY struct edge *badsegmenttraverse() { struct edge *newseg; do { newseg = (struct edge *) traverse(&badsegments); if (newseg == (struct edge *) NULL) { return (struct edge *) NULL; } } while (newseg->shorient == -1); /* Skip dead ones. */ return newseg; } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* getpoint() Get a specific point, by number, from the list. */ /* */ /* The first point is number 'firstnumber'. */ /* */ /* Note that this takes O(n) time (with a small constant, if POINTPERBLOCK */ /* is large). I don't care to take the trouble to make it work in constant */ /* time. */ /* */ /*****************************************************************************/ point getpoint(number) int number; { VOID **getblock; point foundpoint; unsigned long alignptr; int current; getblock = points.firstblock; current = firstnumber; /* Find the right block. */ while (current + points.itemsperblock <= number) { getblock = (VOID **) *getblock; current += points.itemsperblock; } /* Now find the right point. */ alignptr = (unsigned long) (getblock + 1); foundpoint = (point) (alignptr + (unsigned long) points.alignbytes - (alignptr % (unsigned long) points.alignbytes)); while (current < number) { foundpoint += points.itemwords; current++; } return foundpoint; } /*****************************************************************************/ /* */ /* triangledeinit() Free all remaining allocated memory. */ /* */ /*****************************************************************************/ void triangledeinit() { pooldeinit(&triangles); free(dummytribase); if (useshelles) { pooldeinit(&shelles); free(dummyshbase); } pooldeinit(&points); #ifndef CDT_ONLY if (quality) { pooldeinit(&badsegments); if ((minangle > 0.0) || vararea || fixedarea) { pooldeinit(&badtriangles); } } #endif /* not CDT_ONLY */ } /** **/ /** **/ /********* Memory management routines end here *********/ /********* Constructors begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* maketriangle() Create a new triangle with orientation zero. */ /* */ /*****************************************************************************/ void maketriangle(newtriedge) struct triedge *newtriedge; { int i; newtriedge->tri = (triangle *) poolalloc(&triangles); /* Initialize the three adjoining triangles to be "outer space". */ newtriedge->tri[0] = (triangle) dummytri; newtriedge->tri[1] = (triangle) dummytri; newtriedge->tri[2] = (triangle) dummytri; /* Three NULL vertex points. */ newtriedge->tri[3] = (triangle) NULL; newtriedge->tri[4] = (triangle) NULL; newtriedge->tri[5] = (triangle) NULL; /* Initialize the three adjoining shell edges to be the omnipresent */ /* shell edge. */ if (useshelles) { newtriedge->tri[6] = (triangle) dummysh; newtriedge->tri[7] = (triangle) dummysh; newtriedge->tri[8] = (triangle) dummysh; } for (i = 0; i < eextras; i++) { setelemattribute(*newtriedge, i, 0.0); } if (vararea) { setareabound(*newtriedge, -1.0); } newtriedge->orient = 0; } /*****************************************************************************/ /* */ /* makeshelle() Create a new shell edge with orientation zero. */ /* */ /*****************************************************************************/ void makeshelle(newedge) struct edge *newedge; { newedge->sh = (shelle *) poolalloc(&shelles); /* Initialize the two adjoining shell edges to be the omnipresent */ /* shell edge. */ newedge->sh[0] = (shelle) dummysh; newedge->sh[1] = (shelle) dummysh; /* Two NULL vertex points. */ newedge->sh[2] = (shelle) NULL; newedge->sh[3] = (shelle) NULL; /* Initialize the two adjoining triangles to be "outer space". */ newedge->sh[4] = (shelle) dummytri; newedge->sh[5] = (shelle) dummytri; /* Set the boundary marker to zero. */ setmark(*newedge, 0); newedge->shorient = 0; } /** **/ /** **/ /********* Constructors end here *********/ /********* Determinant evaluation routines begin here *********/ /** **/ /** **/ /* The adaptive exact arithmetic geometric predicates implemented herein are */ /* described in detail in my Technical Report CMU-CS-96-140. The complete */ /* reference is given in the header. */ /* Which of the following two methods of finding the absolute values is */ /* fastest is compiler-dependent. A few compilers can inline and optimize */ /* the fabs() call; but most will incur the overhead of a function call, */ /* which is disastrously slow. A faster way on IEEE machines might be to */ /* mask the appropriate bit, but that's difficult to do in C. */ #define Absolute(a) ((a) >= 0.0 ? (a) : -(a)) /* #define Absolute(a) fabs(a) */ /* Many of the operations are broken up into two pieces, a main part that */ /* performs an approximate operation, and a "tail" that computes the */ /* roundoff error of that operation. */ /* */ /* The operations Fast_Two_Sum(), Fast_Two_Diff(), Two_Sum(), Two_Diff(), */ /* Split(), and Two_Product() are all implemented as described in the */ /* reference. Each of these macros requires certain variables to be */ /* defined in the calling routine. The variables `bvirt', `c', `abig', */ /* `_i', `_j', `_k', `_l', `_m', and `_n' are declared `INEXACT' because */ /* they store the result of an operation that may incur roundoff error. */ /* The input parameter `x' (or the highest numbered `x_' parameter) must */ /* also be declared `INEXACT'. */ #define Fast_Two_Sum_Tail(a, b, x, y) \ bvirt = x - a; \ y = b - bvirt #define Fast_Two_Sum(a, b, x, y) \ x = (REAL) (a + b); \ Fast_Two_Sum_Tail(a, b, x, y) #define Two_Sum_Tail(a, b, x, y) \ bvirt = (REAL) (x - a); \ avirt = x - bvirt; \ bround = b - bvirt; \ around = a - avirt; \ y = around + bround #define Two_Sum(a, b, x, y) \ x = (REAL) (a + b); \ Two_Sum_Tail(a, b, x, y) #define Two_Diff_Tail(a, b, x, y) \ bvirt = (REAL) (a - x); \ avirt = x + bvirt; \ bround = bvirt - b; \ around = a - avirt; \ y = around + bround #define Two_Diff(a, b, x, y) \ x = (REAL) (a - b); \ Two_Diff_Tail(a, b, x, y) #define Split(a, ahi, alo) \ c = (REAL) (splitter * a); \ abig = (REAL) (c - a); \ ahi = c - abig; \ alo = a - ahi #define Two_Product_Tail(a, b, x, y) \ Split(a, ahi, alo); \ Split(b, bhi, blo); \ err1 = x - (ahi * bhi); \ err2 = err1 - (alo * bhi); \ err3 = err2 - (ahi * blo); \ y = (alo * blo) - err3 #define Two_Product(a, b, x, y) \ x = (REAL) (a * b); \ Two_Product_Tail(a, b, x, y) /* Two_Product_Presplit() is Two_Product() where one of the inputs has */ /* already been split. Avoids redundant splitting. */ #define Two_Product_Presplit(a, b, bhi, blo, x, y) \ x = (REAL) (a * b); \ Split(a, ahi, alo); \ err1 = x - (ahi * bhi); \ err2 = err1 - (alo * bhi); \ err3 = err2 - (ahi * blo); \ y = (alo * blo) - err3 /* Square() can be done more quickly than Two_Product(). */ #define Square_Tail(a, x, y) \ Split(a, ahi, alo); \ err1 = x - (ahi * ahi); \ err3 = err1 - ((ahi + ahi) * alo); \ y = (alo * alo) - err3 #define Square(a, x, y) \ x = (REAL) (a * a); \ Square_Tail(a, x, y) /* Macros for summing expansions of various fixed lengths. These are all */ /* unrolled versions of Expansion_Sum(). */ #define Two_One_Sum(a1, a0, b, x2, x1, x0) \ Two_Sum(a0, b , _i, x0); \ Two_Sum(a1, _i, x2, x1) #define Two_One_Diff(a1, a0, b, x2, x1, x0) \ Two_Diff(a0, b , _i, x0); \ Two_Sum( a1, _i, x2, x1) #define Two_Two_Sum(a1, a0, b1, b0, x3, x2, x1, x0) \ Two_One_Sum(a1, a0, b0, _j, _0, x0); \ Two_One_Sum(_j, _0, b1, x3, x2, x1) #define Two_Two_Diff(a1, a0, b1, b0, x3, x2, x1, x0) \ Two_One_Diff(a1, a0, b0, _j, _0, x0); \ Two_One_Diff(_j, _0, b1, x3, x2, x1) /*****************************************************************************/ /* */ /* exactinit() Initialize the variables used for exact arithmetic. */ /* */ /* `epsilon' is the largest power of two such that 1.0 + epsilon = 1.0 in */ /* floating-point arithmetic. `epsilon' bounds the relative roundoff */ /* error. It is used for floating-point error analysis. */ /* */ /* `splitter' is used to split floating-point numbers into two half- */ /* length significands for exact multiplication. */ /* */ /* I imagine that a highly optimizing compiler might be too smart for its */ /* own good, and somehow cause this routine to fail, if it pretends that */ /* floating-point arithmetic is too much like real arithmetic. */ /* */ /* Don't change this routine unless you fully understand it. */ /* */ /*****************************************************************************/ void exactinit() { REAL half; REAL check, lastcheck; int every_other; every_other = 1; half = 0.5; epsilon = 1.0; splitter = 1.0; check = 1.0; /* Repeatedly divide `epsilon' by two until it is too small to add to */ /* one without causing roundoff. (Also check if the sum is equal to */ /* the previous sum, for machines that round up instead of using exact */ /* rounding. Not that these routines will work on such machines anyway. */ do { lastcheck = check; epsilon *= half; if (every_other) { splitter *= 2.0; } every_other = !every_other; check = 1.0 + epsilon; } while ((check != 1.0) && (check != lastcheck)); splitter += 1.0; if (verbose > 1) { printf("Floating point roundoff is of magnitude %.17g\n", epsilon); printf("Floating point splitter is %.17g\n", splitter); } /* Error bounds for orientation and incircle tests. */ resulterrbound = (3.0 + 8.0 * epsilon) * epsilon; ccwerrboundA = (3.0 + 16.0 * epsilon) * epsilon; ccwerrboundB = (2.0 + 12.0 * epsilon) * epsilon; ccwerrboundC = (9.0 + 64.0 * epsilon) * epsilon * epsilon; iccerrboundA = (10.0 + 96.0 * epsilon) * epsilon; iccerrboundB = (4.0 + 48.0 * epsilon) * epsilon; iccerrboundC = (44.0 + 576.0 * epsilon) * epsilon * epsilon; } /*****************************************************************************/ /* */ /* fast_expansion_sum_zeroelim() Sum two expansions, eliminating zero */ /* components from the output expansion. */ /* */ /* Sets h = e + f. See my Robust Predicates paper for details. */ /* */ /* If round-to-even is used (as with IEEE 754), maintains the strongly */ /* nonoverlapping property. (That is, if e is strongly nonoverlapping, h */ /* will be also.) Does NOT maintain the nonoverlapping or nonadjacent */ /* properties. */ /* */ /*****************************************************************************/ int fast_expansion_sum_zeroelim(elen, e, flen, f, h) /* h cannot be e or f. */ int elen; REAL *e; int flen; REAL *f; REAL *h; { REAL Q; INEXACT REAL Qnew; INEXACT REAL hh; INEXACT REAL bvirt; REAL avirt, bround, around; int eindex, findex, hindex; REAL enow, fnow; enow = e[0]; fnow = f[0]; eindex = findex = 0; if ((fnow > enow) == (fnow > -enow)) { Q = enow; enow = e[++eindex]; } else { Q = fnow; fnow = f[++findex]; } hindex = 0; if ((eindex < elen) && (findex < flen)) { if ((fnow > enow) == (fnow > -enow)) { Fast_Two_Sum(enow, Q, Qnew, hh); enow = e[++eindex]; } else { Fast_Two_Sum(fnow, Q, Qnew, hh); fnow = f[++findex]; } Q = Qnew; if (hh != 0.0) { h[hindex++] = hh; } while ((eindex < elen) && (findex < flen)) { if ((fnow > enow) == (fnow > -enow)) { Two_Sum(Q, enow, Qnew, hh); enow = e[++eindex]; } else { Two_Sum(Q, fnow, Qnew, hh); fnow = f[++findex]; } Q = Qnew; if (hh != 0.0) { h[hindex++] = hh; } } } while (eindex < elen) { Two_Sum(Q, enow, Qnew, hh); enow = e[++eindex]; Q = Qnew; if (hh != 0.0) { h[hindex++] = hh; } } while (findex < flen) { Two_Sum(Q, fnow, Qnew, hh); fnow = f[++findex]; Q = Qnew; if (hh != 0.0) { h[hindex++] = hh; } } if ((Q != 0.0) || (hindex == 0)) { h[hindex++] = Q; } return hindex; } /*****************************************************************************/ /* */ /* scale_expansion_zeroelim() Multiply an expansion by a scalar, */ /* eliminating zero components from the */ /* output expansion. */ /* */ /* Sets h = be. See my Robust Predicates paper for details. */ /* */ /* Maintains the nonoverlapping property. If round-to-even is used (as */ /* with IEEE 754), maintains the strongly nonoverlapping and nonadjacent */ /* properties as well. (That is, if e has one of these properties, so */ /* will h.) */ /* */ /*****************************************************************************/ int scale_expansion_zeroelim(elen, e, b, h) /* e and h cannot be the same. */ int elen; REAL *e; REAL b; REAL *h; { INEXACT REAL Q, sum; REAL hh; INEXACT REAL product1; REAL product0; int eindex, hindex; REAL enow; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; Split(b, bhi, blo); Two_Product_Presplit(e[0], b, bhi, blo, Q, hh); hindex = 0; if (hh != 0) { h[hindex++] = hh; } for (eindex = 1; eindex < elen; eindex++) { enow = e[eindex]; Two_Product_Presplit(enow, b, bhi, blo, product1, product0); Two_Sum(Q, product0, sum, hh); if (hh != 0) { h[hindex++] = hh; } Fast_Two_Sum(product1, sum, Q, hh); if (hh != 0) { h[hindex++] = hh; } } if ((Q != 0.0) || (hindex == 0)) { h[hindex++] = Q; } return hindex; } /*****************************************************************************/ /* */ /* estimate() Produce a one-word estimate of an expansion's value. */ /* */ /* See my Robust Predicates paper for details. */ /* */ /*****************************************************************************/ REAL estimate(elen, e) int elen; REAL *e; { REAL Q; int eindex; Q = e[0]; for (eindex = 1; eindex < elen; eindex++) { Q += e[eindex]; } return Q; } /*****************************************************************************/ /* */ /* counterclockwise() Return a positive value if the points pa, pb, and */ /* pc occur in counterclockwise order; a negative */ /* value if they occur in clockwise order; and zero */ /* if they are collinear. The result is also a rough */ /* approximation of twice the signed area of the */ /* triangle defined by the three points. */ /* */ /* Uses exact arithmetic if necessary to ensure a correct answer. The */ /* result returned is the determinant of a matrix. This determinant is */ /* computed adaptively, in the sense that exact arithmetic is used only to */ /* the degree it is needed to ensure that the returned value has the */ /* correct sign. Hence, this function is usually quite fast, but will run */ /* more slowly when the input points are collinear or nearly so. */ /* */ /* See my Robust Predicates paper for details. */ /* */ /*****************************************************************************/ REAL counterclockwiseadapt(pa, pb, pc, detsum) point pa; point pb; point pc; REAL detsum; { INEXACT REAL acx, acy, bcx, bcy; REAL acxtail, acytail, bcxtail, bcytail; INEXACT REAL detleft, detright; REAL detlefttail, detrighttail; REAL det, errbound; REAL B[4], C1[8], C2[12], D[16]; INEXACT REAL B3; int C1length, C2length, Dlength; REAL u[4]; INEXACT REAL u3; INEXACT REAL s1, t1; REAL s0, t0; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j; REAL _0; acx = (REAL) (pa[0] - pc[0]); bcx = (REAL) (pb[0] - pc[0]); acy = (REAL) (pa[1] - pc[1]); bcy = (REAL) (pb[1] - pc[1]); Two_Product(acx, bcy, detleft, detlefttail); Two_Product(acy, bcx, detright, detrighttail); Two_Two_Diff(detleft, detlefttail, detright, detrighttail, B3, B[2], B[1], B[0]); B[3] = B3; det = estimate(4, B); errbound = ccwerrboundB * detsum; if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Diff_Tail(pa[0], pc[0], acx, acxtail); Two_Diff_Tail(pb[0], pc[0], bcx, bcxtail); Two_Diff_Tail(pa[1], pc[1], acy, acytail); Two_Diff_Tail(pb[1], pc[1], bcy, bcytail); if ((acxtail == 0.0) && (acytail == 0.0) && (bcxtail == 0.0) && (bcytail == 0.0)) { return det; } errbound = ccwerrboundC * detsum + resulterrbound * Absolute(det); det += (acx * bcytail + bcy * acxtail) - (acy * bcxtail + bcx * acytail); if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Product(acxtail, bcy, s1, s0); Two_Product(acytail, bcx, t1, t0); Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]); u[3] = u3; C1length = fast_expansion_sum_zeroelim(4, B, 4, u, C1); Two_Product(acx, bcytail, s1, s0); Two_Product(acy, bcxtail, t1, t0); Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]); u[3] = u3; C2length = fast_expansion_sum_zeroelim(C1length, C1, 4, u, C2); Two_Product(acxtail, bcytail, s1, s0); Two_Product(acytail, bcxtail, t1, t0); Two_Two_Diff(s1, s0, t1, t0, u3, u[2], u[1], u[0]); u[3] = u3; Dlength = fast_expansion_sum_zeroelim(C2length, C2, 4, u, D); return(D[Dlength - 1]); } REAL counterclockwise(pa, pb, pc) point pa; point pb; point pc; { REAL detleft, detright, det; REAL detsum, errbound; counterclockcount++; detleft = (pa[0] - pc[0]) * (pb[1] - pc[1]); detright = (pa[1] - pc[1]) * (pb[0] - pc[0]); det = detleft - detright; if (noexact) { return det; } if (detleft > 0.0) { if (detright <= 0.0) { return det; } else { detsum = detleft + detright; } } else if (detleft < 0.0) { if (detright >= 0.0) { return det; } else { detsum = -detleft - detright; } } else { return det; } errbound = ccwerrboundA * detsum; if ((det >= errbound) || (-det >= errbound)) { return det; } return counterclockwiseadapt(pa, pb, pc, detsum); } /*****************************************************************************/ /* */ /* incircle() Return a positive value if the point pd lies inside the */ /* circle passing through pa, pb, and pc; a negative value if */ /* it lies outside; and zero if the four points are cocircular.*/ /* The points pa, pb, and pc must be in counterclockwise */ /* order, or the sign of the result will be reversed. */ /* */ /* Uses exact arithmetic if necessary to ensure a correct answer. The */ /* result returned is the determinant of a matrix. This determinant is */ /* computed adaptively, in the sense that exact arithmetic is used only to */ /* the degree it is needed to ensure that the returned value has the */ /* correct sign. Hence, this function is usually quite fast, but will run */ /* more slowly when the input points are cocircular or nearly so. */ /* */ /* See my Robust Predicates paper for details. */ /* */ /*****************************************************************************/ REAL incircleadapt(pa, pb, pc, pd, permanent) point pa; point pb; point pc; point pd; REAL permanent; { INEXACT REAL adx, bdx, cdx, ady, bdy, cdy; REAL det, errbound; INEXACT REAL bdxcdy1, cdxbdy1, cdxady1, adxcdy1, adxbdy1, bdxady1; REAL bdxcdy0, cdxbdy0, cdxady0, adxcdy0, adxbdy0, bdxady0; REAL bc[4], ca[4], ab[4]; INEXACT REAL bc3, ca3, ab3; REAL axbc[8], axxbc[16], aybc[8], ayybc[16], adet[32]; int axbclen, axxbclen, aybclen, ayybclen, alen; REAL bxca[8], bxxca[16], byca[8], byyca[16], bdet[32]; int bxcalen, bxxcalen, bycalen, byycalen, blen; REAL cxab[8], cxxab[16], cyab[8], cyyab[16], cdet[32]; int cxablen, cxxablen, cyablen, cyyablen, clen; REAL abdet[64]; int ablen; REAL fin1[1152], fin2[1152]; REAL *finnow, *finother, *finswap; int finlength; REAL adxtail, bdxtail, cdxtail, adytail, bdytail, cdytail; INEXACT REAL adxadx1, adyady1, bdxbdx1, bdybdy1, cdxcdx1, cdycdy1; REAL adxadx0, adyady0, bdxbdx0, bdybdy0, cdxcdx0, cdycdy0; REAL aa[4], bb[4], cc[4]; INEXACT REAL aa3, bb3, cc3; INEXACT REAL ti1, tj1; REAL ti0, tj0; REAL u[4], v[4]; INEXACT REAL u3, v3; REAL temp8[8], temp16a[16], temp16b[16], temp16c[16]; REAL temp32a[32], temp32b[32], temp48[48], temp64[64]; int temp8len, temp16alen, temp16blen, temp16clen; int temp32alen, temp32blen, temp48len, temp64len; REAL axtbb[8], axtcc[8], aytbb[8], aytcc[8]; int axtbblen, axtcclen, aytbblen, aytcclen; REAL bxtaa[8], bxtcc[8], bytaa[8], bytcc[8]; int bxtaalen, bxtcclen, bytaalen, bytcclen; REAL cxtaa[8], cxtbb[8], cytaa[8], cytbb[8]; int cxtaalen, cxtbblen, cytaalen, cytbblen; REAL axtbc[8], aytbc[8], bxtca[8], bytca[8], cxtab[8], cytab[8]; int axtbclen, aytbclen, bxtcalen, bytcalen, cxtablen, cytablen; REAL axtbct[16], aytbct[16], bxtcat[16], bytcat[16], cxtabt[16], cytabt[16]; int axtbctlen, aytbctlen, bxtcatlen, bytcatlen, cxtabtlen, cytabtlen; REAL axtbctt[8], aytbctt[8], bxtcatt[8]; REAL bytcatt[8], cxtabtt[8], cytabtt[8]; int axtbcttlen, aytbcttlen, bxtcattlen, bytcattlen, cxtabttlen, cytabttlen; REAL abt[8], bct[8], cat[8]; int abtlen, bctlen, catlen; REAL abtt[4], bctt[4], catt[4]; int abttlen, bcttlen, cattlen; INEXACT REAL abtt3, bctt3, catt3; REAL negate; INEXACT REAL bvirt; REAL avirt, bround, around; INEXACT REAL c; INEXACT REAL abig; REAL ahi, alo, bhi, blo; REAL err1, err2, err3; INEXACT REAL _i, _j; REAL _0; adx = (REAL) (pa[0] - pd[0]); bdx = (REAL) (pb[0] - pd[0]); cdx = (REAL) (pc[0] - pd[0]); ady = (REAL) (pa[1] - pd[1]); bdy = (REAL) (pb[1] - pd[1]); cdy = (REAL) (pc[1] - pd[1]); Two_Product(bdx, cdy, bdxcdy1, bdxcdy0); Two_Product(cdx, bdy, cdxbdy1, cdxbdy0); Two_Two_Diff(bdxcdy1, bdxcdy0, cdxbdy1, cdxbdy0, bc3, bc[2], bc[1], bc[0]); bc[3] = bc3; axbclen = scale_expansion_zeroelim(4, bc, adx, axbc); axxbclen = scale_expansion_zeroelim(axbclen, axbc, adx, axxbc); aybclen = scale_expansion_zeroelim(4, bc, ady, aybc); ayybclen = scale_expansion_zeroelim(aybclen, aybc, ady, ayybc); alen = fast_expansion_sum_zeroelim(axxbclen, axxbc, ayybclen, ayybc, adet); Two_Product(cdx, ady, cdxady1, cdxady0); Two_Product(adx, cdy, adxcdy1, adxcdy0); Two_Two_Diff(cdxady1, cdxady0, adxcdy1, adxcdy0, ca3, ca[2], ca[1], ca[0]); ca[3] = ca3; bxcalen = scale_expansion_zeroelim(4, ca, bdx, bxca); bxxcalen = scale_expansion_zeroelim(bxcalen, bxca, bdx, bxxca); bycalen = scale_expansion_zeroelim(4, ca, bdy, byca); byycalen = scale_expansion_zeroelim(bycalen, byca, bdy, byyca); blen = fast_expansion_sum_zeroelim(bxxcalen, bxxca, byycalen, byyca, bdet); Two_Product(adx, bdy, adxbdy1, adxbdy0); Two_Product(bdx, ady, bdxady1, bdxady0); Two_Two_Diff(adxbdy1, adxbdy0, bdxady1, bdxady0, ab3, ab[2], ab[1], ab[0]); ab[3] = ab3; cxablen = scale_expansion_zeroelim(4, ab, cdx, cxab); cxxablen = scale_expansion_zeroelim(cxablen, cxab, cdx, cxxab); cyablen = scale_expansion_zeroelim(4, ab, cdy, cyab); cyyablen = scale_expansion_zeroelim(cyablen, cyab, cdy, cyyab); clen = fast_expansion_sum_zeroelim(cxxablen, cxxab, cyyablen, cyyab, cdet); ablen = fast_expansion_sum_zeroelim(alen, adet, blen, bdet, abdet); finlength = fast_expansion_sum_zeroelim(ablen, abdet, clen, cdet, fin1); det = estimate(finlength, fin1); errbound = iccerrboundB * permanent; if ((det >= errbound) || (-det >= errbound)) { return det; } Two_Diff_Tail(pa[0], pd[0], adx, adxtail); Two_Diff_Tail(pa[1], pd[1], ady, adytail); Two_Diff_Tail(pb[0], pd[0], bdx, bdxtail); Two_Diff_Tail(pb[1], pd[1], bdy, bdytail); Two_Diff_Tail(pc[0], pd[0], cdx, cdxtail); Two_Diff_Tail(pc[1], pd[1], cdy, cdytail); if ((adxtail == 0.0) && (bdxtail == 0.0) && (cdxtail == 0.0) && (adytail == 0.0) && (bdytail == 0.0) && (cdytail == 0.0)) { return det; } errbound = iccerrboundC * permanent + resulterrbound * Absolute(det); det += ((adx * adx + ady * ady) * ((bdx * cdytail + cdy * bdxtail) - (bdy * cdxtail + cdx * bdytail)) + 2.0 * (adx * adxtail + ady * adytail) * (bdx * cdy - bdy * cdx)) + ((bdx * bdx + bdy * bdy) * ((cdx * adytail + ady * cdxtail) - (cdy * adxtail + adx * cdytail)) + 2.0 * (bdx * bdxtail + bdy * bdytail) * (cdx * ady - cdy * adx)) + ((cdx * cdx + cdy * cdy) * ((adx * bdytail + bdy * adxtail) - (ady * bdxtail + bdx * adytail)) + 2.0 * (cdx * cdxtail + cdy * cdytail) * (adx * bdy - ady * bdx)); if ((det >= errbound) || (-det >= errbound)) { return det; } finnow = fin1; finother = fin2; if ((bdxtail != 0.0) || (bdytail != 0.0) || (cdxtail != 0.0) || (cdytail != 0.0)) { Square(adx, adxadx1, adxadx0); Square(ady, adyady1, adyady0); Two_Two_Sum(adxadx1, adxadx0, adyady1, adyady0, aa3, aa[2], aa[1], aa[0]); aa[3] = aa3; } if ((cdxtail != 0.0) || (cdytail != 0.0) || (adxtail != 0.0) || (adytail != 0.0)) { Square(bdx, bdxbdx1, bdxbdx0); Square(bdy, bdybdy1, bdybdy0); Two_Two_Sum(bdxbdx1, bdxbdx0, bdybdy1, bdybdy0, bb3, bb[2], bb[1], bb[0]); bb[3] = bb3; } if ((adxtail != 0.0) || (adytail != 0.0) || (bdxtail != 0.0) || (bdytail != 0.0)) { Square(cdx, cdxcdx1, cdxcdx0); Square(cdy, cdycdy1, cdycdy0); Two_Two_Sum(cdxcdx1, cdxcdx0, cdycdy1, cdycdy0, cc3, cc[2], cc[1], cc[0]); cc[3] = cc3; } if (adxtail != 0.0) { axtbclen = scale_expansion_zeroelim(4, bc, adxtail, axtbc); temp16alen = scale_expansion_zeroelim(axtbclen, axtbc, 2.0 * adx, temp16a); axtcclen = scale_expansion_zeroelim(4, cc, adxtail, axtcc); temp16blen = scale_expansion_zeroelim(axtcclen, axtcc, bdy, temp16b); axtbblen = scale_expansion_zeroelim(4, bb, adxtail, axtbb); temp16clen = scale_expansion_zeroelim(axtbblen, axtbb, -cdy, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (adytail != 0.0) { aytbclen = scale_expansion_zeroelim(4, bc, adytail, aytbc); temp16alen = scale_expansion_zeroelim(aytbclen, aytbc, 2.0 * ady, temp16a); aytbblen = scale_expansion_zeroelim(4, bb, adytail, aytbb); temp16blen = scale_expansion_zeroelim(aytbblen, aytbb, cdx, temp16b); aytcclen = scale_expansion_zeroelim(4, cc, adytail, aytcc); temp16clen = scale_expansion_zeroelim(aytcclen, aytcc, -bdx, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdxtail != 0.0) { bxtcalen = scale_expansion_zeroelim(4, ca, bdxtail, bxtca); temp16alen = scale_expansion_zeroelim(bxtcalen, bxtca, 2.0 * bdx, temp16a); bxtaalen = scale_expansion_zeroelim(4, aa, bdxtail, bxtaa); temp16blen = scale_expansion_zeroelim(bxtaalen, bxtaa, cdy, temp16b); bxtcclen = scale_expansion_zeroelim(4, cc, bdxtail, bxtcc); temp16clen = scale_expansion_zeroelim(bxtcclen, bxtcc, -ady, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdytail != 0.0) { bytcalen = scale_expansion_zeroelim(4, ca, bdytail, bytca); temp16alen = scale_expansion_zeroelim(bytcalen, bytca, 2.0 * bdy, temp16a); bytcclen = scale_expansion_zeroelim(4, cc, bdytail, bytcc); temp16blen = scale_expansion_zeroelim(bytcclen, bytcc, adx, temp16b); bytaalen = scale_expansion_zeroelim(4, aa, bdytail, bytaa); temp16clen = scale_expansion_zeroelim(bytaalen, bytaa, -cdx, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdxtail != 0.0) { cxtablen = scale_expansion_zeroelim(4, ab, cdxtail, cxtab); temp16alen = scale_expansion_zeroelim(cxtablen, cxtab, 2.0 * cdx, temp16a); cxtbblen = scale_expansion_zeroelim(4, bb, cdxtail, cxtbb); temp16blen = scale_expansion_zeroelim(cxtbblen, cxtbb, ady, temp16b); cxtaalen = scale_expansion_zeroelim(4, aa, cdxtail, cxtaa); temp16clen = scale_expansion_zeroelim(cxtaalen, cxtaa, -bdy, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdytail != 0.0) { cytablen = scale_expansion_zeroelim(4, ab, cdytail, cytab); temp16alen = scale_expansion_zeroelim(cytablen, cytab, 2.0 * cdy, temp16a); cytaalen = scale_expansion_zeroelim(4, aa, cdytail, cytaa); temp16blen = scale_expansion_zeroelim(cytaalen, cytaa, bdx, temp16b); cytbblen = scale_expansion_zeroelim(4, bb, cdytail, cytbb); temp16clen = scale_expansion_zeroelim(cytbblen, cytbb, -adx, temp16c); temp32alen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16clen, temp16c, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; } if ((adxtail != 0.0) || (adytail != 0.0)) { if ((bdxtail != 0.0) || (bdytail != 0.0) || (cdxtail != 0.0) || (cdytail != 0.0)) { Two_Product(bdxtail, cdy, ti1, ti0); Two_Product(bdx, cdytail, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0]); u[3] = u3; negate = -bdy; Two_Product(cdxtail, negate, ti1, ti0); negate = -bdytail; Two_Product(cdx, negate, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0]); v[3] = v3; bctlen = fast_expansion_sum_zeroelim(4, u, 4, v, bct); Two_Product(bdxtail, cdytail, ti1, ti0); Two_Product(cdxtail, bdytail, tj1, tj0); Two_Two_Diff(ti1, ti0, tj1, tj0, bctt3, bctt[2], bctt[1], bctt[0]); bctt[3] = bctt3; bcttlen = 4; } else { bct[0] = 0.0; bctlen = 1; bctt[0] = 0.0; bcttlen = 1; } if (adxtail != 0.0) { temp16alen = scale_expansion_zeroelim(axtbclen, axtbc, adxtail, temp16a); axtbctlen = scale_expansion_zeroelim(bctlen, bct, adxtail, axtbct); temp32alen = scale_expansion_zeroelim(axtbctlen, axtbct, 2.0 * adx, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; if (bdytail != 0.0) { temp8len = scale_expansion_zeroelim(4, cc, adxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, bdytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdytail != 0.0) { temp8len = scale_expansion_zeroelim(4, bb, -adxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, cdytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim(axtbctlen, axtbct, adxtail, temp32a); axtbcttlen = scale_expansion_zeroelim(bcttlen, bctt, adxtail, axtbctt); temp16alen = scale_expansion_zeroelim(axtbcttlen, axtbctt, 2.0 * adx, temp16a); temp16blen = scale_expansion_zeroelim(axtbcttlen, axtbctt, adxtail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } if (adytail != 0.0) { temp16alen = scale_expansion_zeroelim(aytbclen, aytbc, adytail, temp16a); aytbctlen = scale_expansion_zeroelim(bctlen, bct, adytail, aytbct); temp32alen = scale_expansion_zeroelim(aytbctlen, aytbct, 2.0 * ady, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim(aytbctlen, aytbct, adytail, temp32a); aytbcttlen = scale_expansion_zeroelim(bcttlen, bctt, adytail, aytbctt); temp16alen = scale_expansion_zeroelim(aytbcttlen, aytbctt, 2.0 * ady, temp16a); temp16blen = scale_expansion_zeroelim(aytbcttlen, aytbctt, adytail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } } if ((bdxtail != 0.0) || (bdytail != 0.0)) { if ((cdxtail != 0.0) || (cdytail != 0.0) || (adxtail != 0.0) || (adytail != 0.0)) { Two_Product(cdxtail, ady, ti1, ti0); Two_Product(cdx, adytail, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0]); u[3] = u3; negate = -cdy; Two_Product(adxtail, negate, ti1, ti0); negate = -cdytail; Two_Product(adx, negate, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0]); v[3] = v3; catlen = fast_expansion_sum_zeroelim(4, u, 4, v, cat); Two_Product(cdxtail, adytail, ti1, ti0); Two_Product(adxtail, cdytail, tj1, tj0); Two_Two_Diff(ti1, ti0, tj1, tj0, catt3, catt[2], catt[1], catt[0]); catt[3] = catt3; cattlen = 4; } else { cat[0] = 0.0; catlen = 1; catt[0] = 0.0; cattlen = 1; } if (bdxtail != 0.0) { temp16alen = scale_expansion_zeroelim(bxtcalen, bxtca, bdxtail, temp16a); bxtcatlen = scale_expansion_zeroelim(catlen, cat, bdxtail, bxtcat); temp32alen = scale_expansion_zeroelim(bxtcatlen, bxtcat, 2.0 * bdx, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; if (cdytail != 0.0) { temp8len = scale_expansion_zeroelim(4, aa, bdxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, cdytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } if (adytail != 0.0) { temp8len = scale_expansion_zeroelim(4, cc, -bdxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, adytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim(bxtcatlen, bxtcat, bdxtail, temp32a); bxtcattlen = scale_expansion_zeroelim(cattlen, catt, bdxtail, bxtcatt); temp16alen = scale_expansion_zeroelim(bxtcattlen, bxtcatt, 2.0 * bdx, temp16a); temp16blen = scale_expansion_zeroelim(bxtcattlen, bxtcatt, bdxtail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdytail != 0.0) { temp16alen = scale_expansion_zeroelim(bytcalen, bytca, bdytail, temp16a); bytcatlen = scale_expansion_zeroelim(catlen, cat, bdytail, bytcat); temp32alen = scale_expansion_zeroelim(bytcatlen, bytcat, 2.0 * bdy, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim(bytcatlen, bytcat, bdytail, temp32a); bytcattlen = scale_expansion_zeroelim(cattlen, catt, bdytail, bytcatt); temp16alen = scale_expansion_zeroelim(bytcattlen, bytcatt, 2.0 * bdy, temp16a); temp16blen = scale_expansion_zeroelim(bytcattlen, bytcatt, bdytail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } } if ((cdxtail != 0.0) || (cdytail != 0.0)) { if ((adxtail != 0.0) || (adytail != 0.0) || (bdxtail != 0.0) || (bdytail != 0.0)) { Two_Product(adxtail, bdy, ti1, ti0); Two_Product(adx, bdytail, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, u3, u[2], u[1], u[0]); u[3] = u3; negate = -ady; Two_Product(bdxtail, negate, ti1, ti0); negate = -adytail; Two_Product(bdx, negate, tj1, tj0); Two_Two_Sum(ti1, ti0, tj1, tj0, v3, v[2], v[1], v[0]); v[3] = v3; abtlen = fast_expansion_sum_zeroelim(4, u, 4, v, abt); Two_Product(adxtail, bdytail, ti1, ti0); Two_Product(bdxtail, adytail, tj1, tj0); Two_Two_Diff(ti1, ti0, tj1, tj0, abtt3, abtt[2], abtt[1], abtt[0]); abtt[3] = abtt3; abttlen = 4; } else { abt[0] = 0.0; abtlen = 1; abtt[0] = 0.0; abttlen = 1; } if (cdxtail != 0.0) { temp16alen = scale_expansion_zeroelim(cxtablen, cxtab, cdxtail, temp16a); cxtabtlen = scale_expansion_zeroelim(abtlen, abt, cdxtail, cxtabt); temp32alen = scale_expansion_zeroelim(cxtabtlen, cxtabt, 2.0 * cdx, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; if (adytail != 0.0) { temp8len = scale_expansion_zeroelim(4, bb, cdxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, adytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } if (bdytail != 0.0) { temp8len = scale_expansion_zeroelim(4, aa, -cdxtail, temp8); temp16alen = scale_expansion_zeroelim(temp8len, temp8, bdytail, temp16a); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp16alen, temp16a, finother); finswap = finnow; finnow = finother; finother = finswap; } temp32alen = scale_expansion_zeroelim(cxtabtlen, cxtabt, cdxtail, temp32a); cxtabttlen = scale_expansion_zeroelim(abttlen, abtt, cdxtail, cxtabtt); temp16alen = scale_expansion_zeroelim(cxtabttlen, cxtabtt, 2.0 * cdx, temp16a); temp16blen = scale_expansion_zeroelim(cxtabttlen, cxtabtt, cdxtail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } if (cdytail != 0.0) { temp16alen = scale_expansion_zeroelim(cytablen, cytab, cdytail, temp16a); cytabtlen = scale_expansion_zeroelim(abtlen, abt, cdytail, cytabt); temp32alen = scale_expansion_zeroelim(cytabtlen, cytabt, 2.0 * cdy, temp32a); temp48len = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp32alen, temp32a, temp48); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp48len, temp48, finother); finswap = finnow; finnow = finother; finother = finswap; temp32alen = scale_expansion_zeroelim(cytabtlen, cytabt, cdytail, temp32a); cytabttlen = scale_expansion_zeroelim(abttlen, abtt, cdytail, cytabtt); temp16alen = scale_expansion_zeroelim(cytabttlen, cytabtt, 2.0 * cdy, temp16a); temp16blen = scale_expansion_zeroelim(cytabttlen, cytabtt, cdytail, temp16b); temp32blen = fast_expansion_sum_zeroelim(temp16alen, temp16a, temp16blen, temp16b, temp32b); temp64len = fast_expansion_sum_zeroelim(temp32alen, temp32a, temp32blen, temp32b, temp64); finlength = fast_expansion_sum_zeroelim(finlength, finnow, temp64len, temp64, finother); finswap = finnow; finnow = finother; finother = finswap; } } return finnow[finlength - 1]; } REAL incircle(pa, pb, pc, pd) point pa; point pb; point pc; point pd; { REAL adx, bdx, cdx, ady, bdy, cdy; REAL bdxcdy, cdxbdy, cdxady, adxcdy, adxbdy, bdxady; REAL alift, blift, clift; REAL det; REAL permanent, errbound; incirclecount++; adx = pa[0] - pd[0]; bdx = pb[0] - pd[0]; cdx = pc[0] - pd[0]; ady = pa[1] - pd[1]; bdy = pb[1] - pd[1]; cdy = pc[1] - pd[1]; bdxcdy = bdx * cdy; cdxbdy = cdx * bdy; alift = adx * adx + ady * ady; cdxady = cdx * ady; adxcdy = adx * cdy; blift = bdx * bdx + bdy * bdy; adxbdy = adx * bdy; bdxady = bdx * ady; clift = cdx * cdx + cdy * cdy; det = alift * (bdxcdy - cdxbdy) + blift * (cdxady - adxcdy) + clift * (adxbdy - bdxady); if (noexact) { return det; } permanent = (Absolute(bdxcdy) + Absolute(cdxbdy)) * alift + (Absolute(cdxady) + Absolute(adxcdy)) * blift + (Absolute(adxbdy) + Absolute(bdxady)) * clift; errbound = iccerrboundA * permanent; if ((det > errbound) || (-det > errbound)) { return det; } return incircleadapt(pa, pb, pc, pd, permanent); } /** **/ /** **/ /********* Determinant evaluation routines end here *********/ /*****************************************************************************/ /* */ /* triangleinit() Initialize some variables. */ /* */ /*****************************************************************************/ void triangleinit() { points.maxitems = triangles.maxitems = shelles.maxitems = viri.maxitems = badsegments.maxitems = badtriangles.maxitems = splaynodes.maxitems = 0l; points.itembytes = triangles.itembytes = shelles.itembytes = viri.itembytes = badsegments.itembytes = badtriangles.itembytes = splaynodes.itembytes = 0; recenttri.tri = (triangle *) NULL; /* No triangle has been visited yet. */ samples = 1; /* Point location should take at least one sample. */ checksegments = 0; /* There are no segments in the triangulation yet. */ incirclecount = counterclockcount = hyperbolacount = 0; circumcentercount = circletopcount = 0; randomseed = 1; exactinit(); /* Initialize exact arithmetic constants. */ } /*****************************************************************************/ /* */ /* randomnation() Generate a random number between 0 and `choices' - 1. */ /* */ /* This is a simple linear congruential random number generator. Hence, it */ /* is a bad random number generator, but good enough for most randomized */ /* geometric algorithms. */ /* */ /*****************************************************************************/ unsigned long randomnation(choices) unsigned int choices; { randomseed = (randomseed * 1366l + 150889l) % 714025l; return randomseed / (714025l / choices + 1); } /********* Mesh quality testing routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* checkmesh() Test the mesh for topological consistency. */ /* */ /*****************************************************************************/ #ifndef REDUCED void checkmesh() { struct triedge triangleloop; struct triedge oppotri, oppooppotri; point triorg, tridest, triapex; point oppoorg, oppodest; int horrors; int saveexact; triangle ptr; /* Temporary variable used by sym(). */ /* Temporarily turn on exact arithmetic if it's off. */ saveexact = noexact; noexact = 0; if (!quiet) { printf(" Checking consistency of mesh...\n"); } horrors = 0; /* Run through the list of triangles, checking each one. */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); while (triangleloop.tri != (triangle *) NULL) { /* Check all three edges of the triangle. */ for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { org(triangleloop, triorg); dest(triangleloop, tridest); if (triangleloop.orient == 0) { /* Only test for inversion once. */ /* Test if the triangle is flat or inverted. */ apex(triangleloop, triapex); if (counterclockwise(triorg, tridest, triapex) <= 0.0) { printf(" !! !! Inverted "); printtriangle(&triangleloop); horrors++; } } /* Find the neighboring triangle on this edge. */ sym(triangleloop, oppotri); if (oppotri.tri != dummytri) { /* Check that the triangle's neighbor knows it's a neighbor. */ sym(oppotri, oppooppotri); if ((triangleloop.tri != oppooppotri.tri) || (triangleloop.orient != oppooppotri.orient)) { printf(" !! !! Asymmetric triangle-triangle bond:\n"); if (triangleloop.tri == oppooppotri.tri) { printf(" (Right triangle, wrong orientation)\n"); } printf(" First "); printtriangle(&triangleloop); printf(" Second (nonreciprocating) "); printtriangle(&oppotri); horrors++; } /* Check that both triangles agree on the identities */ /* of their shared vertices. */ org(oppotri, oppoorg); dest(oppotri, oppodest); if ((triorg != oppodest) || (tridest != oppoorg)) { printf(" !! !! Mismatched edge coordinates between two triangles:\n" ); printf(" First mismatched "); printtriangle(&triangleloop); printf(" Second mismatched "); printtriangle(&oppotri); horrors++; } } } triangleloop.tri = triangletraverse(); } if (horrors == 0) { if (!quiet) { printf(" In my studied opinion, the mesh appears to be consistent.\n"); } } else if (horrors == 1) { printf(" !! !! !! !! Precisely one festering wound discovered.\n"); } else { printf(" !! !! !! !! %d abominations witnessed.\n", horrors); } /* Restore the status of exact arithmetic. */ noexact = saveexact; } #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* checkdelaunay() Ensure that the mesh is (constrained) Delaunay. */ /* */ /*****************************************************************************/ #ifndef REDUCED void checkdelaunay() { struct triedge triangleloop; struct triedge oppotri; struct edge opposhelle; point triorg, tridest, triapex; point oppoapex; int shouldbedelaunay; int horrors; int saveexact; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Temporarily turn on exact arithmetic if it's off. */ saveexact = noexact; noexact = 0; if (!quiet) { printf(" Checking Delaunay property of mesh...\n"); } horrors = 0; /* Run through the list of triangles, checking each one. */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); while (triangleloop.tri != (triangle *) NULL) { /* Check all three edges of the triangle. */ for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { org(triangleloop, triorg); dest(triangleloop, tridest); apex(triangleloop, triapex); sym(triangleloop, oppotri); apex(oppotri, oppoapex); /* Only test that the edge is locally Delaunay if there is an */ /* adjoining triangle whose pointer is larger (to ensure that */ /* each pair isn't tested twice). */ shouldbedelaunay = (oppotri.tri != dummytri) && (triapex != (point) NULL) && (oppoapex != (point) NULL) && (triangleloop.tri < oppotri.tri); if (checksegments && shouldbedelaunay) { /* If a shell edge separates the triangles, then the edge is */ /* constrained, so no local Delaunay test should be done. */ tspivot(triangleloop, opposhelle); if (opposhelle.sh != dummysh){ shouldbedelaunay = 0; } } if (shouldbedelaunay) { if (incircle(triorg, tridest, triapex, oppoapex) > 0.0) { printf(" !! !! Non-Delaunay pair of triangles:\n"); printf(" First non-Delaunay "); printtriangle(&triangleloop); printf(" Second non-Delaunay "); printtriangle(&oppotri); horrors++; } } } triangleloop.tri = triangletraverse(); } if (horrors == 0) { if (!quiet) { printf( " By virtue of my perceptive intelligence, I declare the mesh Delaunay.\n"); } } else if (horrors == 1) { printf( " !! !! !! !! Precisely one terrifying transgression identified.\n"); } else { printf(" !! !! !! !! %d obscenities viewed with horror.\n", horrors); } /* Restore the status of exact arithmetic. */ noexact = saveexact; } #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* enqueuebadtri() Add a bad triangle to the end of a queue. */ /* */ /* The queue is actually a set of 64 queues. I use multiple queues to give */ /* priority to smaller angles. I originally implemented a heap, but the */ /* queues are (to my surprise) much faster. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void enqueuebadtri(instri, angle, insapex, insorg, insdest) struct triedge *instri; REAL angle; point insapex; point insorg; point insdest; { struct badface *newface; int queuenumber; if (verbose > 2) { printf(" Queueing bad triangle:\n"); printf(" (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", insorg[0], insorg[1], insdest[0], insdest[1], insapex[0], insapex[1]); } /* Allocate space for the bad triangle. */ newface = (struct badface *) poolalloc(&badtriangles); triedgecopy(*instri, newface->badfacetri); newface->key = angle; newface->faceapex = insapex; newface->faceorg = insorg; newface->facedest = insdest; newface->nextface = (struct badface *) NULL; /* Determine the appropriate queue to put the bad triangle into. */ if (angle > 0.6) { queuenumber = (int) (160.0 * (angle - 0.6)); if (queuenumber > 63) { queuenumber = 63; } } else { /* It's not a bad angle; put the triangle in the lowest-priority queue. */ queuenumber = 0; } /* Add the triangle to the end of a queue. */ *queuetail[queuenumber] = newface; /* Maintain a pointer to the NULL pointer at the end of the queue. */ queuetail[queuenumber] = &newface->nextface; } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* dequeuebadtri() Remove a triangle from the front of the queue. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY struct badface *dequeuebadtri() { struct badface *result; int queuenumber; /* Look for a nonempty queue. */ for (queuenumber = 63; queuenumber >= 0; queuenumber--) { result = queuefront[queuenumber]; if (result != (struct badface *) NULL) { /* Remove the triangle from the queue. */ queuefront[queuenumber] = result->nextface; /* Maintain a pointer to the NULL pointer at the end of the queue. */ if (queuefront[queuenumber] == (struct badface *) NULL) { queuetail[queuenumber] = &queuefront[queuenumber]; } return result; } } return (struct badface *) NULL; } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* checkedge4encroach() Check a segment to see if it is encroached; add */ /* it to the list if it is. */ /* */ /* An encroached segment is an unflippable edge that has a point in its */ /* diametral circle (that is, it faces an angle greater than 90 degrees). */ /* This definition is due to Ruppert. */ /* */ /* Returns a nonzero value if the edge is encroached. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY int checkedge4encroach(testedge) struct edge *testedge; { struct triedge neighbortri; struct edge testsym; struct edge *badedge; int addtolist; int sides; point eorg, edest, eapex; triangle ptr; /* Temporary variable used by stpivot(). */ addtolist = 0; sides = 0; sorg(*testedge, eorg); sdest(*testedge, edest); /* Check one neighbor of the shell edge. */ stpivot(*testedge, neighbortri); /* Does the neighbor exist, or is this a boundary edge? */ if (neighbortri.tri != dummytri) { sides++; /* Find a vertex opposite this edge. */ apex(neighbortri, eapex); /* Check whether the vertex is inside the diametral circle of the */ /* shell edge. Pythagoras' Theorem is used to check whether the */ /* angle at the vertex is greater than 90 degrees. */ if (eapex[0] * (eorg[0] + edest[0]) + eapex[1] * (eorg[1] + edest[1]) > eapex[0] * eapex[0] + eorg[0] * edest[0] + eapex[1] * eapex[1] + eorg[1] * edest[1]) { addtolist = 1; } } /* Check the other neighbor of the shell edge. */ ssym(*testedge, testsym); stpivot(testsym, neighbortri); /* Does the neighbor exist, or is this a boundary edge? */ if (neighbortri.tri != dummytri) { sides++; /* Find the other vertex opposite this edge. */ apex(neighbortri, eapex); /* Check whether the vertex is inside the diametral circle of the */ /* shell edge. Pythagoras' Theorem is used to check whether the */ /* angle at the vertex is greater than 90 degrees. */ if (eapex[0] * (eorg[0] + edest[0]) + eapex[1] * (eorg[1] + edest[1]) > eapex[0] * eapex[0] + eorg[0] * edest[0] + eapex[1] * eapex[1] + eorg[1] * edest[1]) { addtolist += 2; } } if (addtolist && (!nobisect || ((nobisect == 1) && (sides == 2)))) { if (verbose > 2) { printf(" Queueing encroached segment (%.12g, %.12g) (%.12g, %.12g).\n", eorg[0], eorg[1], edest[0], edest[1]); } /* Add the shell edge to the list of encroached segments. */ /* Be sure to get the orientation right. */ badedge = (struct edge *) poolalloc(&badsegments); if (addtolist == 1) { shellecopy(*testedge, *badedge); } else { shellecopy(testsym, *badedge); } } return addtolist; } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* testtriangle() Test a face for quality measures. */ /* */ /* Tests a triangle to see if it satisfies the minimum angle condition and */ /* the maximum area condition. Triangles that aren't up to spec are added */ /* to the bad triangle queue. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void testtriangle(testtri) struct triedge *testtri; { struct triedge sametesttri; struct edge edge1, edge2; point torg, tdest, tapex; point anglevertex; REAL dxod, dyod, dxda, dyda, dxao, dyao; REAL dxod2, dyod2, dxda2, dyda2, dxao2, dyao2; REAL apexlen, orglen, destlen; REAL angle; REAL area; shelle sptr; /* Temporary variable used by tspivot(). */ org(*testtri, torg); dest(*testtri, tdest); apex(*testtri, tapex); dxod = torg[0] - tdest[0]; dyod = torg[1] - tdest[1]; dxda = tdest[0] - tapex[0]; dyda = tdest[1] - tapex[1]; dxao = tapex[0] - torg[0]; dyao = tapex[1] - torg[1]; dxod2 = dxod * dxod; dyod2 = dyod * dyod; dxda2 = dxda * dxda; dyda2 = dyda * dyda; dxao2 = dxao * dxao; dyao2 = dyao * dyao; /* Find the lengths of the triangle's three edges. */ apexlen = dxod2 + dyod2; orglen = dxda2 + dyda2; destlen = dxao2 + dyao2; if ((apexlen < orglen) && (apexlen < destlen)) { /* The edge opposite the apex is shortest. */ /* Find the square of the cosine of the angle at the apex. */ angle = dxda * dxao + dyda * dyao; angle = angle * angle / (orglen * destlen); anglevertex = tapex; lnext(*testtri, sametesttri); tspivot(sametesttri, edge1); lnextself(sametesttri); tspivot(sametesttri, edge2); } else if (orglen < destlen) { /* The edge opposite the origin is shortest. */ /* Find the square of the cosine of the angle at the origin. */ angle = dxod * dxao + dyod * dyao; angle = angle * angle / (apexlen * destlen); anglevertex = torg; tspivot(*testtri, edge1); lprev(*testtri, sametesttri); tspivot(sametesttri, edge2); } else { /* The edge opposite the destination is shortest. */ /* Find the square of the cosine of the angle at the destination. */ angle = dxod * dxda + dyod * dyda; angle = angle * angle / (apexlen * orglen); anglevertex = tdest; tspivot(*testtri, edge1); lnext(*testtri, sametesttri); tspivot(sametesttri, edge2); } /* Check if both edges that form the angle are segments. */ if ((edge1.sh != dummysh) && (edge2.sh != dummysh)) { /* The angle is a segment intersection. */ if ((angle > 0.9924) && !quiet) { /* Roughly 5 degrees. */ if (angle > 1.0) { /* Beware of a floating exception in acos(). */ angle = 1.0; } /* Find the actual angle in degrees, for printing. */ angle = acos(sqrt(angle)) * (180.0 / PI); printf( "Warning: Small angle (%.4g degrees) between segments at point\n", angle); printf(" (%.12g, %.12g)\n", anglevertex[0], anglevertex[1]); } /* Don't add this bad triangle to the list; there's nothing that */ /* can be done about a small angle between two segments. */ angle = 0.0; } /* Check whether the angle is smaller than permitted. */ if (angle > goodangle) { /* Add this triangle to the list of bad triangles. */ enqueuebadtri(testtri, angle, tapex, torg, tdest); return; } if (vararea || fixedarea) { /* Check whether the area is larger than permitted. */ area = 0.5 * (dxod * dyda - dyod * dxda); if (fixedarea && (area > maxarea)) { /* Add this triangle to the list of bad triangles. */ enqueuebadtri(testtri, angle, tapex, torg, tdest); } else if (vararea) { /* Nonpositive area constraints are treated as unconstrained. */ if ((area > areabound(*testtri)) && (areabound(*testtri) > 0.0)) { /* Add this triangle to the list of bad triangles. */ enqueuebadtri(testtri, angle, tapex, torg, tdest); } } } } #endif /* not CDT_ONLY */ /** **/ /** **/ /********* Mesh quality testing routines end here *********/ /********* Point location routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* makepointmap() Construct a mapping from points to triangles to improve */ /* the speed of point location for segment insertion. */ /* */ /* Traverses all the triangles, and provides each corner of each triangle */ /* with a pointer to that triangle. Of course, pointers will be */ /* overwritten by other pointers because (almost) each point is a corner */ /* of several triangles, but in the end every point will point to some */ /* triangle that contains it. */ /* */ /*****************************************************************************/ void makepointmap() { struct triedge triangleloop; point triorg; if (verbose) { printf(" Constructing mapping from points to triangles.\n"); } traversalinit(&triangles); triangleloop.tri = triangletraverse(); while (triangleloop.tri != (triangle *) NULL) { /* Check all three points of the triangle. */ for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { org(triangleloop, triorg); setpoint2tri(triorg, encode(triangleloop)); } triangleloop.tri = triangletraverse(); } } /*****************************************************************************/ /* */ /* preciselocate() Find a triangle or edge containing a given point. */ /* */ /* Begins its search from `searchtri'. It is important that `searchtri' */ /* be a handle with the property that `searchpoint' is strictly to the left */ /* of the edge denoted by `searchtri', or is collinear with that edge and */ /* does not intersect that edge. (In particular, `searchpoint' should not */ /* be the origin or destination of that edge.) */ /* */ /* These conditions are imposed because preciselocate() is normally used in */ /* one of two situations: */ /* */ /* (1) To try to find the location to insert a new point. Normally, we */ /* know an edge that the point is strictly to the left of. In the */ /* incremental Delaunay algorithm, that edge is a bounding box edge. */ /* In Ruppert's Delaunay refinement algorithm for quality meshing, */ /* that edge is the shortest edge of the triangle whose circumcenter */ /* is being inserted. */ /* */ /* (2) To try to find an existing point. In this case, any edge on the */ /* convex hull is a good starting edge. The possibility that the */ /* vertex one seeks is an endpoint of the starting edge must be */ /* screened out before preciselocate() is called. */ /* */ /* On completion, `searchtri' is a triangle that contains `searchpoint'. */ /* */ /* This implementation differs from that given by Guibas and Stolfi. It */ /* walks from triangle to triangle, crossing an edge only if `searchpoint' */ /* is on the other side of the line containing that edge. After entering */ /* a triangle, there are two edges by which one can leave that triangle. */ /* If both edges are valid (`searchpoint' is on the other side of both */ /* edges), one of the two is chosen by drawing a line perpendicular to */ /* the entry edge (whose endpoints are `forg' and `fdest') passing through */ /* `fapex'. Depending on which side of this perpendicular `searchpoint' */ /* falls on, an exit edge is chosen. */ /* */ /* This implementation is empirically faster than the Guibas and Stolfi */ /* point location routine (which I originally used), which tends to spiral */ /* in toward its target. */ /* */ /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */ /* is a handle whose origin is the existing vertex. */ /* */ /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */ /* handle whose primary edge is the edge on which the point lies. */ /* */ /* Returns INTRIANGLE if the point lies strictly within a triangle. */ /* `searchtri' is a handle on the triangle that contains the point. */ /* */ /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */ /* handle whose primary edge the point is to the right of. This might */ /* occur when the circumcenter of a triangle falls just slightly outside */ /* the mesh due to floating-point roundoff error. It also occurs when */ /* seeking a hole or region point that a foolish user has placed outside */ /* the mesh. */ /* */ /* WARNING: This routine is designed for convex triangulations, and will */ /* not generally work after the holes and concavities have been carved. */ /* However, it can still be used to find the circumcenter of a triangle, as */ /* long as the search is begun from the triangle in question. */ /* */ /*****************************************************************************/ enum locateresult preciselocate(searchpoint, searchtri) point searchpoint; struct triedge *searchtri; { struct triedge backtracktri; point forg, fdest, fapex; point swappoint; REAL orgorient, destorient; int moveleft; triangle ptr; /* Temporary variable used by sym(). */ if (verbose > 2) { printf(" Searching for point (%.12g, %.12g).\n", searchpoint[0], searchpoint[1]); } /* Where are we? */ org(*searchtri, forg); dest(*searchtri, fdest); apex(*searchtri, fapex); while (1) { if (verbose > 2) { printf(" At (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", forg[0], forg[1], fdest[0], fdest[1], fapex[0], fapex[1]); } /* Check whether the apex is the point we seek. */ if ((fapex[0] == searchpoint[0]) && (fapex[1] == searchpoint[1])) { lprevself(*searchtri); return ONVERTEX; } /* Does the point lie on the other side of the line defined by the */ /* triangle edge opposite the triangle's destination? */ destorient = counterclockwise(forg, fapex, searchpoint); /* Does the point lie on the other side of the line defined by the */ /* triangle edge opposite the triangle's origin? */ orgorient = counterclockwise(fapex, fdest, searchpoint); if (destorient > 0.0) { if (orgorient > 0.0) { /* Move left if the inner product of (fapex - searchpoint) and */ /* (fdest - forg) is positive. This is equivalent to drawing */ /* a line perpendicular to the line (forg, fdest) passing */ /* through `fapex', and determining which side of this line */ /* `searchpoint' falls on. */ moveleft = (fapex[0] - searchpoint[0]) * (fdest[0] - forg[0]) + (fapex[1] - searchpoint[1]) * (fdest[1] - forg[1]) > 0.0; } else { moveleft = 1; } } else { if (orgorient > 0.0) { moveleft = 0; } else { /* The point we seek must be on the boundary of or inside this */ /* triangle. */ if (destorient == 0.0) { lprevself(*searchtri); return ONEDGE; } if (orgorient == 0.0) { lnextself(*searchtri); return ONEDGE; } return INTRIANGLE; } } /* Move to another triangle. Leave a trace `backtracktri' in case */ /* floating-point roundoff or some such bogey causes us to walk */ /* off a boundary of the triangulation. We can just bounce off */ /* the boundary as if it were an elastic band. */ if (moveleft) { lprev(*searchtri, backtracktri); fdest = fapex; } else { lnext(*searchtri, backtracktri); forg = fapex; } sym(backtracktri, *searchtri); /* Check for walking off the edge. */ if (searchtri->tri == dummytri) { /* Turn around. */ triedgecopy(backtracktri, *searchtri); swappoint = forg; forg = fdest; fdest = swappoint; apex(*searchtri, fapex); /* Check if the point really is beyond the triangulation boundary. */ destorient = counterclockwise(forg, fapex, searchpoint); orgorient = counterclockwise(fapex, fdest, searchpoint); if ((orgorient < 0.0) && (destorient < 0.0)) { return OUTSIDE; } } else { apex(*searchtri, fapex); } } } /*****************************************************************************/ /* */ /* locate() Find a triangle or edge containing a given point. */ /* */ /* Searching begins from one of: the input `searchtri', a recently */ /* encountered triangle `recenttri', or from a triangle chosen from a */ /* random sample. The choice is made by determining which triangle's */ /* origin is closest to the point we are searcing for. Normally, */ /* `searchtri' should be a handle on the convex hull of the triangulation. */ /* */ /* Details on the random sampling method can be found in the Mucke, Saias, */ /* and Zhu paper cited in the header of this code. */ /* */ /* On completion, `searchtri' is a triangle that contains `searchpoint'. */ /* */ /* Returns ONVERTEX if the point lies on an existing vertex. `searchtri' */ /* is a handle whose origin is the existing vertex. */ /* */ /* Returns ONEDGE if the point lies on a mesh edge. `searchtri' is a */ /* handle whose primary edge is the edge on which the point lies. */ /* */ /* Returns INTRIANGLE if the point lies strictly within a triangle. */ /* `searchtri' is a handle on the triangle that contains the point. */ /* */ /* Returns OUTSIDE if the point lies outside the mesh. `searchtri' is a */ /* handle whose primary edge the point is to the right of. This might */ /* occur when the circumcenter of a triangle falls just slightly outside */ /* the mesh due to floating-point roundoff error. It also occurs when */ /* seeking a hole or region point that a foolish user has placed outside */ /* the mesh. */ /* */ /* WARNING: This routine is designed for convex triangulations, and will */ /* not generally work after the holes and concavities have been carved. */ /* */ /*****************************************************************************/ enum locateresult locate(searchpoint, searchtri) point searchpoint; struct triedge *searchtri; { VOID **sampleblock; triangle *firsttri; struct triedge sampletri; point torg, tdest; unsigned long alignptr; REAL searchdist, dist; REAL ahead; long sampleblocks, samplesperblock, samplenum; long triblocks; long i, j; triangle ptr; /* Temporary variable used by sym(). */ if (verbose > 2) { printf(" Randomly sampling for a triangle near point (%.12g, %.12g).\n", searchpoint[0], searchpoint[1]); } /* Record the distance from the suggested starting triangle to the */ /* point we seek. */ org(*searchtri, torg); searchdist = (searchpoint[0] - torg[0]) * (searchpoint[0] - torg[0]) + (searchpoint[1] - torg[1]) * (searchpoint[1] - torg[1]); if (verbose > 2) { printf(" Boundary triangle has origin (%.12g, %.12g).\n", torg[0], torg[1]); } /* If a recently encountered triangle has been recorded and has not been */ /* deallocated, test it as a good starting point. */ if (recenttri.tri != (triangle *) NULL) { if (recenttri.tri[3] != (triangle) NULL) { org(recenttri, torg); if ((torg[0] == searchpoint[0]) && (torg[1] == searchpoint[1])) { triedgecopy(recenttri, *searchtri); return ONVERTEX; } dist = (searchpoint[0] - torg[0]) * (searchpoint[0] - torg[0]) + (searchpoint[1] - torg[1]) * (searchpoint[1] - torg[1]); if (dist < searchdist) { triedgecopy(recenttri, *searchtri); searchdist = dist; if (verbose > 2) { printf(" Choosing recent triangle with origin (%.12g, %.12g).\n", torg[0], torg[1]); } } } } /* The number of random samples taken is proportional to the cube root of */ /* the number of triangles in the mesh. The next bit of code assumes */ /* that the number of triangles increases monotonically. */ while (SAMPLEFACTOR * samples * samples * samples < triangles.items) { samples++; } triblocks = (triangles.maxitems + TRIPERBLOCK - 1) / TRIPERBLOCK; samplesperblock = 1 + (samples / triblocks); sampleblocks = samples / samplesperblock; sampleblock = triangles.firstblock; sampletri.orient = 0; for (i = 0; i < sampleblocks; i++) { alignptr = (unsigned long) (sampleblock + 1); firsttri = (triangle *) (alignptr + (unsigned long) triangles.alignbytes - (alignptr % (unsigned long) triangles.alignbytes)); for (j = 0; j < samplesperblock; j++) { if (i == triblocks - 1) { samplenum = randomnation((int) (triangles.maxitems - (i * TRIPERBLOCK))); } else { samplenum = randomnation(TRIPERBLOCK); } sampletri.tri = (triangle *) (firsttri + (samplenum * triangles.itemwords)); if (sampletri.tri[3] != (triangle) NULL) { org(sampletri, torg); dist = (searchpoint[0] - torg[0]) * (searchpoint[0] - torg[0]) + (searchpoint[1] - torg[1]) * (searchpoint[1] - torg[1]); if (dist < searchdist) { triedgecopy(sampletri, *searchtri); searchdist = dist; if (verbose > 2) { printf(" Choosing triangle with origin (%.12g, %.12g).\n", torg[0], torg[1]); } } } } sampleblock = (VOID **) *sampleblock; } /* Where are we? */ org(*searchtri, torg); dest(*searchtri, tdest); /* Check the starting triangle's vertices. */ if ((torg[0] == searchpoint[0]) && (torg[1] == searchpoint[1])) { return ONVERTEX; } if ((tdest[0] == searchpoint[0]) && (tdest[1] == searchpoint[1])) { lnextself(*searchtri); return ONVERTEX; } /* Orient `searchtri' to fit the preconditions of calling preciselocate(). */ ahead = counterclockwise(torg, tdest, searchpoint); if (ahead < 0.0) { /* Turn around so that `searchpoint' is to the left of the */ /* edge specified by `searchtri'. */ symself(*searchtri); } else if (ahead == 0.0) { /* Check if `searchpoint' is between `torg' and `tdest'. */ if (((torg[0] < searchpoint[0]) == (searchpoint[0] < tdest[0])) && ((torg[1] < searchpoint[1]) == (searchpoint[1] < tdest[1]))) { return ONEDGE; } } return preciselocate(searchpoint, searchtri); } /** **/ /** **/ /********* Point location routines end here *********/ /********* Mesh transformation routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* insertshelle() Create a new shell edge and insert it between two */ /* triangles. */ /* */ /* The new shell edge is inserted at the edge described by the handle */ /* `tri'. Its vertices are properly initialized. The marker `shellemark' */ /* is applied to the shell edge and, if appropriate, its vertices. */ /* */ /*****************************************************************************/ void insertshelle(tri, shellemark) struct triedge *tri; /* Edge at which to insert the new shell edge. */ int shellemark; /* Marker for the new shell edge. */ { struct triedge oppotri; struct edge newshelle; point triorg, tridest; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Mark points if possible. */ org(*tri, triorg); dest(*tri, tridest); if (pointmark(triorg) == 0) { setpointmark(triorg, shellemark); } if (pointmark(tridest) == 0) { setpointmark(tridest, shellemark); } /* Check if there's already a shell edge here. */ tspivot(*tri, newshelle); if (newshelle.sh == dummysh) { /* Make new shell edge and initialize its vertices. */ makeshelle(&newshelle); setsorg(newshelle, tridest); setsdest(newshelle, triorg); /* Bond new shell edge to the two triangles it is sandwiched between. */ /* Note that the facing triangle `oppotri' might be equal to */ /* `dummytri' (outer space), but the new shell edge is bonded to it */ /* all the same. */ tsbond(*tri, newshelle); sym(*tri, oppotri); ssymself(newshelle); tsbond(oppotri, newshelle); setmark(newshelle, shellemark); if (verbose > 2) { printf(" Inserting new "); printshelle(&newshelle); } } else { if (mark(newshelle) == 0) { setmark(newshelle, shellemark); } } } /*****************************************************************************/ /* */ /* Terminology */ /* */ /* A "local transformation" replaces a small set of triangles with another */ /* set of triangles. This may or may not involve inserting or deleting a */ /* point. */ /* */ /* The term "casing" is used to describe the set of triangles that are */ /* attached to the triangles being transformed, but are not transformed */ /* themselves. Think of the casing as a fixed hollow structure inside */ /* which all the action happens. A "casing" is only defined relative to */ /* a single transformation; each occurrence of a transformation will */ /* involve a different casing. */ /* */ /* A "shell" is similar to a "casing". The term "shell" describes the set */ /* of shell edges (if any) that are attached to the triangles being */ /* transformed. However, I sometimes use "shell" to refer to a single */ /* shell edge, so don't get confused. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* flip() Transform two triangles to two different triangles by flipping */ /* an edge within a quadrilateral. */ /* */ /* Imagine the original triangles, abc and bad, oriented so that the */ /* shared edge ab lies in a horizontal plane, with the point b on the left */ /* and the point a on the right. The point c lies below the edge, and the */ /* point d lies above the edge. The `flipedge' handle holds the edge ab */ /* of triangle abc, and is directed left, from vertex a to vertex b. */ /* */ /* The triangles abc and bad are deleted and replaced by the triangles cdb */ /* and dca. The triangles that represent abc and bad are NOT deallocated; */ /* they are reused for dca and cdb, respectively. Hence, any handles that */ /* may have held the original triangles are still valid, although not */ /* directed as they were before. */ /* */ /* Upon completion of this routine, the `flipedge' handle holds the edge */ /* dc of triangle dca, and is directed down, from vertex d to vertex c. */ /* (Hence, the two triangles have rotated counterclockwise.) */ /* */ /* WARNING: This transformation is geometrically valid only if the */ /* quadrilateral adbc is convex. Furthermore, this transformation is */ /* valid only if there is not a shell edge between the triangles abc and */ /* bad. This routine does not check either of these preconditions, and */ /* it is the responsibility of the calling routine to ensure that they are */ /* met. If they are not, the streets shall be filled with wailing and */ /* gnashing of teeth. */ /* */ /*****************************************************************************/ void flip(flipedge) struct triedge *flipedge; /* Handle for the triangle abc. */ { struct triedge botleft, botright; struct triedge topleft, topright; struct triedge top; struct triedge botlcasing, botrcasing; struct triedge toplcasing, toprcasing; struct edge botlshelle, botrshelle; struct edge toplshelle, toprshelle; point leftpoint, rightpoint, botpoint; point farpoint; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ /* Identify the vertices of the quadrilateral. */ org(*flipedge, rightpoint); dest(*flipedge, leftpoint); apex(*flipedge, botpoint); sym(*flipedge, top); #ifdef SELF_CHECK if (top.tri == dummytri) { printf("Internal error in flip(): Attempt to flip on boundary.\n"); lnextself(*flipedge); return; } if (checksegments) { tspivot(*flipedge, toplshelle); if (toplshelle.sh != dummysh) { printf("Internal error in flip(): Attempt to flip a segment.\n"); lnextself(*flipedge); return; } } #endif /* SELF_CHECK */ apex(top, farpoint); /* Identify the casing of the quadrilateral. */ lprev(top, topleft); sym(topleft, toplcasing); lnext(top, topright); sym(topright, toprcasing); lnext(*flipedge, botleft); sym(botleft, botlcasing); lprev(*flipedge, botright); sym(botright, botrcasing); /* Rotate the quadrilateral one-quarter turn counterclockwise. */ bond(topleft, botlcasing); bond(botleft, botrcasing); bond(botright, toprcasing); bond(topright, toplcasing); if (checksegments) { /* Check for shell edges and rebond them to the quadrilateral. */ tspivot(topleft, toplshelle); tspivot(botleft, botlshelle); tspivot(botright, botrshelle); tspivot(topright, toprshelle); if (toplshelle.sh == dummysh) { tsdissolve(topright); } else { tsbond(topright, toplshelle); } if (botlshelle.sh == dummysh) { tsdissolve(topleft); } else { tsbond(topleft, botlshelle); } if (botrshelle.sh == dummysh) { tsdissolve(botleft); } else { tsbond(botleft, botrshelle); } if (toprshelle.sh == dummysh) { tsdissolve(botright); } else { tsbond(botright, toprshelle); } } /* New point assignments for the rotated quadrilateral. */ setorg(*flipedge, farpoint); setdest(*flipedge, botpoint); setapex(*flipedge, rightpoint); setorg(top, botpoint); setdest(top, farpoint); setapex(top, leftpoint); if (verbose > 2) { printf(" Edge flip results in left "); lnextself(topleft); printtriangle(&topleft); printf(" and right "); printtriangle(flipedge); } } /*****************************************************************************/ /* */ /* insertsite() Insert a vertex into a Delaunay triangulation, */ /* performing flips as necessary to maintain the Delaunay */ /* property. */ /* */ /* The point `insertpoint' is located. If `searchtri.tri' is not NULL, */ /* the search for the containing triangle begins from `searchtri'. If */ /* `searchtri.tri' is NULL, a full point location procedure is called. */ /* If `insertpoint' is found inside a triangle, the triangle is split into */ /* three; if `insertpoint' lies on an edge, the edge is split in two, */ /* thereby splitting the two adjacent triangles into four. Edge flips are */ /* used to restore the Delaunay property. If `insertpoint' lies on an */ /* existing vertex, no action is taken, and the value DUPLICATEPOINT is */ /* returned. On return, `searchtri' is set to a handle whose origin is the */ /* existing vertex. */ /* */ /* Normally, the parameter `splitedge' is set to NULL, implying that no */ /* segment should be split. In this case, if `insertpoint' is found to */ /* lie on a segment, no action is taken, and the value VIOLATINGPOINT is */ /* returned. On return, `searchtri' is set to a handle whose primary edge */ /* is the violated segment. */ /* */ /* If the calling routine wishes to split a segment by inserting a point in */ /* it, the parameter `splitedge' should be that segment. In this case, */ /* `searchtri' MUST be the triangle handle reached by pivoting from that */ /* segment; no point location is done. */ /* */ /* `segmentflaws' and `triflaws' are flags that indicate whether or not */ /* there should be checks for the creation of encroached segments or bad */ /* quality faces. If a newly inserted point encroaches upon segments, */ /* these segments are added to the list of segments to be split if */ /* `segmentflaws' is set. If bad triangles are created, these are added */ /* to the queue if `triflaws' is set. */ /* */ /* If a duplicate point or violated segment does not prevent the point */ /* from being inserted, the return value will be ENCROACHINGPOINT if the */ /* point encroaches upon a segment (and checking is enabled), or */ /* SUCCESSFULPOINT otherwise. In either case, `searchtri' is set to a */ /* handle whose origin is the newly inserted vertex. */ /* */ /* insertsite() does not use flip() for reasons of speed; some */ /* information can be reused from edge flip to edge flip, like the */ /* locations of shell edges. */ /* */ /*****************************************************************************/ enum insertsiteresult insertsite(insertpoint, searchtri, splitedge, segmentflaws, triflaws) point insertpoint; struct triedge *searchtri; struct edge *splitedge; int segmentflaws; int triflaws; { struct triedge horiz; struct triedge top; struct triedge botleft, botright; struct triedge topleft, topright; struct triedge newbotleft, newbotright; struct triedge newtopright; struct triedge botlcasing, botrcasing; struct triedge toplcasing, toprcasing; struct triedge testtri; struct edge botlshelle, botrshelle; struct edge toplshelle, toprshelle; struct edge brokenshelle; struct edge checkshelle; struct edge rightedge; struct edge newedge; struct edge *encroached; point first; point leftpoint, rightpoint, botpoint, toppoint, farpoint; REAL attrib; REAL area; enum insertsiteresult success; enum locateresult intersect; int doflip; int mirrorflag; int i; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by spivot() and tspivot(). */ if (verbose > 1) { printf(" Inserting (%.12g, %.12g).\n", insertpoint[0], insertpoint[1]); } if (splitedge == (struct edge *) NULL) { /* Find the location of the point to be inserted. Check if a good */ /* starting triangle has already been provided by the caller. */ if (searchtri->tri == (triangle *) NULL) { /* Find a boundary triangle. */ horiz.tri = dummytri; horiz.orient = 0; symself(horiz); /* Search for a triangle containing `insertpoint'. */ intersect = locate(insertpoint, &horiz); } else { /* Start searching from the triangle provided by the caller. */ triedgecopy(*searchtri, horiz); intersect = preciselocate(insertpoint, &horiz); } } else { /* The calling routine provides the edge in which the point is inserted. */ triedgecopy(*searchtri, horiz); intersect = ONEDGE; } if (intersect == ONVERTEX) { /* There's already a vertex there. Return in `searchtri' a triangle */ /* whose origin is the existing vertex. */ triedgecopy(horiz, *searchtri); triedgecopy(horiz, recenttri); return DUPLICATEPOINT; } if ((intersect == ONEDGE) || (intersect == OUTSIDE)) { /* The vertex falls on an edge or boundary. */ if (checksegments && (splitedge == (struct edge *) NULL)) { /* Check whether the vertex falls on a shell edge. */ tspivot(horiz, brokenshelle); if (brokenshelle.sh != dummysh) { /* The vertex falls on a shell edge. */ if (segmentflaws) { if (nobisect == 0) { /* Add the shell edge to the list of encroached segments. */ encroached = (struct edge *) poolalloc(&badsegments); shellecopy(brokenshelle, *encroached); } else if ((nobisect == 1) && (intersect == ONEDGE)) { /* This segment may be split only if it is an internal boundary. */ sym(horiz, testtri); if (testtri.tri != dummytri) { /* Add the shell edge to the list of encroached segments. */ encroached = (struct edge *) poolalloc(&badsegments); shellecopy(brokenshelle, *encroached); } } } /* Return a handle whose primary edge contains the point, */ /* which has not been inserted. */ triedgecopy(horiz, *searchtri); triedgecopy(horiz, recenttri); return VIOLATINGPOINT; } } /* Insert the point on an edge, dividing one triangle into two (if */ /* the edge lies on a boundary) or two triangles into four. */ lprev(horiz, botright); sym(botright, botrcasing); sym(horiz, topright); /* Is there a second triangle? (Or does this edge lie on a boundary?) */ mirrorflag = topright.tri != dummytri; if (mirrorflag) { lnextself(topright); sym(topright, toprcasing); maketriangle(&newtopright); } else { /* Splitting the boundary edge increases the number of boundary edges. */ hullsize++; } maketriangle(&newbotright); /* Set the vertices of changed and new triangles. */ org(horiz, rightpoint); dest(horiz, leftpoint); apex(horiz, botpoint); setorg(newbotright, botpoint); setdest(newbotright, rightpoint); setapex(newbotright, insertpoint); setorg(horiz, insertpoint); for (i = 0; i < eextras; i++) { /* Set the element attributes of a new triangle. */ setelemattribute(newbotright, i, elemattribute(botright, i)); } if (vararea) { /* Set the area constraint of a new triangle. */ setareabound(newbotright, areabound(botright)); } if (mirrorflag) { dest(topright, toppoint); setorg(newtopright, rightpoint); setdest(newtopright, toppoint); setapex(newtopright, insertpoint); setorg(topright, insertpoint); for (i = 0; i < eextras; i++) { /* Set the element attributes of another new triangle. */ setelemattribute(newtopright, i, elemattribute(topright, i)); } if (vararea) { /* Set the area constraint of another new triangle. */ setareabound(newtopright, areabound(topright)); } } /* There may be shell edges that need to be bonded */ /* to the new triangle(s). */ if (checksegments) { tspivot(botright, botrshelle); if (botrshelle.sh != dummysh) { tsdissolve(botright); tsbond(newbotright, botrshelle); } if (mirrorflag) { tspivot(topright, toprshelle); if (toprshelle.sh != dummysh) { tsdissolve(topright); tsbond(newtopright, toprshelle); } } } /* Bond the new triangle(s) to the surrounding triangles. */ bond(newbotright, botrcasing); lprevself(newbotright); bond(newbotright, botright); lprevself(newbotright); if (mirrorflag) { bond(newtopright, toprcasing); lnextself(newtopright); bond(newtopright, topright); lnextself(newtopright); bond(newtopright, newbotright); } if (splitedge != (struct edge *) NULL) { /* Split the shell edge into two. */ setsdest(*splitedge, insertpoint); ssymself(*splitedge); spivot(*splitedge, rightedge); insertshelle(&newbotright, mark(*splitedge)); tspivot(newbotright, newedge); sbond(*splitedge, newedge); ssymself(newedge); sbond(newedge, rightedge); ssymself(*splitedge); } #ifdef SELF_CHECK if (counterclockwise(rightpoint, leftpoint, botpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle prior to edge point insertion (bottom).\n"); } if (mirrorflag) { if (counterclockwise(leftpoint, rightpoint, toppoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle prior to edge point insertion (top).\n"); } if (counterclockwise(rightpoint, toppoint, insertpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle after edge point insertion (top right).\n" ); } if (counterclockwise(toppoint, leftpoint, insertpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle after edge point insertion (top left).\n" ); } } if (counterclockwise(leftpoint, botpoint, insertpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle after edge point insertion (bottom left).\n" ); } if (counterclockwise(botpoint, rightpoint, insertpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf( " Clockwise triangle after edge point insertion (bottom right).\n"); } #endif /* SELF_CHECK */ if (verbose > 2) { printf(" Updating bottom left "); printtriangle(&botright); if (mirrorflag) { printf(" Updating top left "); printtriangle(&topright); printf(" Creating top right "); printtriangle(&newtopright); } printf(" Creating bottom right "); printtriangle(&newbotright); } /* Position `horiz' on the first edge to check for */ /* the Delaunay property. */ lnextself(horiz); } else { /* Insert the point in a triangle, splitting it into three. */ lnext(horiz, botleft); lprev(horiz, botright); sym(botleft, botlcasing); sym(botright, botrcasing); maketriangle(&newbotleft); maketriangle(&newbotright); /* Set the vertices of changed and new triangles. */ org(horiz, rightpoint); dest(horiz, leftpoint); apex(horiz, botpoint); setorg(newbotleft, leftpoint); setdest(newbotleft, botpoint); setapex(newbotleft, insertpoint); setorg(newbotright, botpoint); setdest(newbotright, rightpoint); setapex(newbotright, insertpoint); setapex(horiz, insertpoint); for (i = 0; i < eextras; i++) { /* Set the element attributes of the new triangles. */ attrib = elemattribute(horiz, i); setelemattribute(newbotleft, i, attrib); setelemattribute(newbotright, i, attrib); } if (vararea) { /* Set the area constraint of the new triangles. */ area = areabound(horiz); setareabound(newbotleft, area); setareabound(newbotright, area); } /* There may be shell edges that need to be bonded */ /* to the new triangles. */ if (checksegments) { tspivot(botleft, botlshelle); if (botlshelle.sh != dummysh) { tsdissolve(botleft); tsbond(newbotleft, botlshelle); } tspivot(botright, botrshelle); if (botrshelle.sh != dummysh) { tsdissolve(botright); tsbond(newbotright, botrshelle); } } /* Bond the new triangles to the surrounding triangles. */ bond(newbotleft, botlcasing); bond(newbotright, botrcasing); lnextself(newbotleft); lprevself(newbotright); bond(newbotleft, newbotright); lnextself(newbotleft); bond(botleft, newbotleft); lprevself(newbotright); bond(botright, newbotright); #ifdef SELF_CHECK if (counterclockwise(rightpoint, leftpoint, botpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle prior to point insertion.\n"); } if (counterclockwise(rightpoint, leftpoint, insertpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle after point insertion (top).\n"); } if (counterclockwise(leftpoint, botpoint, insertpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle after point insertion (left).\n"); } if (counterclockwise(botpoint, rightpoint, insertpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle after point insertion (right).\n"); } #endif /* SELF_CHECK */ if (verbose > 2) { printf(" Updating top "); printtriangle(&horiz); printf(" Creating left "); printtriangle(&newbotleft); printf(" Creating right "); printtriangle(&newbotright); } } /* The insertion is successful by default, unless an encroached */ /* edge is found. */ success = SUCCESSFULPOINT; /* Circle around the newly inserted vertex, checking each edge opposite */ /* it for the Delaunay property. Non-Delaunay edges are flipped. */ /* `horiz' is always the edge being checked. `first' marks where to */ /* stop circling. */ org(horiz, first); rightpoint = first; dest(horiz, leftpoint); /* Circle until finished. */ while (1) { /* By default, the edge will be flipped. */ doflip = 1; if (checksegments) { /* Check for a segment, which cannot be flipped. */ tspivot(horiz, checkshelle); if (checkshelle.sh != dummysh) { /* The edge is a segment and cannot be flipped. */ doflip = 0; #ifndef CDT_ONLY if (segmentflaws) { /* Does the new point encroach upon this segment? */ if (checkedge4encroach(&checkshelle)) { success = ENCROACHINGPOINT; } } #endif /* not CDT_ONLY */ } } if (doflip) { /* Check if the edge is a boundary edge. */ sym(horiz, top); if (top.tri == dummytri) { /* The edge is a boundary edge and cannot be flipped. */ doflip = 0; } else { /* Find the point on the other side of the edge. */ apex(top, farpoint); /* In the incremental Delaunay triangulation algorithm, any of */ /* `leftpoint', `rightpoint', and `farpoint' could be vertices */ /* of the triangular bounding box. These vertices must be */ /* treated as if they are infinitely distant, even though their */ /* "coordinates" are not. */ if ((leftpoint == infpoint1) || (leftpoint == infpoint2) || (leftpoint == infpoint3)) { /* `leftpoint' is infinitely distant. Check the convexity of */ /* the boundary of the triangulation. 'farpoint' might be */ /* infinite as well, but trust me, this same condition */ /* should be applied. */ doflip = counterclockwise(insertpoint, rightpoint, farpoint) > 0.0; } else if ((rightpoint == infpoint1) || (rightpoint == infpoint2) || (rightpoint == infpoint3)) { /* `rightpoint' is infinitely distant. Check the convexity of */ /* the boundary of the triangulation. 'farpoint' might be */ /* infinite as well, but trust me, this same condition */ /* should be applied. */ doflip = counterclockwise(farpoint, leftpoint, insertpoint) > 0.0; } else if ((farpoint == infpoint1) || (farpoint == infpoint2) || (farpoint == infpoint3)) { /* `farpoint' is infinitely distant and cannot be inside */ /* the circumcircle of the triangle `horiz'. */ doflip = 0; } else { /* Test whether the edge is locally Delaunay. */ doflip = incircle(leftpoint, insertpoint, rightpoint, farpoint) > 0.0; } if (doflip) { /* We made it! Flip the edge `horiz' by rotating its containing */ /* quadrilateral (the two triangles adjacent to `horiz'). */ /* Identify the casing of the quadrilateral. */ lprev(top, topleft); sym(topleft, toplcasing); lnext(top, topright); sym(topright, toprcasing); lnext(horiz, botleft); sym(botleft, botlcasing); lprev(horiz, botright); sym(botright, botrcasing); /* Rotate the quadrilateral one-quarter turn counterclockwise. */ bond(topleft, botlcasing); bond(botleft, botrcasing); bond(botright, toprcasing); bond(topright, toplcasing); if (checksegments) { /* Check for shell edges and rebond them to the quadrilateral. */ tspivot(topleft, toplshelle); tspivot(botleft, botlshelle); tspivot(botright, botrshelle); tspivot(topright, toprshelle); if (toplshelle.sh == dummysh) { tsdissolve(topright); } else { tsbond(topright, toplshelle); } if (botlshelle.sh == dummysh) { tsdissolve(topleft); } else { tsbond(topleft, botlshelle); } if (botrshelle.sh == dummysh) { tsdissolve(botleft); } else { tsbond(botleft, botrshelle); } if (toprshelle.sh == dummysh) { tsdissolve(botright); } else { tsbond(botright, toprshelle); } } /* New point assignments for the rotated quadrilateral. */ setorg(horiz, farpoint); setdest(horiz, insertpoint); setapex(horiz, rightpoint); setorg(top, insertpoint); setdest(top, farpoint); setapex(top, leftpoint); for (i = 0; i < eextras; i++) { /* Take the average of the two triangles' attributes. */ attrib = 0.5 * (elemattribute(top, i) + elemattribute(horiz, i)); setelemattribute(top, i, attrib); setelemattribute(horiz, i, attrib); } if (vararea) { if ((areabound(top) <= 0.0) || (areabound(horiz) <= 0.0)) { area = -1.0; } else { /* Take the average of the two triangles' area constraints. */ /* This prevents small area constraints from migrating a */ /* long, long way from their original location due to flips. */ area = 0.5 * (areabound(top) + areabound(horiz)); } setareabound(top, area); setareabound(horiz, area); } #ifdef SELF_CHECK if (insertpoint != (point) NULL) { if (counterclockwise(leftpoint, insertpoint, rightpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle prior to edge flip (bottom).\n"); } /* The following test has been removed because constrainededge() */ /* sometimes generates inverted triangles that insertsite() */ /* removes. */ /* if (counterclockwise(rightpoint, farpoint, leftpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle prior to edge flip (top).\n"); } */ if (counterclockwise(farpoint, leftpoint, insertpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle after edge flip (left).\n"); } if (counterclockwise(insertpoint, rightpoint, farpoint) < 0.0) { printf("Internal error in insertsite():\n"); printf(" Clockwise triangle after edge flip (right).\n"); } } #endif /* SELF_CHECK */ if (verbose > 2) { printf(" Edge flip results in left "); lnextself(topleft); printtriangle(&topleft); printf(" and right "); printtriangle(&horiz); } /* On the next iterations, consider the two edges that were */ /* exposed (this is, are now visible to the newly inserted */ /* point) by the edge flip. */ lprevself(horiz); leftpoint = farpoint; } } } if (!doflip) { /* The handle `horiz' is accepted as locally Delaunay. */ #ifndef CDT_ONLY if (triflaws) { /* Check the triangle `horiz' for quality. */ testtriangle(&horiz); } #endif /* not CDT_ONLY */ /* Look for the next edge around the newly inserted point. */ lnextself(horiz); sym(horiz, testtri); /* Check for finishing a complete revolution about the new point, or */ /* falling off the edge of the triangulation. The latter will */ /* happen when a point is inserted at a boundary. */ if ((leftpoint == first) || (testtri.tri == dummytri)) { /* We're done. Return a triangle whose origin is the new point. */ lnext(horiz, *searchtri); lnext(horiz, recenttri); return success; } /* Finish finding the next edge around the newly inserted point. */ lnext(testtri, horiz); rightpoint = leftpoint; dest(horiz, leftpoint); } } } /*****************************************************************************/ /* */ /* triangulatepolygon() Find the Delaunay triangulation of a polygon that */ /* has a certain "nice" shape. This includes the */ /* polygons that result from deletion of a point or */ /* insertion of a segment. */ /* */ /* This is a conceptually difficult routine. The starting assumption is */ /* that we have a polygon with n sides. n - 1 of these sides are currently */ /* represented as edges in the mesh. One side, called the "base", need not */ /* be. */ /* */ /* Inside the polygon is a structure I call a "fan", consisting of n - 1 */ /* triangles that share a common origin. For each of these triangles, the */ /* edge opposite the origin is one of the sides of the polygon. The */ /* primary edge of each triangle is the edge directed from the origin to */ /* the destination; note that this is not the same edge that is a side of */ /* the polygon. `firstedge' is the primary edge of the first triangle. */ /* From there, the triangles follow in counterclockwise order about the */ /* polygon, until `lastedge', the primary edge of the last triangle. */ /* `firstedge' and `lastedge' are probably connected to other triangles */ /* beyond the extremes of the fan, but their identity is not important, as */ /* long as the fan remains connected to them. */ /* */ /* Imagine the polygon oriented so that its base is at the bottom. This */ /* puts `firstedge' on the far right, and `lastedge' on the far left. */ /* The right vertex of the base is the destination of `firstedge', and the */ /* left vertex of the base is the apex of `lastedge'. */ /* */ /* The challenge now is to find the right sequence of edge flips to */ /* transform the fan into a Delaunay triangulation of the polygon. Each */ /* edge flip effectively removes one triangle from the fan, committing it */ /* to the polygon. The resulting polygon has one fewer edge. If `doflip' */ /* is set, the final flip will be performed, resulting in a fan of one */ /* (useless?) triangle. If `doflip' is not set, the final flip is not */ /* performed, resulting in a fan of two triangles, and an unfinished */ /* triangular polygon that is not yet filled out with a single triangle. */ /* On completion of the routine, `lastedge' is the last remaining triangle, */ /* or the leftmost of the last two. */ /* */ /* Although the flips are performed in the order described above, the */ /* decisions about what flips to perform are made in precisely the reverse */ /* order. The recursive triangulatepolygon() procedure makes a decision, */ /* uses up to two recursive calls to triangulate the "subproblems" */ /* (polygons with fewer edges), and then performs an edge flip. */ /* */ /* The "decision" it makes is which vertex of the polygon should be */ /* connected to the base. This decision is made by testing every possible */ /* vertex. Once the best vertex is found, the two edges that connect this */ /* vertex to the base become the bases for two smaller polygons. These */ /* are triangulated recursively. Unfortunately, this approach can take */ /* O(n^2) time not only in the worst case, but in many common cases. It's */ /* rarely a big deal for point deletion, where n is rarely larger than ten, */ /* but it could be a big deal for segment insertion, especially if there's */ /* a lot of long segments that each cut many triangles. I ought to code */ /* a faster algorithm some time. */ /* */ /* The `edgecount' parameter is the number of sides of the polygon, */ /* including its base. `triflaws' is a flag that determines whether the */ /* new triangles should be tested for quality, and enqueued if they are */ /* bad. */ /* */ /*****************************************************************************/ void triangulatepolygon(firstedge, lastedge, edgecount, doflip, triflaws) struct triedge *firstedge; struct triedge *lastedge; int edgecount; int doflip; int triflaws; { struct triedge testtri; struct triedge besttri; struct triedge tempedge; point leftbasepoint, rightbasepoint; point testpoint; point bestpoint; int bestnumber; int i; triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */ /* Identify the base vertices. */ apex(*lastedge, leftbasepoint); dest(*firstedge, rightbasepoint); if (verbose > 2) { printf(" Triangulating interior polygon at edge\n"); printf(" (%.12g, %.12g) (%.12g, %.12g)\n", leftbasepoint[0], leftbasepoint[1], rightbasepoint[0], rightbasepoint[1]); } /* Find the best vertex to connect the base to. */ onext(*firstedge, besttri); dest(besttri, bestpoint); triedgecopy(besttri, testtri); bestnumber = 1; for (i = 2; i <= edgecount - 2; i++) { onextself(testtri); dest(testtri, testpoint); /* Is this a better vertex? */ if (incircle(leftbasepoint, rightbasepoint, bestpoint, testpoint) > 0.0) { triedgecopy(testtri, besttri); bestpoint = testpoint; bestnumber = i; } } if (verbose > 2) { printf(" Connecting edge to (%.12g, %.12g)\n", bestpoint[0], bestpoint[1]); } if (bestnumber > 1) { /* Recursively triangulate the smaller polygon on the right. */ oprev(besttri, tempedge); triangulatepolygon(firstedge, &tempedge, bestnumber + 1, 1, triflaws); } if (bestnumber < edgecount - 2) { /* Recursively triangulate the smaller polygon on the left. */ sym(besttri, tempedge); triangulatepolygon(&besttri, lastedge, edgecount - bestnumber, 1, triflaws); /* Find `besttri' again; it may have been lost to edge flips. */ sym(tempedge, besttri); } if (doflip) { /* Do one final edge flip. */ flip(&besttri); #ifndef CDT_ONLY if (triflaws) { /* Check the quality of the newly committed triangle. */ sym(besttri, testtri); testtriangle(&testtri); } #endif /* not CDT_ONLY */ } /* Return the base triangle. */ triedgecopy(besttri, *lastedge); } /*****************************************************************************/ /* */ /* deletesite() Delete a vertex from a Delaunay triangulation, ensuring */ /* that the triangulation remains Delaunay. */ /* */ /* The origin of `deltri' is deleted. The union of the triangles adjacent */ /* to this point is a polygon, for which the Delaunay triangulation is */ /* found. Two triangles are removed from the mesh. */ /* */ /* Only interior points that do not lie on segments (shell edges) or */ /* boundaries may be deleted. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void deletesite(deltri) struct triedge *deltri; { struct triedge countingtri; struct triedge firstedge, lastedge; struct triedge deltriright; struct triedge lefttri, righttri; struct triedge leftcasing, rightcasing; struct edge leftshelle, rightshelle; point delpoint; point neworg; int edgecount; triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */ shelle sptr; /* Temporary variable used by tspivot(). */ org(*deltri, delpoint); if (verbose > 1) { printf(" Deleting (%.12g, %.12g).\n", delpoint[0], delpoint[1]); } pointdealloc(delpoint); /* Count the degree of the point being deleted. */ onext(*deltri, countingtri); edgecount = 1; while (!triedgeequal(*deltri, countingtri)) { #ifdef SELF_CHECK if (countingtri.tri == dummytri) { printf("Internal error in deletesite():\n"); printf(" Attempt to delete boundary point.\n"); internalerror(); } #endif /* SELF_CHECK */ edgecount++; onextself(countingtri); } #ifdef SELF_CHECK if (edgecount < 3) { printf("Internal error in deletesite():\n Point has degree %d.\n", edgecount); internalerror(); } #endif /* SELF_CHECK */ if (edgecount > 3) { /* Triangulate the polygon defined by the union of all triangles */ /* adjacent to the point being deleted. Check the quality of */ /* the resulting triangles. */ onext(*deltri, firstedge); oprev(*deltri, lastedge); triangulatepolygon(&firstedge, &lastedge, edgecount, 0, !nobisect); } /* Splice out two triangles. */ lprev(*deltri, deltriright); dnext(*deltri, lefttri); sym(lefttri, leftcasing); oprev(deltriright, righttri); sym(righttri, rightcasing); bond(*deltri, leftcasing); bond(deltriright, rightcasing); tspivot(lefttri, leftshelle); if (leftshelle.sh != dummysh) { tsbond(*deltri, leftshelle); } tspivot(righttri, rightshelle); if (rightshelle.sh != dummysh) { tsbond(deltriright, rightshelle); } /* Set the new origin of `deltri' and check its quality. */ org(lefttri, neworg); setorg(*deltri, neworg); if (!nobisect) { testtriangle(deltri); } /* Delete the two spliced-out triangles. */ triangledealloc(lefttri.tri); triangledealloc(righttri.tri); } #endif /* not CDT_ONLY */ /** **/ /** **/ /********* Mesh transformation routines end here *********/ /********* Divide-and-conquer Delaunay triangulation begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* The divide-and-conquer bounding box */ /* */ /* I originally implemented the divide-and-conquer and incremental Delaunay */ /* triangulations using the edge-based data structure presented by Guibas */ /* and Stolfi. Switching to a triangle-based data structure doubled the */ /* speed. However, I had to think of a few extra tricks to maintain the */ /* elegance of the original algorithms. */ /* */ /* The "bounding box" used by my variant of the divide-and-conquer */ /* algorithm uses one triangle for each edge of the convex hull of the */ /* triangulation. These bounding triangles all share a common apical */ /* vertex, which is represented by NULL and which represents nothing. */ /* The bounding triangles are linked in a circular fan about this NULL */ /* vertex, and the edges on the convex hull of the triangulation appear */ /* opposite the NULL vertex. You might find it easiest to imagine that */ /* the NULL vertex is a point in 3D space behind the center of the */ /* triangulation, and that the bounding triangles form a sort of cone. */ /* */ /* This bounding box makes it easy to represent degenerate cases. For */ /* instance, the triangulation of two vertices is a single edge. This edge */ /* is represented by two bounding box triangles, one on each "side" of the */ /* edge. These triangles are also linked together in a fan about the NULL */ /* vertex. */ /* */ /* The bounding box also makes it easy to traverse the convex hull, as the */ /* divide-and-conquer algorithm needs to do. */ /* */ /*****************************************************************************/ /*****************************************************************************/ /* */ /* pointsort() Sort an array of points by x-coordinate, using the */ /* y-coordinate as a secondary key. */ /* */ /* Uses quicksort. Randomized O(n log n) time. No, I did not make any of */ /* the usual quicksort mistakes. */ /* */ /*****************************************************************************/ void pointsort(sortarray, arraysize) point *sortarray; int arraysize; { int left, right; int pivot; REAL pivotx, pivoty; point temp; if (arraysize == 2) { /* Recursive base case. */ if ((sortarray[0][0] > sortarray[1][0]) || ((sortarray[0][0] == sortarray[1][0]) && (sortarray[0][1] > sortarray[1][1]))) { temp = sortarray[1]; sortarray[1] = sortarray[0]; sortarray[0] = temp; } return; } /* Choose a random pivot to split the array. */ pivot = (int) randomnation(arraysize); pivotx = sortarray[pivot][0]; pivoty = sortarray[pivot][1]; /* Split the array. */ left = -1; right = arraysize; while (left < right) { /* Search for a point whose x-coordinate is too large for the left. */ do { left++; } while ((left <= right) && ((sortarray[left][0] < pivotx) || ((sortarray[left][0] == pivotx) && (sortarray[left][1] < pivoty)))); /* Search for a point whose x-coordinate is too small for the right. */ do { right--; } while ((left <= right) && ((sortarray[right][0] > pivotx) || ((sortarray[right][0] == pivotx) && (sortarray[right][1] > pivoty)))); if (left < right) { /* Swap the left and right points. */ temp = sortarray[left]; sortarray[left] = sortarray[right]; sortarray[right] = temp; } } if (left > 1) { /* Recursively sort the left subset. */ pointsort(sortarray, left); } if (right < arraysize - 2) { /* Recursively sort the right subset. */ pointsort(&sortarray[right + 1], arraysize - right - 1); } } /*****************************************************************************/ /* */ /* pointmedian() An order statistic algorithm, almost. Shuffles an array */ /* of points so that the first `median' points occur */ /* lexicographically before the remaining points. */ /* */ /* Uses the x-coordinate as the primary key if axis == 0; the y-coordinate */ /* if axis == 1. Very similar to the pointsort() procedure, but runs in */ /* randomized linear time. */ /* */ /*****************************************************************************/ void pointmedian(sortarray, arraysize, median, axis) point *sortarray; int arraysize; int median; int axis; { int left, right; int pivot; REAL pivot1, pivot2; point temp; if (arraysize == 2) { /* Recursive base case. */ if ((sortarray[0][axis] > sortarray[1][axis]) || ((sortarray[0][axis] == sortarray[1][axis]) && (sortarray[0][1 - axis] > sortarray[1][1 - axis]))) { temp = sortarray[1]; sortarray[1] = sortarray[0]; sortarray[0] = temp; } return; } /* Choose a random pivot to split the array. */ pivot = (int) randomnation(arraysize); pivot1 = sortarray[pivot][axis]; pivot2 = sortarray[pivot][1 - axis]; /* Split the array. */ left = -1; right = arraysize; while (left < right) { /* Search for a point whose x-coordinate is too large for the left. */ do { left++; } while ((left <= right) && ((sortarray[left][axis] < pivot1) || ((sortarray[left][axis] == pivot1) && (sortarray[left][1 - axis] < pivot2)))); /* Search for a point whose x-coordinate is too small for the right. */ do { right--; } while ((left <= right) && ((sortarray[right][axis] > pivot1) || ((sortarray[right][axis] == pivot1) && (sortarray[right][1 - axis] > pivot2)))); if (left < right) { /* Swap the left and right points. */ temp = sortarray[left]; sortarray[left] = sortarray[right]; sortarray[right] = temp; } } /* Unlike in pointsort(), at most one of the following */ /* conditionals is true. */ if (left > median) { /* Recursively shuffle the left subset. */ pointmedian(sortarray, left, median, axis); } if (right < median - 1) { /* Recursively shuffle the right subset. */ pointmedian(&sortarray[right + 1], arraysize - right - 1, median - right - 1, axis); } } /*****************************************************************************/ /* */ /* alternateaxes() Sorts the points as appropriate for the divide-and- */ /* conquer algorithm with alternating cuts. */ /* */ /* Partitions by x-coordinate if axis == 0; by y-coordinate if axis == 1. */ /* For the base case, subsets containing only two or three points are */ /* always sorted by x-coordinate. */ /* */ /*****************************************************************************/ void alternateaxes(sortarray, arraysize, axis) point *sortarray; int arraysize; int axis; { int divider; divider = arraysize >> 1; if (arraysize <= 3) { /* Recursive base case: subsets of two or three points will be */ /* handled specially, and should always be sorted by x-coordinate. */ axis = 0; } /* Partition with a horizontal or vertical cut. */ pointmedian(sortarray, arraysize, divider, axis); /* Recursively partition the subsets with a cross cut. */ if (arraysize - divider >= 2) { if (divider >= 2) { alternateaxes(sortarray, divider, 1 - axis); } alternateaxes(&sortarray[divider], arraysize - divider, 1 - axis); } } /*****************************************************************************/ /* */ /* mergehulls() Merge two adjacent Delaunay triangulations into a */ /* single Delaunay triangulation. */ /* */ /* This is similar to the algorithm given by Guibas and Stolfi, but uses */ /* a triangle-based, rather than edge-based, data structure. */ /* */ /* The algorithm walks up the gap between the two triangulations, knitting */ /* them together. As they are merged, some of their bounding triangles */ /* are converted into real triangles of the triangulation. The procedure */ /* pulls each hull's bounding triangles apart, then knits them together */ /* like the teeth of two gears. The Delaunay property determines, at each */ /* step, whether the next "tooth" is a bounding triangle of the left hull */ /* or the right. When a bounding triangle becomes real, its apex is */ /* changed from NULL to a real point. */ /* */ /* Only two new triangles need to be allocated. These become new bounding */ /* triangles at the top and bottom of the seam. They are used to connect */ /* the remaining bounding triangles (those that have not been converted */ /* into real triangles) into a single fan. */ /* */ /* On entry, `farleft' and `innerleft' are bounding triangles of the left */ /* triangulation. The origin of `farleft' is the leftmost vertex, and */ /* the destination of `innerleft' is the rightmost vertex of the */ /* triangulation. Similarly, `innerright' and `farright' are bounding */ /* triangles of the right triangulation. The origin of `innerright' and */ /* destination of `farright' are the leftmost and rightmost vertices. */ /* */ /* On completion, the origin of `farleft' is the leftmost vertex of the */ /* merged triangulation, and the destination of `farright' is the rightmost */ /* vertex. */ /* */ /*****************************************************************************/ void mergehulls(farleft, innerleft, innerright, farright, axis) struct triedge *farleft; struct triedge *innerleft; struct triedge *innerright; struct triedge *farright; int axis; { struct triedge leftcand, rightcand; struct triedge baseedge; struct triedge nextedge; struct triedge sidecasing, topcasing, outercasing; struct triedge checkedge; point innerleftdest; point innerrightorg; point innerleftapex, innerrightapex; point farleftpt, farrightpt; point farleftapex, farrightapex; point lowerleft, lowerright; point upperleft, upperright; point nextapex; point checkvertex; int changemade; int badedge; int leftfinished, rightfinished; triangle ptr; /* Temporary variable used by sym(). */ dest(*innerleft, innerleftdest); apex(*innerleft, innerleftapex); org(*innerright, innerrightorg); apex(*innerright, innerrightapex); /* Special treatment for horizontal cuts. */ if (dwyer && (axis == 1)) { org(*farleft, farleftpt); apex(*farleft, farleftapex); dest(*farright, farrightpt); apex(*farright, farrightapex); /* The pointers to the extremal points are shifted to point to the */ /* topmost and bottommost point of each hull, rather than the */ /* leftmost and rightmost points. */ while (farleftapex[1] < farleftpt[1]) { lnextself(*farleft); symself(*farleft); farleftpt = farleftapex; apex(*farleft, farleftapex); } sym(*innerleft, checkedge); apex(checkedge, checkvertex); while (checkvertex[1] > innerleftdest[1]) { lnext(checkedge, *innerleft); innerleftapex = innerleftdest; innerleftdest = checkvertex; sym(*innerleft, checkedge); apex(checkedge, checkvertex); } while (innerrightapex[1] < innerrightorg[1]) { lnextself(*innerright); symself(*innerright); innerrightorg = innerrightapex; apex(*innerright, innerrightapex); } sym(*farright, checkedge); apex(checkedge, checkvertex); while (checkvertex[1] > farrightpt[1]) { lnext(checkedge, *farright); farrightapex = farrightpt; farrightpt = checkvertex; sym(*farright, checkedge); apex(checkedge, checkvertex); } } /* Find a line tangent to and below both hulls. */ do { changemade = 0; /* Make innerleftdest the "bottommost" point of the left hull. */ if (counterclockwise(innerleftdest, innerleftapex, innerrightorg) > 0.0) { lprevself(*innerleft); symself(*innerleft); innerleftdest = innerleftapex; apex(*innerleft, innerleftapex); changemade = 1; } /* Make innerrightorg the "bottommost" point of the right hull. */ if (counterclockwise(innerrightapex, innerrightorg, innerleftdest) > 0.0) { lnextself(*innerright); symself(*innerright); innerrightorg = innerrightapex; apex(*innerright, innerrightapex); changemade = 1; } } while (changemade); /* Find the two candidates to be the next "gear tooth". */ sym(*innerleft, leftcand); sym(*innerright, rightcand); /* Create the bottom new bounding triangle. */ maketriangle(&baseedge); /* Connect it to the bounding boxes of the left and right triangulations. */ bond(baseedge, *innerleft); lnextself(baseedge); bond(baseedge, *innerright); lnextself(baseedge); setorg(baseedge, innerrightorg); setdest(baseedge, innerleftdest); /* Apex is intentionally left NULL. */ if (verbose > 2) { printf(" Creating base bounding "); printtriangle(&baseedge); } /* Fix the extreme triangles if necessary. */ org(*farleft, farleftpt); if (innerleftdest == farleftpt) { lnext(baseedge, *farleft); } dest(*farright, farrightpt); if (innerrightorg == farrightpt) { lprev(baseedge, *farright); } /* The vertices of the current knitting edge. */ lowerleft = innerleftdest; lowerright = innerrightorg; /* The candidate vertices for knitting. */ apex(leftcand, upperleft); apex(rightcand, upperright); /* Walk up the gap between the two triangulations, knitting them together. */ while (1) { /* Have we reached the top? (This isn't quite the right question, */ /* because even though the left triangulation might seem finished now, */ /* moving up on the right triangulation might reveal a new point of */ /* the left triangulation. And vice-versa.) */ leftfinished = counterclockwise(upperleft, lowerleft, lowerright) <= 0.0; rightfinished = counterclockwise(upperright, lowerleft, lowerright) <= 0.0; if (leftfinished && rightfinished) { /* Create the top new bounding triangle. */ maketriangle(&nextedge); setorg(nextedge, lowerleft); setdest(nextedge, lowerright); /* Apex is intentionally left NULL. */ /* Connect it to the bounding boxes of the two triangulations. */ bond(nextedge, baseedge); lnextself(nextedge); bond(nextedge, rightcand); lnextself(nextedge); bond(nextedge, leftcand); if (verbose > 2) { printf(" Creating top bounding "); printtriangle(&baseedge); } /* Special treatment for horizontal cuts. */ if (dwyer && (axis == 1)) { org(*farleft, farleftpt); apex(*farleft, farleftapex); dest(*farright, farrightpt); apex(*farright, farrightapex); sym(*farleft, checkedge); apex(checkedge, checkvertex); /* The pointers to the extremal points are restored to the leftmost */ /* and rightmost points (rather than topmost and bottommost). */ while (checkvertex[0] < farleftpt[0]) { lprev(checkedge, *farleft); farleftapex = farleftpt; farleftpt = checkvertex; sym(*farleft, checkedge); apex(checkedge, checkvertex); } while (farrightapex[0] > farrightpt[0]) { lprevself(*farright); symself(*farright); farrightpt = farrightapex; apex(*farright, farrightapex); } } return; } /* Consider eliminating edges from the left triangulation. */ if (!leftfinished) { /* What vertex would be exposed if an edge were deleted? */ lprev(leftcand, nextedge); symself(nextedge); apex(nextedge, nextapex); /* If nextapex is NULL, then no vertex would be exposed; the */ /* triangulation would have been eaten right through. */ if (nextapex != (point) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(lowerleft, lowerright, upperleft, nextapex) > 0.0; while (badedge) { /* Eliminate the edge with an edge flip. As a result, the */ /* left triangulation will have one more boundary triangle. */ lnextself(nextedge); sym(nextedge, topcasing); lnextself(nextedge); sym(nextedge, sidecasing); bond(nextedge, topcasing); bond(leftcand, sidecasing); lnextself(leftcand); sym(leftcand, outercasing); lprevself(nextedge); bond(nextedge, outercasing); /* Correct the vertices to reflect the edge flip. */ setorg(leftcand, lowerleft); setdest(leftcand, NULL); setapex(leftcand, nextapex); setorg(nextedge, NULL); setdest(nextedge, upperleft); setapex(nextedge, nextapex); /* Consider the newly exposed vertex. */ upperleft = nextapex; /* What vertex would be exposed if another edge were deleted? */ triedgecopy(sidecasing, nextedge); apex(nextedge, nextapex); if (nextapex != (point) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(lowerleft, lowerright, upperleft, nextapex) > 0.0; } else { /* Avoid eating right through the triangulation. */ badedge = 0; } } } } /* Consider eliminating edges from the right triangulation. */ if (!rightfinished) { /* What vertex would be exposed if an edge were deleted? */ lnext(rightcand, nextedge); symself(nextedge); apex(nextedge, nextapex); /* If nextapex is NULL, then no vertex would be exposed; the */ /* triangulation would have been eaten right through. */ if (nextapex != (point) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(lowerleft, lowerright, upperright, nextapex) > 0.0; while (badedge) { /* Eliminate the edge with an edge flip. As a result, the */ /* right triangulation will have one more boundary triangle. */ lprevself(nextedge); sym(nextedge, topcasing); lprevself(nextedge); sym(nextedge, sidecasing); bond(nextedge, topcasing); bond(rightcand, sidecasing); lprevself(rightcand); sym(rightcand, outercasing); lnextself(nextedge); bond(nextedge, outercasing); /* Correct the vertices to reflect the edge flip. */ setorg(rightcand, NULL); setdest(rightcand, lowerright); setapex(rightcand, nextapex); setorg(nextedge, upperright); setdest(nextedge, NULL); setapex(nextedge, nextapex); /* Consider the newly exposed vertex. */ upperright = nextapex; /* What vertex would be exposed if another edge were deleted? */ triedgecopy(sidecasing, nextedge); apex(nextedge, nextapex); if (nextapex != (point) NULL) { /* Check whether the edge is Delaunay. */ badedge = incircle(lowerleft, lowerright, upperright, nextapex) > 0.0; } else { /* Avoid eating right through the triangulation. */ badedge = 0; } } } } if (leftfinished || (!rightfinished && (incircle(upperleft, lowerleft, lowerright, upperright) > 0.0))) { /* Knit the triangulations, adding an edge from `lowerleft' */ /* to `upperright'. */ bond(baseedge, rightcand); lprev(rightcand, baseedge); setdest(baseedge, lowerleft); lowerright = upperright; sym(baseedge, rightcand); apex(rightcand, upperright); } else { /* Knit the triangulations, adding an edge from `upperleft' */ /* to `lowerright'. */ bond(baseedge, leftcand); lnext(leftcand, baseedge); setorg(baseedge, lowerright); lowerleft = upperleft; sym(baseedge, leftcand); apex(leftcand, upperleft); } if (verbose > 2) { printf(" Connecting "); printtriangle(&baseedge); } } } /*****************************************************************************/ /* */ /* divconqrecurse() Recursively form a Delaunay triangulation by the */ /* divide-and-conquer method. */ /* */ /* Recursively breaks down the problem into smaller pieces, which are */ /* knitted together by mergehulls(). The base cases (problems of two or */ /* three points) are handled specially here. */ /* */ /* On completion, `farleft' and `farright' are bounding triangles such that */ /* the origin of `farleft' is the leftmost vertex (breaking ties by */ /* choosing the highest leftmost vertex), and the destination of */ /* `farright' is the rightmost vertex (breaking ties by choosing the */ /* lowest rightmost vertex). */ /* */ /*****************************************************************************/ void divconqrecurse(sortarray, vertices, axis, farleft, farright) point *sortarray; int vertices; int axis; struct triedge *farleft; struct triedge *farright; { struct triedge midtri, tri1, tri2, tri3; struct triedge innerleft, innerright; REAL area; int divider; if (verbose > 2) { printf(" Triangulating %d points.\n", vertices); } if (vertices == 2) { /* The triangulation of two vertices is an edge. An edge is */ /* represented by two bounding triangles. */ maketriangle(farleft); setorg(*farleft, sortarray[0]); setdest(*farleft, sortarray[1]); /* The apex is intentionally left NULL. */ maketriangle(farright); setorg(*farright, sortarray[1]); setdest(*farright, sortarray[0]); /* The apex is intentionally left NULL. */ bond(*farleft, *farright); lprevself(*farleft); lnextself(*farright); bond(*farleft, *farright); lprevself(*farleft); lnextself(*farright); bond(*farleft, *farright); if (verbose > 2) { printf(" Creating "); printtriangle(farleft); printf(" Creating "); printtriangle(farright); } /* Ensure that the origin of `farleft' is sortarray[0]. */ lprev(*farright, *farleft); return; } else if (vertices == 3) { /* The triangulation of three vertices is either a triangle (with */ /* three bounding triangles) or two edges (with four bounding */ /* triangles). In either case, four triangles are created. */ maketriangle(&midtri); maketriangle(&tri1); maketriangle(&tri2); maketriangle(&tri3); area = counterclockwise(sortarray[0], sortarray[1], sortarray[2]); if (area == 0.0) { /* Three collinear points; the triangulation is two edges. */ setorg(midtri, sortarray[0]); setdest(midtri, sortarray[1]); setorg(tri1, sortarray[1]); setdest(tri1, sortarray[0]); setorg(tri2, sortarray[2]); setdest(tri2, sortarray[1]); setorg(tri3, sortarray[1]); setdest(tri3, sortarray[2]); /* All apices are intentionally left NULL. */ bond(midtri, tri1); bond(tri2, tri3); lnextself(midtri); lprevself(tri1); lnextself(tri2); lprevself(tri3); bond(midtri, tri3); bond(tri1, tri2); lnextself(midtri); lprevself(tri1); lnextself(tri2); lprevself(tri3); bond(midtri, tri1); bond(tri2, tri3); /* Ensure that the origin of `farleft' is sortarray[0]. */ triedgecopy(tri1, *farleft); /* Ensure that the destination of `farright' is sortarray[2]. */ triedgecopy(tri2, *farright); } else { /* The three points are not collinear; the triangulation is one */ /* triangle, namely `midtri'. */ setorg(midtri, sortarray[0]); setdest(tri1, sortarray[0]); setorg(tri3, sortarray[0]); /* Apices of tri1, tri2, and tri3 are left NULL. */ if (area > 0.0) { /* The vertices are in counterclockwise order. */ setdest(midtri, sortarray[1]); setorg(tri1, sortarray[1]); setdest(tri2, sortarray[1]); setapex(midtri, sortarray[2]); setorg(tri2, sortarray[2]); setdest(tri3, sortarray[2]); } else { /* The vertices are in clockwise order. */ setdest(midtri, sortarray[2]); setorg(tri1, sortarray[2]); setdest(tri2, sortarray[2]); setapex(midtri, sortarray[1]); setorg(tri2, sortarray[1]); setdest(tri3, sortarray[1]); } /* The topology does not depend on how the vertices are ordered. */ bond(midtri, tri1); lnextself(midtri); bond(midtri, tri2); lnextself(midtri); bond(midtri, tri3); lprevself(tri1); lnextself(tri2); bond(tri1, tri2); lprevself(tri1); lprevself(tri3); bond(tri1, tri3); lnextself(tri2); lprevself(tri3); bond(tri2, tri3); /* Ensure that the origin of `farleft' is sortarray[0]. */ triedgecopy(tri1, *farleft); /* Ensure that the destination of `farright' is sortarray[2]. */ if (area > 0.0) { triedgecopy(tri2, *farright); } else { lnext(*farleft, *farright); } } if (verbose > 2) { printf(" Creating "); printtriangle(&midtri); printf(" Creating "); printtriangle(&tri1); printf(" Creating "); printtriangle(&tri2); printf(" Creating "); printtriangle(&tri3); } return; } else { /* Split the vertices in half. */ divider = vertices >> 1; /* Recursively triangulate each half. */ divconqrecurse(sortarray, divider, 1 - axis, farleft, &innerleft); divconqrecurse(&sortarray[divider], vertices - divider, 1 - axis, &innerright, farright); if (verbose > 1) { printf(" Joining triangulations with %d and %d vertices.\n", divider, vertices - divider); } /* Merge the two triangulations into one. */ mergehulls(farleft, &innerleft, &innerright, farright, axis); } } long removeghosts(startghost) struct triedge *startghost; { struct triedge searchedge; struct triedge dissolveedge; struct triedge deadtri; point markorg; long hullsize; triangle ptr; /* Temporary variable used by sym(). */ if (verbose) { printf(" Removing ghost triangles.\n"); } /* Find an edge on the convex hull to start point location from. */ lprev(*startghost, searchedge); symself(searchedge); dummytri[0] = encode(searchedge); /* Remove the bounding box and count the convex hull edges. */ triedgecopy(*startghost, dissolveedge); hullsize = 0; do { hullsize++; lnext(dissolveedge, deadtri); lprevself(dissolveedge); symself(dissolveedge); /* If no PSLG is involved, set the boundary markers of all the points */ /* on the convex hull. If a PSLG is used, this step is done later. */ if (!poly) { /* Watch out for the case where all the input points are collinear. */ if (dissolveedge.tri != dummytri) { org(dissolveedge, markorg); if (pointmark(markorg) == 0) { setpointmark(markorg, 1); } } } /* Remove a bounding triangle from a convex hull triangle. */ dissolve(dissolveedge); /* Find the next bounding triangle. */ sym(deadtri, dissolveedge); /* Delete the bounding triangle. */ triangledealloc(deadtri.tri); } while (!triedgeequal(dissolveedge, *startghost)); return hullsize; } /*****************************************************************************/ /* */ /* divconqdelaunay() Form a Delaunay triangulation by the divide-and- */ /* conquer method. */ /* */ /* Sorts the points, calls a recursive procedure to triangulate them, and */ /* removes the bounding box, setting boundary markers as appropriate. */ /* */ /*****************************************************************************/ long divconqdelaunay() { point *sortarray; struct triedge hullleft, hullright; int divider; int i, j; /* Allocate an array of pointers to points for sorting. */ sortarray = (point *) malloc(inpoints * sizeof(point)); if (sortarray == (point *) NULL) { printf("Error: Out of memory.\n"); exit(1); } traversalinit(&points); for (i = 0; i < inpoints; i++) { sortarray[i] = pointtraverse(); } if (verbose) { printf(" Sorting points.\n"); } /* Sort the points. */ pointsort(sortarray, inpoints); /* Discard duplicate points, which can really mess up the algorithm. */ i = 0; for (j = 1; j < inpoints; j++) { if ((sortarray[i][0] == sortarray[j][0]) && (sortarray[i][1] == sortarray[j][1])) { if (!quiet) { printf( "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n", sortarray[j][0], sortarray[j][1]); } /* Commented out - would eliminate point from output .node file, but causes a failure if some segment has this point as an endpoint. setpointmark(sortarray[j], DEADPOINT); */ } else { i++; sortarray[i] = sortarray[j]; } } i++; if (dwyer) { /* Re-sort the array of points to accommodate alternating cuts. */ divider = i >> 1; if (i - divider >= 2) { if (divider >= 2) { alternateaxes(sortarray, divider, 1); } alternateaxes(&sortarray[divider], i - divider, 1); } } if (verbose) { printf(" Forming triangulation.\n"); } /* Form the Delaunay triangulation. */ divconqrecurse(sortarray, i, 0, &hullleft, &hullright); free(sortarray); return removeghosts(&hullleft); } /** **/ /** **/ /********* Divide-and-conquer Delaunay triangulation ends here *********/ /********* Incremental Delaunay triangulation begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* boundingbox() Form an "infinite" bounding triangle to insert points */ /* into. */ /* */ /* The points at "infinity" are assigned finite coordinates, which are used */ /* by the point location routines, but (mostly) ignored by the Delaunay */ /* edge flip routines. */ /* */ /*****************************************************************************/ #ifndef REDUCED void boundingbox() { struct triedge inftri; /* Handle for the triangular bounding box. */ REAL width; if (verbose) { printf(" Creating triangular bounding box.\n"); } /* Find the width (or height, whichever is larger) of the triangulation. */ width = xmax - xmin; if (ymax - ymin > width) { width = ymax - ymin; } if (width == 0.0) { width = 1.0; } /* Create the vertices of the bounding box. */ infpoint1 = (point) malloc(points.itembytes); infpoint2 = (point) malloc(points.itembytes); infpoint3 = (point) malloc(points.itembytes); if ((infpoint1 == (point) NULL) || (infpoint2 == (point) NULL) || (infpoint3 == (point) NULL)) { printf("Error: Out of memory.\n"); exit(1); } infpoint1[0] = xmin - 50.0 * width; infpoint1[1] = ymin - 40.0 * width; infpoint2[0] = xmax + 50.0 * width; infpoint2[1] = ymin - 40.0 * width; infpoint3[0] = 0.5 * (xmin + xmax); infpoint3[1] = ymax + 60.0 * width; /* Create the bounding box. */ maketriangle(&inftri); setorg(inftri, infpoint1); setdest(inftri, infpoint2); setapex(inftri, infpoint3); /* Link dummytri to the bounding box so we can always find an */ /* edge to begin searching (point location) from. */ dummytri[0] = (triangle) inftri.tri; if (verbose > 2) { printf(" Creating "); printtriangle(&inftri); } } #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* removebox() Remove the "infinite" bounding triangle, setting boundary */ /* markers as appropriate. */ /* */ /* The triangular bounding box has three boundary triangles (one for each */ /* side of the bounding box), and a bunch of triangles fanning out from */ /* the three bounding box vertices (one triangle for each edge of the */ /* convex hull of the inner mesh). This routine removes these triangles. */ /* */ /*****************************************************************************/ #ifndef REDUCED long removebox() { struct triedge deadtri; struct triedge searchedge; struct triedge checkedge; struct triedge nextedge, finaledge, dissolveedge; point markorg; long hullsize; triangle ptr; /* Temporary variable used by sym(). */ if (verbose) { printf(" Removing triangular bounding box.\n"); } /* Find a boundary triangle. */ nextedge.tri = dummytri; nextedge.orient = 0; symself(nextedge); /* Mark a place to stop. */ lprev(nextedge, finaledge); lnextself(nextedge); symself(nextedge); /* Find a triangle (on the boundary of the point set) that isn't */ /* a bounding box triangle. */ lprev(nextedge, searchedge); symself(searchedge); /* Check whether nextedge is another boundary triangle */ /* adjacent to the first one. */ lnext(nextedge, checkedge); symself(checkedge); if (checkedge.tri == dummytri) { /* Go on to the next triangle. There are only three boundary */ /* triangles, and this next triangle cannot be the third one, */ /* so it's safe to stop here. */ lprevself(searchedge); symself(searchedge); } /* Find a new boundary edge to search from, as the current search */ /* edge lies on a bounding box triangle and will be deleted. */ dummytri[0] = encode(searchedge); hullsize = -2l; while (!triedgeequal(nextedge, finaledge)) { hullsize++; lprev(nextedge, dissolveedge); symself(dissolveedge); /* If not using a PSLG, the vertices should be marked now. */ /* (If using a PSLG, markhull() will do the job.) */ if (!poly) { /* Be careful! One must check for the case where all the input */ /* points are collinear, and thus all the triangles are part of */ /* the bounding box. Otherwise, the setpointmark() call below */ /* will cause a bad pointer reference. */ if (dissolveedge.tri != dummytri) { org(dissolveedge, markorg); if (pointmark(markorg) == 0) { setpointmark(markorg, 1); } } } /* Disconnect the bounding box triangle from the mesh triangle. */ dissolve(dissolveedge); lnext(nextedge, deadtri); sym(deadtri, nextedge); /* Get rid of the bounding box triangle. */ triangledealloc(deadtri.tri); /* Do we need to turn the corner? */ if (nextedge.tri == dummytri) { /* Turn the corner. */ triedgecopy(dissolveedge, nextedge); } } triangledealloc(finaledge.tri); free(infpoint1); /* Deallocate the bounding box vertices. */ free(infpoint2); free(infpoint3); return hullsize; } #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* incrementaldelaunay() Form a Delaunay triangulation by incrementally */ /* adding vertices. */ /* */ /*****************************************************************************/ #ifndef REDUCED long incrementaldelaunay() { struct triedge starttri; point pointloop; int i; /* Create a triangular bounding box. */ boundingbox(); if (verbose) { printf(" Incrementally inserting points.\n"); } traversalinit(&points); pointloop = pointtraverse(); i = 1; while (pointloop != (point) NULL) { /* Find a boundary triangle to search from. */ starttri.tri = (triangle *) NULL; if (insertsite(pointloop, &starttri, (struct edge *) NULL, 0, 0) == DUPLICATEPOINT) { if (!quiet) { printf( "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n", pointloop[0], pointloop[1]); } /* Commented out - would eliminate point from output .node file. setpointmark(pointloop, DEADPOINT); */ } pointloop = pointtraverse(); i++; } /* Remove the bounding box. */ return removebox(); } #endif /* not REDUCED */ /** **/ /** **/ /********* Incremental Delaunay triangulation ends here *********/ /********* Sweepline Delaunay triangulation begins here *********/ /** **/ /** **/ #ifndef REDUCED void eventheapinsert(heap, heapsize, newevent) struct event **heap; int heapsize; struct event *newevent; { REAL eventx, eventy; int eventnum; int parent; int notdone; eventx = newevent->xkey; eventy = newevent->ykey; eventnum = heapsize; notdone = eventnum > 0; while (notdone) { parent = (eventnum - 1) >> 1; if ((heap[parent]->ykey < eventy) || ((heap[parent]->ykey == eventy) && (heap[parent]->xkey <= eventx))) { notdone = 0; } else { heap[eventnum] = heap[parent]; heap[eventnum]->heapposition = eventnum; eventnum = parent; notdone = eventnum > 0; } } heap[eventnum] = newevent; newevent->heapposition = eventnum; } #endif /* not REDUCED */ #ifndef REDUCED void eventheapify(heap, heapsize, eventnum) struct event **heap; int heapsize; int eventnum; { struct event *thisevent; REAL eventx, eventy; int leftchild, rightchild; int smallest; int notdone; thisevent = heap[eventnum]; eventx = thisevent->xkey; eventy = thisevent->ykey; leftchild = 2 * eventnum + 1; notdone = leftchild < heapsize; while (notdone) { if ((heap[leftchild]->ykey < eventy) || ((heap[leftchild]->ykey == eventy) && (heap[leftchild]->xkey < eventx))) { smallest = leftchild; } else { smallest = eventnum; } rightchild = leftchild + 1; if (rightchild < heapsize) { if ((heap[rightchild]->ykey < heap[smallest]->ykey) || ((heap[rightchild]->ykey == heap[smallest]->ykey) && (heap[rightchild]->xkey < heap[smallest]->xkey))) { smallest = rightchild; } } if (smallest == eventnum) { notdone = 0; } else { heap[eventnum] = heap[smallest]; heap[eventnum]->heapposition = eventnum; heap[smallest] = thisevent; thisevent->heapposition = smallest; eventnum = smallest; leftchild = 2 * eventnum + 1; notdone = leftchild < heapsize; } } } #endif /* not REDUCED */ #ifndef REDUCED void eventheapdelete(heap, heapsize, eventnum) struct event **heap; int heapsize; int eventnum; { struct event *moveevent; REAL eventx, eventy; int parent; int notdone; moveevent = heap[heapsize - 1]; if (eventnum > 0) { eventx = moveevent->xkey; eventy = moveevent->ykey; do { parent = (eventnum - 1) >> 1; if ((heap[parent]->ykey < eventy) || ((heap[parent]->ykey == eventy) && (heap[parent]->xkey <= eventx))) { notdone = 0; } else { heap[eventnum] = heap[parent]; heap[eventnum]->heapposition = eventnum; eventnum = parent; notdone = eventnum > 0; } } while (notdone); } heap[eventnum] = moveevent; moveevent->heapposition = eventnum; eventheapify(heap, heapsize - 1, eventnum); } #endif /* not REDUCED */ #ifndef REDUCED void createeventheap(eventheap, events, freeevents) struct event ***eventheap; struct event **events; struct event **freeevents; { point thispoint; int maxevents; int i; maxevents = (3 * inpoints) / 2; *eventheap = (struct event **) malloc(maxevents * sizeof(struct event *)); if (*eventheap == (struct event **) NULL) { printf("Error: Out of memory.\n"); exit(1); } *events = (struct event *) malloc(maxevents * sizeof(struct event)); if (*events == (struct event *) NULL) { printf("Error: Out of memory.\n"); exit(1); } traversalinit(&points); for (i = 0; i < inpoints; i++) { thispoint = pointtraverse(); (*events)[i].eventptr = (VOID *) thispoint; (*events)[i].xkey = thispoint[0]; (*events)[i].ykey = thispoint[1]; eventheapinsert(*eventheap, i, *events + i); } *freeevents = (struct event *) NULL; for (i = maxevents - 1; i >= inpoints; i--) { (*events)[i].eventptr = (VOID *) *freeevents; *freeevents = *events + i; } } #endif /* not REDUCED */ #ifndef REDUCED int rightofhyperbola(fronttri, newsite) struct triedge *fronttri; point newsite; { point leftpoint, rightpoint; REAL dxa, dya, dxb, dyb; hyperbolacount++; dest(*fronttri, leftpoint); apex(*fronttri, rightpoint); if ((leftpoint[1] < rightpoint[1]) || ((leftpoint[1] == rightpoint[1]) && (leftpoint[0] < rightpoint[0]))) { if (newsite[0] >= rightpoint[0]) { return 1; } } else { if (newsite[0] <= leftpoint[0]) { return 0; } } dxa = leftpoint[0] - newsite[0]; dya = leftpoint[1] - newsite[1]; dxb = rightpoint[0] - newsite[0]; dyb = rightpoint[1] - newsite[1]; return dya * (dxb * dxb + dyb * dyb) > dyb * (dxa * dxa + dya * dya); } #endif /* not REDUCED */ #ifndef REDUCED REAL circletop(pa, pb, pc, ccwabc) point pa; point pb; point pc; REAL ccwabc; { REAL xac, yac, xbc, ybc, xab, yab; REAL aclen2, bclen2, ablen2; circletopcount++; xac = pa[0] - pc[0]; yac = pa[1] - pc[1]; xbc = pb[0] - pc[0]; ybc = pb[1] - pc[1]; xab = pa[0] - pb[0]; yab = pa[1] - pb[1]; aclen2 = xac * xac + yac * yac; bclen2 = xbc * xbc + ybc * ybc; ablen2 = xab * xab + yab * yab; return pc[1] + (xac * bclen2 - xbc * aclen2 + sqrt(aclen2 * bclen2 * ablen2)) / (2.0 * ccwabc); } #endif /* not REDUCED */ #ifndef REDUCED void check4deadevent(checktri, freeevents, eventheap, heapsize) struct triedge *checktri; struct event **freeevents; struct event **eventheap; int *heapsize; { struct event *deadevent; point eventpoint; int eventnum; org(*checktri, eventpoint); if (eventpoint != (point) NULL) { deadevent = (struct event *) eventpoint; eventnum = deadevent->heapposition; deadevent->eventptr = (VOID *) *freeevents; *freeevents = deadevent; eventheapdelete(eventheap, *heapsize, eventnum); (*heapsize)--; setorg(*checktri, NULL); } } #endif /* not REDUCED */ #ifndef REDUCED struct splaynode *splay(splaytree, searchpoint, searchtri) struct splaynode *splaytree; point searchpoint; struct triedge *searchtri; { struct splaynode *child, *grandchild; struct splaynode *lefttree, *righttree; struct splaynode *leftright; point checkpoint; int rightofroot, rightofchild; if (splaytree == (struct splaynode *) NULL) { return (struct splaynode *) NULL; } dest(splaytree->keyedge, checkpoint); if (checkpoint == splaytree->keydest) { rightofroot = rightofhyperbola(&splaytree->keyedge, searchpoint); if (rightofroot) { triedgecopy(splaytree->keyedge, *searchtri); child = splaytree->rchild; } else { child = splaytree->lchild; } if (child == (struct splaynode *) NULL) { return splaytree; } dest(child->keyedge, checkpoint); if (checkpoint != child->keydest) { child = splay(child, searchpoint, searchtri); if (child == (struct splaynode *) NULL) { if (rightofroot) { splaytree->rchild = (struct splaynode *) NULL; } else { splaytree->lchild = (struct splaynode *) NULL; } return splaytree; } } rightofchild = rightofhyperbola(&child->keyedge, searchpoint); if (rightofchild) { triedgecopy(child->keyedge, *searchtri); grandchild = splay(child->rchild, searchpoint, searchtri); child->rchild = grandchild; } else { grandchild = splay(child->lchild, searchpoint, searchtri); child->lchild = grandchild; } if (grandchild == (struct splaynode *) NULL) { if (rightofroot) { splaytree->rchild = child->lchild; child->lchild = splaytree; } else { splaytree->lchild = child->rchild; child->rchild = splaytree; } return child; } if (rightofchild) { if (rightofroot) { splaytree->rchild = child->lchild; child->lchild = splaytree; } else { splaytree->lchild = grandchild->rchild; grandchild->rchild = splaytree; } child->rchild = grandchild->lchild; grandchild->lchild = child; } else { if (rightofroot) { splaytree->rchild = grandchild->lchild; grandchild->lchild = splaytree; } else { splaytree->lchild = child->rchild; child->rchild = splaytree; } child->lchild = grandchild->rchild; grandchild->rchild = child; } return grandchild; } else { lefttree = splay(splaytree->lchild, searchpoint, searchtri); righttree = splay(splaytree->rchild, searchpoint, searchtri); pooldealloc(&splaynodes, (VOID *) splaytree); if (lefttree == (struct splaynode *) NULL) { return righttree; } else if (righttree == (struct splaynode *) NULL) { return lefttree; } else if (lefttree->rchild == (struct splaynode *) NULL) { lefttree->rchild = righttree->lchild; righttree->lchild = lefttree; return righttree; } else if (righttree->lchild == (struct splaynode *) NULL) { righttree->lchild = lefttree->rchild; lefttree->rchild = righttree; return lefttree; } else { /* printf("Holy Toledo!!!\n"); */ leftright = lefttree->rchild; while (leftright->rchild != (struct splaynode *) NULL) { leftright = leftright->rchild; } leftright->rchild = righttree; return lefttree; } } } #endif /* not REDUCED */ #ifndef REDUCED struct splaynode *splayinsert(splayroot, newkey, searchpoint) struct splaynode *splayroot; struct triedge *newkey; point searchpoint; { struct splaynode *newsplaynode; newsplaynode = (struct splaynode *) poolalloc(&splaynodes); triedgecopy(*newkey, newsplaynode->keyedge); dest(*newkey, newsplaynode->keydest); if (splayroot == (struct splaynode *) NULL) { newsplaynode->lchild = (struct splaynode *) NULL; newsplaynode->rchild = (struct splaynode *) NULL; } else if (rightofhyperbola(&splayroot->keyedge, searchpoint)) { newsplaynode->lchild = splayroot; newsplaynode->rchild = splayroot->rchild; splayroot->rchild = (struct splaynode *) NULL; } else { newsplaynode->lchild = splayroot->lchild; newsplaynode->rchild = splayroot; splayroot->lchild = (struct splaynode *) NULL; } return newsplaynode; } #endif /* not REDUCED */ #ifndef REDUCED struct splaynode *circletopinsert(splayroot, newkey, pa, pb, pc, topy) struct splaynode *splayroot; struct triedge *newkey; point pa; point pb; point pc; REAL topy; { REAL ccwabc; REAL xac, yac, xbc, ybc; REAL aclen2, bclen2; REAL searchpoint[2]; struct triedge dummytri; ccwabc = counterclockwise(pa, pb, pc); xac = pa[0] - pc[0]; yac = pa[1] - pc[1]; xbc = pb[0] - pc[0]; ybc = pb[1] - pc[1]; aclen2 = xac * xac + yac * yac; bclen2 = xbc * xbc + ybc * ybc; searchpoint[0] = pc[0] - (yac * bclen2 - ybc * aclen2) / (2.0 * ccwabc); searchpoint[1] = topy; return splayinsert(splay(splayroot, (point) searchpoint, &dummytri), newkey, (point) searchpoint); } #endif /* not REDUCED */ #ifndef REDUCED struct splaynode *frontlocate(splayroot, bottommost, searchpoint, searchtri, farright) struct splaynode *splayroot; struct triedge *bottommost; point searchpoint; struct triedge *searchtri; int *farright; { int farrightflag; triangle ptr; /* Temporary variable used by onext(). */ triedgecopy(*bottommost, *searchtri); splayroot = splay(splayroot, searchpoint, searchtri); farrightflag = 0; while (!farrightflag && rightofhyperbola(searchtri, searchpoint)) { onextself(*searchtri); farrightflag = triedgeequal(*searchtri, *bottommost); } *farright = farrightflag; return splayroot; } #endif /* not REDUCED */ #ifndef REDUCED long sweeplinedelaunay() { struct event **eventheap; struct event *events; struct event *freeevents; struct event *nextevent; struct event *newevent; struct splaynode *splayroot; struct triedge bottommost; struct triedge searchtri; struct triedge fliptri; struct triedge lefttri, righttri, farlefttri, farrighttri; struct triedge inserttri; point firstpoint, secondpoint; point nextpoint, lastpoint; point connectpoint; point leftpoint, midpoint, rightpoint; REAL lefttest, righttest; int heapsize; int check4events, farrightflag; triangle ptr; /* Temporary variable used by sym(), onext(), and oprev(). */ poolinit(&splaynodes, sizeof(struct splaynode), SPLAYNODEPERBLOCK, POINTER, 0); splayroot = (struct splaynode *) NULL; if (verbose) { printf(" Placing points in event heap.\n"); } createeventheap(&eventheap, &events, &freeevents); heapsize = inpoints; if (verbose) { printf(" Forming triangulation.\n"); } maketriangle(&lefttri); maketriangle(&righttri); bond(lefttri, righttri); lnextself(lefttri); lprevself(righttri); bond(lefttri, righttri); lnextself(lefttri); lprevself(righttri); bond(lefttri, righttri); firstpoint = (point) eventheap[0]->eventptr; eventheap[0]->eventptr = (VOID *) freeevents; freeevents = eventheap[0]; eventheapdelete(eventheap, heapsize, 0); heapsize--; do { if (heapsize == 0) { printf("Error: Input points are all identical.\n"); exit(1); } secondpoint = (point) eventheap[0]->eventptr; eventheap[0]->eventptr = (VOID *) freeevents; freeevents = eventheap[0]; eventheapdelete(eventheap, heapsize, 0); heapsize--; if ((firstpoint[0] == secondpoint[0]) && (firstpoint[1] == secondpoint[1])) { printf( "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n", secondpoint[0], secondpoint[1]); /* Commented out - would eliminate point from output .node file. setpointmark(secondpoint, DEADPOINT); */ } } while ((firstpoint[0] == secondpoint[0]) && (firstpoint[1] == secondpoint[1])); setorg(lefttri, firstpoint); setdest(lefttri, secondpoint); setorg(righttri, secondpoint); setdest(righttri, firstpoint); lprev(lefttri, bottommost); lastpoint = secondpoint; while (heapsize > 0) { nextevent = eventheap[0]; eventheapdelete(eventheap, heapsize, 0); heapsize--; check4events = 1; if (nextevent->xkey < xmin) { decode(nextevent->eventptr, fliptri); oprev(fliptri, farlefttri); check4deadevent(&farlefttri, &freeevents, eventheap, &heapsize); onext(fliptri, farrighttri); check4deadevent(&farrighttri, &freeevents, eventheap, &heapsize); if (triedgeequal(farlefttri, bottommost)) { lprev(fliptri, bottommost); } flip(&fliptri); setapex(fliptri, NULL); lprev(fliptri, lefttri); lnext(fliptri, righttri); sym(lefttri, farlefttri); if (randomnation(SAMPLERATE) == 0) { symself(fliptri); dest(fliptri, leftpoint); apex(fliptri, midpoint); org(fliptri, rightpoint); splayroot = circletopinsert(splayroot, &lefttri, leftpoint, midpoint, rightpoint, nextevent->ykey); } } else { nextpoint = (point) nextevent->eventptr; if ((nextpoint[0] == lastpoint[0]) && (nextpoint[1] == lastpoint[1])) { printf( "Warning: A duplicate point at (%.12g, %.12g) appeared and was ignored.\n", nextpoint[0], nextpoint[1]); /* Commented out - would eliminate point from output .node file. setpointmark(nextpoint, DEADPOINT); */ check4events = 0; } else { lastpoint = nextpoint; splayroot = frontlocate(splayroot, &bottommost, nextpoint, &searchtri, &farrightflag); /* triedgecopy(bottommost, searchtri); farrightflag = 0; while (!farrightflag && rightofhyperbola(&searchtri, nextpoint)) { onextself(searchtri); farrightflag = triedgeequal(searchtri, bottommost); } */ check4deadevent(&searchtri, &freeevents, eventheap, &heapsize); triedgecopy(searchtri, farrighttri); sym(searchtri, farlefttri); maketriangle(&lefttri); maketriangle(&righttri); dest(farrighttri, connectpoint); setorg(lefttri, connectpoint); setdest(lefttri, nextpoint); setorg(righttri, nextpoint); setdest(righttri, connectpoint); bond(lefttri, righttri); lnextself(lefttri); lprevself(righttri); bond(lefttri, righttri); lnextself(lefttri); lprevself(righttri); bond(lefttri, farlefttri); bond(righttri, farrighttri); if (!farrightflag && triedgeequal(farrighttri, bottommost)) { triedgecopy(lefttri, bottommost); } if (randomnation(SAMPLERATE) == 0) { splayroot = splayinsert(splayroot, &lefttri, nextpoint); } else if (randomnation(SAMPLERATE) == 0) { lnext(righttri, inserttri); splayroot = splayinsert(splayroot, &inserttri, nextpoint); } } } nextevent->eventptr = (VOID *) freeevents; freeevents = nextevent; if (check4events) { apex(farlefttri, leftpoint); dest(lefttri, midpoint); apex(lefttri, rightpoint); lefttest = counterclockwise(leftpoint, midpoint, rightpoint); if (lefttest > 0.0) { newevent = freeevents; freeevents = (struct event *) freeevents->eventptr; newevent->xkey = xminextreme; newevent->ykey = circletop(leftpoint, midpoint, rightpoint, lefttest); newevent->eventptr = (VOID *) encode(lefttri); eventheapinsert(eventheap, heapsize, newevent); heapsize++; setorg(lefttri, newevent); } apex(righttri, leftpoint); org(righttri, midpoint); apex(farrighttri, rightpoint); righttest = counterclockwise(leftpoint, midpoint, rightpoint); if (righttest > 0.0) { newevent = freeevents; freeevents = (struct event *) freeevents->eventptr; newevent->xkey = xminextreme; newevent->ykey = circletop(leftpoint, midpoint, rightpoint, righttest); newevent->eventptr = (VOID *) encode(farrighttri); eventheapinsert(eventheap, heapsize, newevent); heapsize++; setorg(farrighttri, newevent); } } } pooldeinit(&splaynodes); lprevself(bottommost); return removeghosts(&bottommost); } #endif /* not REDUCED */ /** **/ /** **/ /********* Sweepline Delaunay triangulation ends here *********/ /********* General mesh construction routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* delaunay() Form a Delaunay triangulation. */ /* */ /*****************************************************************************/ long delaunay() { eextras = 0; initializetrisegpools(); #ifdef REDUCED if (!quiet) { printf( "Constructing Delaunay triangulation by divide-and-conquer method.\n"); } return divconqdelaunay(); #else /* not REDUCED */ if (!quiet) { printf("Constructing Delaunay triangulation "); if (incremental) { printf("by incremental method.\n"); } else if (sweepline) { printf("by sweepline method.\n"); } else { printf("by divide-and-conquer method.\n"); } } if (incremental) { return incrementaldelaunay(); } else if (sweepline) { return sweeplinedelaunay(); } else { return divconqdelaunay(); } #endif /* not REDUCED */ } /*****************************************************************************/ /* */ /* reconstruct() Reconstruct a triangulation from its .ele (and possibly */ /* .poly) file. Used when the -r switch is used. */ /* */ /* Reads an .ele file and reconstructs the original mesh. If the -p switch */ /* is used, this procedure will also read a .poly file and reconstruct the */ /* shell edges of the original mesh. If the -a switch is used, this */ /* procedure will also read an .area file and set a maximum area constraint */ /* on each triangle. */ /* */ /* Points that are not corners of triangles, such as nodes on edges of */ /* subparametric elements, are discarded. */ /* */ /* This routine finds the adjacencies between triangles (and shell edges) */ /* by forming one stack of triangles for each vertex. Each triangle is on */ /* three different stacks simultaneously. Each triangle's shell edge */ /* pointers are used to link the items in each stack. This memory-saving */ /* feature makes the code harder to read. The most important thing to keep */ /* in mind is that each triangle is removed from a stack precisely when */ /* the corresponding pointer is adjusted to refer to a shell edge rather */ /* than the next triangle of the stack. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY #ifdef TRILIBRARY int reconstruct(trianglelist, triangleattriblist, trianglearealist, elements, corners, attribs, segmentlist, segmentmarkerlist, numberofsegments) int *trianglelist; REAL *triangleattriblist; REAL *trianglearealist; int elements; int corners; int attribs; int *segmentlist; int *segmentmarkerlist; int numberofsegments; #else /* not TRILIBRARY */ long reconstruct(elefilename, areafilename, polyfilename, polyfile) char *elefilename; char *areafilename; char *polyfilename; FILE *polyfile; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int pointindex; int attribindex; #else /* not TRILIBRARY */ FILE *elefile; FILE *areafile; char inputline[INPUTLINESIZE]; char *stringptr; int areaelements; #endif /* not TRILIBRARY */ struct triedge triangleloop; struct triedge triangleleft; struct triedge checktri; struct triedge checkleft; struct triedge checkneighbor; struct edge shelleloop; triangle *vertexarray; triangle *prevlink; triangle nexttri; point tdest, tapex; point checkdest, checkapex; point shorg; point killpoint; REAL area; int corner[3]; int end[2]; int killpointindex; int incorners; int segmentmarkers; int boundmarker; int aroundpoint; long hullsize; int notfound; int elementnumber, segmentnumber; int i, j; triangle ptr; /* Temporary variable used by sym(). */ #ifdef TRILIBRARY inelements = elements; incorners = corners; if (incorners < 3) { printf("Error: Triangles must have at least 3 points.\n"); exit(1); } eextras = attribs; #else /* not TRILIBRARY */ /* Read the triangles from an .ele file. */ if (!quiet) { printf("Opening %s.\n", elefilename); } elefile = fopen(elefilename, "r"); if (elefile == (FILE *) NULL) { printf(" Error: Cannot access file %s.\n", elefilename); exit(1); } /* Read number of triangles, number of points per triangle, and */ /* number of triangle attributes from .ele file. */ stringptr = readline(inputline, elefile, elefilename); inelements = (int) strtol (stringptr, &stringptr, 0); stringptr = findfield(stringptr); if (*stringptr == '\0') { incorners = 3; } else { incorners = (int) strtol (stringptr, &stringptr, 0); if (incorners < 3) { printf("Error: Triangles in %s must have at least 3 points.\n", elefilename); exit(1); } } stringptr = findfield(stringptr); if (*stringptr == '\0') { eextras = 0; } else { eextras = (int) strtol (stringptr, &stringptr, 0); } #endif /* not TRILIBRARY */ initializetrisegpools(); /* Create the triangles. */ for (elementnumber = 1; elementnumber <= inelements; elementnumber++) { maketriangle(&triangleloop); /* Mark the triangle as living. */ triangleloop.tri[3] = (triangle) triangleloop.tri; } if (poly) { #ifdef TRILIBRARY insegments = numberofsegments; segmentmarkers = segmentmarkerlist != (int *) NULL; #else /* not TRILIBRARY */ /* Read number of segments and number of segment */ /* boundary markers from .poly file. */ stringptr = readline(inputline, polyfile, inpolyfilename); insegments = (int) strtol (stringptr, &stringptr, 0); stringptr = findfield(stringptr); if (*stringptr == '\0') { segmentmarkers = 0; } else { segmentmarkers = (int) strtol (stringptr, &stringptr, 0); } #endif /* not TRILIBRARY */ /* Create the shell edges. */ for (segmentnumber = 1; segmentnumber <= insegments; segmentnumber++) { makeshelle(&shelleloop); /* Mark the shell edge as living. */ shelleloop.sh[2] = (shelle) shelleloop.sh; } } #ifdef TRILIBRARY pointindex = 0; attribindex = 0; #else /* not TRILIBRARY */ if (vararea) { /* Open an .area file, check for consistency with the .ele file. */ if (!quiet) { printf("Opening %s.\n", areafilename); } areafile = fopen(areafilename, "r"); if (areafile == (FILE *) NULL) { printf(" Error: Cannot access file %s.\n", areafilename); exit(1); } stringptr = readline(inputline, areafile, areafilename); areaelements = (int) strtol (stringptr, &stringptr, 0); if (areaelements != inelements) { printf("Error: %s and %s disagree on number of triangles.\n", elefilename, areafilename); exit(1); } } #endif /* not TRILIBRARY */ if (!quiet) { printf("Reconstructing mesh.\n"); } /* Allocate a temporary array that maps each point to some adjacent */ /* triangle. I took care to allocate all the permanent memory for */ /* triangles and shell edges first. */ vertexarray = (triangle *) malloc(points.items * sizeof(triangle)); if (vertexarray == (triangle *) NULL) { printf("Error: Out of memory.\n"); exit(1); } /* Each point is initially unrepresented. */ for (i = 0; i < points.items; i++) { vertexarray[i] = (triangle) dummytri; } if (verbose) { printf(" Assembling triangles.\n"); } /* Read the triangles from the .ele file, and link */ /* together those that share an edge. */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); elementnumber = firstnumber; while (triangleloop.tri != (triangle *) NULL) { #ifdef TRILIBRARY /* Copy the triangle's three corners. */ for (j = 0; j < 3; j++) { corner[j] = trianglelist[pointindex++]; if ((corner[j] < firstnumber) || (corner[j] >= firstnumber + inpoints)) { printf("Error: Triangle %d has an invalid vertex index.\n", elementnumber); exit(1); } } #else /* not TRILIBRARY */ /* Read triangle number and the triangle's three corners. */ stringptr = readline(inputline, elefile, elefilename); for (j = 0; j < 3; j++) { stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Triangle %d is missing point %d in %s.\n", elementnumber, j + 1, elefilename); exit(1); } else { corner[j] = (int) strtol (stringptr, &stringptr, 0); if ((corner[j] < firstnumber) || (corner[j] >= firstnumber + inpoints)) { printf("Error: Triangle %d has an invalid vertex index.\n", elementnumber); exit(1); } } } #endif /* not TRILIBRARY */ /* Find out about (and throw away) extra nodes. */ for (j = 3; j < incorners; j++) { #ifdef TRILIBRARY killpointindex = trianglelist[pointindex++]; #else /* not TRILIBRARY */ stringptr = findfield(stringptr); if (*stringptr != '\0') { killpointindex = (int) strtol (stringptr, &stringptr, 0); #endif /* not TRILIBRARY */ if ((killpointindex >= firstnumber) && (killpointindex < firstnumber + inpoints)) { /* Delete the non-corner point if it's not already deleted. */ killpoint = getpoint(killpointindex); if (pointmark(killpoint) != DEADPOINT) { pointdealloc(killpoint); } } #ifndef TRILIBRARY } #endif /* not TRILIBRARY */ } /* Read the triangle's attributes. */ for (j = 0; j < eextras; j++) { #ifdef TRILIBRARY setelemattribute(triangleloop, j, triangleattriblist[attribindex++]); #else /* not TRILIBRARY */ stringptr = findfield(stringptr); if (*stringptr == '\0') { setelemattribute(triangleloop, j, 0); } else { setelemattribute(triangleloop, j, (REAL) strtod (stringptr, &stringptr)); } #endif /* not TRILIBRARY */ } if (vararea) { #ifdef TRILIBRARY area = trianglearealist[elementnumber - firstnumber]; #else /* not TRILIBRARY */ /* Read an area constraint from the .area file. */ stringptr = readline(inputline, areafile, areafilename); stringptr = findfield(stringptr); if (*stringptr == '\0') { area = -1.0; /* No constraint on this triangle. */ } else { area = (REAL) strtod(stringptr, &stringptr); } #endif /* not TRILIBRARY */ setareabound(triangleloop, area); } /* Set the triangle's vertices. */ triangleloop.orient = 0; setorg(triangleloop, getpoint(corner[0])); setdest(triangleloop, getpoint(corner[1])); setapex(triangleloop, getpoint(corner[2])); /* Try linking the triangle to others that share these vertices. */ for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { /* Take the number for the origin of triangleloop. */ aroundpoint = corner[triangleloop.orient]; /* Look for other triangles having this vertex. */ nexttri = vertexarray[aroundpoint - firstnumber]; /* Link the current triangle to the next one in the stack. */ triangleloop.tri[6 + triangleloop.orient] = nexttri; /* Push the current triangle onto the stack. */ vertexarray[aroundpoint - firstnumber] = encode(triangleloop); decode(nexttri, checktri); if (checktri.tri != dummytri) { dest(triangleloop, tdest); apex(triangleloop, tapex); /* Look for other triangles that share an edge. */ do { dest(checktri, checkdest); apex(checktri, checkapex); if (tapex == checkdest) { /* The two triangles share an edge; bond them together. */ lprev(triangleloop, triangleleft); bond(triangleleft, checktri); } if (tdest == checkapex) { /* The two triangles share an edge; bond them together. */ lprev(checktri, checkleft); bond(triangleloop, checkleft); } /* Find the next triangle in the stack. */ nexttri = checktri.tri[6 + checktri.orient]; decode(nexttri, checktri); } while (checktri.tri != dummytri); } } triangleloop.tri = triangletraverse(); elementnumber++; } #ifdef TRILIBRARY pointindex = 0; #else /* not TRILIBRARY */ fclose(elefile); if (vararea) { fclose(areafile); } #endif /* not TRILIBRARY */ hullsize = 0; /* Prepare to count the boundary edges. */ if (poly) { if (verbose) { printf(" Marking segments in triangulation.\n"); } /* Read the segments from the .poly file, and link them */ /* to their neighboring triangles. */ boundmarker = 0; traversalinit(&shelles); shelleloop.sh = shelletraverse(); segmentnumber = firstnumber; while (shelleloop.sh != (shelle *) NULL) { #ifdef TRILIBRARY end[0] = segmentlist[pointindex++]; end[1] = segmentlist[pointindex++]; if (segmentmarkers) { boundmarker = segmentmarkerlist[segmentnumber - firstnumber]; } #else /* not TRILIBRARY */ /* Read the endpoints of each segment, and possibly a boundary marker. */ stringptr = readline(inputline, polyfile, inpolyfilename); /* Skip the first (segment number) field. */ stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Segment %d has no endpoints in %s.\n", segmentnumber, polyfilename); exit(1); } else { end[0] = (int) strtol (stringptr, &stringptr, 0); } stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Segment %d is missing its second endpoint in %s.\n", segmentnumber, polyfilename); exit(1); } else { end[1] = (int) strtol (stringptr, &stringptr, 0); } if (segmentmarkers) { stringptr = findfield(stringptr); if (*stringptr == '\0') { boundmarker = 0; } else { boundmarker = (int) strtol (stringptr, &stringptr, 0); } } #endif /* not TRILIBRARY */ for (j = 0; j < 2; j++) { if ((end[j] < firstnumber) || (end[j] >= firstnumber + inpoints)) { printf("Error: Segment %d has an invalid vertex index.\n", segmentnumber); exit(1); } } /* set the shell edge's vertices. */ shelleloop.shorient = 0; setsorg(shelleloop, getpoint(end[0])); setsdest(shelleloop, getpoint(end[1])); setmark(shelleloop, boundmarker); /* Try linking the shell edge to triangles that share these vertices. */ for (shelleloop.shorient = 0; shelleloop.shorient < 2; shelleloop.shorient++) { /* Take the number for the destination of shelleloop. */ aroundpoint = end[1 - shelleloop.shorient]; /* Look for triangles having this vertex. */ prevlink = &vertexarray[aroundpoint - firstnumber]; nexttri = vertexarray[aroundpoint - firstnumber]; decode(nexttri, checktri); sorg(shelleloop, shorg); notfound = 1; /* Look for triangles having this edge. Note that I'm only */ /* comparing each triangle's destination with the shell edge; */ /* each triangle's apex is handled through a different vertex. */ /* Because each triangle appears on three vertices' lists, each */ /* occurrence of a triangle on a list can (and does) represent */ /* an edge. In this way, most edges are represented twice, and */ /* every triangle-segment bond is represented once. */ while (notfound && (checktri.tri != dummytri)) { dest(checktri, checkdest); if (shorg == checkdest) { /* We have a match. Remove this triangle from the list. */ *prevlink = checktri.tri[6 + checktri.orient]; /* Bond the shell edge to the triangle. */ tsbond(checktri, shelleloop); /* Check if this is a boundary edge. */ sym(checktri, checkneighbor); if (checkneighbor.tri == dummytri) { /* The next line doesn't insert a shell edge (because there's */ /* already one there), but it sets the boundary markers of */ /* the existing shell edge and its vertices. */ insertshelle(&checktri, 1); hullsize++; } notfound = 0; } /* Find the next triangle in the stack. */ prevlink = &checktri.tri[6 + checktri.orient]; nexttri = checktri.tri[6 + checktri.orient]; decode(nexttri, checktri); } } shelleloop.sh = shelletraverse(); segmentnumber++; } } /* Mark the remaining edges as not being attached to any shell edge. */ /* Also, count the (yet uncounted) boundary edges. */ for (i = 0; i < points.items; i++) { /* Search the stack of triangles adjacent to a point. */ nexttri = vertexarray[i]; decode(nexttri, checktri); while (checktri.tri != dummytri) { /* Find the next triangle in the stack before this */ /* information gets overwritten. */ nexttri = checktri.tri[6 + checktri.orient]; /* No adjacent shell edge. (This overwrites the stack info.) */ tsdissolve(checktri); sym(checktri, checkneighbor); if (checkneighbor.tri == dummytri) { insertshelle(&checktri, 1); hullsize++; } decode(nexttri, checktri); } } free(vertexarray); return hullsize; } #endif /* not CDT_ONLY */ /** **/ /** **/ /********* General mesh construction routines end here *********/ /********* Segment (shell edge) insertion begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* finddirection() Find the first triangle on the path from one point */ /* to another. */ /* */ /* Finds the triangle that intersects a line segment drawn from the */ /* origin of `searchtri' to the point `endpoint', and returns the result */ /* in `searchtri'. The origin of `searchtri' does not change, even though */ /* the triangle returned may differ from the one passed in. This routine */ /* is used to find the direction to move in to get from one point to */ /* another. */ /* */ /* The return value notes whether the destination or apex of the found */ /* triangle is collinear with the two points in question. */ /* */ /*****************************************************************************/ enum finddirectionresult finddirection(searchtri, endpoint) struct triedge *searchtri; point endpoint; { struct triedge checktri; point startpoint; point leftpoint, rightpoint; REAL leftccw, rightccw; int leftflag, rightflag; triangle ptr; /* Temporary variable used by onext() and oprev(). */ org(*searchtri, startpoint); dest(*searchtri, rightpoint); apex(*searchtri, leftpoint); /* Is `endpoint' to the left? */ leftccw = counterclockwise(endpoint, startpoint, leftpoint); leftflag = leftccw > 0.0; /* Is `endpoint' to the right? */ rightccw = counterclockwise(startpoint, endpoint, rightpoint); rightflag = rightccw > 0.0; if (leftflag && rightflag) { /* `searchtri' faces directly away from `endpoint'. We could go */ /* left or right. Ask whether it's a triangle or a boundary */ /* on the left. */ onext(*searchtri, checktri); if (checktri.tri == dummytri) { leftflag = 0; } else { rightflag = 0; } } while (leftflag) { /* Turn left until satisfied. */ onextself(*searchtri); if (searchtri->tri == dummytri) { printf("Internal error in finddirection(): Unable to find a\n"); printf(" triangle leading from (%.12g, %.12g) to", startpoint[0], startpoint[1]); printf(" (%.12g, %.12g).\n", endpoint[0], endpoint[1]); internalerror(); } apex(*searchtri, leftpoint); rightccw = leftccw; leftccw = counterclockwise(endpoint, startpoint, leftpoint); leftflag = leftccw > 0.0; } while (rightflag) { /* Turn right until satisfied. */ oprevself(*searchtri); if (searchtri->tri == dummytri) { printf("Internal error in finddirection(): Unable to find a\n"); printf(" triangle leading from (%.12g, %.12g) to", startpoint[0], startpoint[1]); printf(" (%.12g, %.12g).\n", endpoint[0], endpoint[1]); internalerror(); } dest(*searchtri, rightpoint); leftccw = rightccw; rightccw = counterclockwise(startpoint, endpoint, rightpoint); rightflag = rightccw > 0.0; } if (leftccw == 0.0) { return LEFTCOLLINEAR; } else if (rightccw == 0.0) { return RIGHTCOLLINEAR; } else { return WITHIN; } } /*****************************************************************************/ /* */ /* segmentintersection() Find the intersection of an existing segment */ /* and a segment that is being inserted. Insert */ /* a point at the intersection, splitting an */ /* existing shell edge. */ /* */ /* The segment being inserted connects the apex of splittri to endpoint2. */ /* splitshelle is the shell edge being split, and MUST be opposite */ /* splittri. Hence, the edge being split connects the origin and */ /* destination of splittri. */ /* */ /* On completion, splittri is a handle having the newly inserted */ /* intersection point as its origin, and endpoint1 as its destination. */ /* */ /*****************************************************************************/ void segmentintersection(splittri, splitshelle, endpoint2) struct triedge *splittri; struct edge *splitshelle; point endpoint2; { point endpoint1; point torg, tdest; point leftpoint, rightpoint; point newpoint; enum insertsiteresult success; enum finddirectionresult collinear; REAL ex, ey; REAL tx, ty; REAL etx, ety; REAL split, denom; int i; triangle ptr; /* Temporary variable used by onext(). */ /* Find the other three segment endpoints. */ apex(*splittri, endpoint1); org(*splittri, torg); dest(*splittri, tdest); /* Segment intersection formulae; see the Antonio reference. */ tx = tdest[0] - torg[0]; ty = tdest[1] - torg[1]; ex = endpoint2[0] - endpoint1[0]; ey = endpoint2[1] - endpoint1[1]; etx = torg[0] - endpoint2[0]; ety = torg[1] - endpoint2[1]; denom = ty * ex - tx * ey; if (denom == 0.0) { printf("Internal error in segmentintersection():"); printf(" Attempt to find intersection of parallel segments.\n"); internalerror(); } split = (ey * etx - ex * ety) / denom; /* Create the new point. */ newpoint = (point) poolalloc(&points); /* Interpolate its coordinate and attributes. */ for (i = 0; i < 2 + nextras; i++) { newpoint[i] = torg[i] + split * (tdest[i] - torg[i]); } setpointmark(newpoint, mark(*splitshelle)); if (verbose > 1) { printf( " Splitting edge (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n", torg[0], torg[1], tdest[0], tdest[1], newpoint[0], newpoint[1]); } /* Insert the intersection point. This should always succeed. */ success = insertsite(newpoint, splittri, splitshelle, 0, 0); if (success != SUCCESSFULPOINT) { printf("Internal error in segmentintersection():\n"); printf(" Failure to split a segment.\n"); internalerror(); } if (steinerleft > 0) { steinerleft--; } /* Inserting the point may have caused edge flips. We wish to rediscover */ /* the edge connecting endpoint1 to the new intersection point. */ collinear = finddirection(splittri, endpoint1); dest(*splittri, rightpoint); apex(*splittri, leftpoint); if ((leftpoint[0] == endpoint1[0]) && (leftpoint[1] == endpoint1[1])) { onextself(*splittri); } else if ((rightpoint[0] != endpoint1[0]) || (rightpoint[1] != endpoint1[1])) { printf("Internal error in segmentintersection():\n"); printf(" Topological inconsistency after splitting a segment.\n"); internalerror(); } /* `splittri' should have destination endpoint1. */ } /*****************************************************************************/ /* */ /* scoutsegment() Scout the first triangle on the path from one endpoint */ /* to another, and check for completion (reaching the */ /* second endpoint), a collinear point, and the */ /* intersection of two segments. */ /* */ /* Returns one if the entire segment is successfully inserted, and zero if */ /* the job must be finished by conformingedge() or constrainededge(). */ /* */ /* If the first triangle on the path has the second endpoint as its */ /* destination or apex, a shell edge is inserted and the job is done. */ /* */ /* If the first triangle on the path has a destination or apex that lies on */ /* the segment, a shell edge is inserted connecting the first endpoint to */ /* the collinear point, and the search is continued from the collinear */ /* point. */ /* */ /* If the first triangle on the path has a shell edge opposite its origin, */ /* then there is a segment that intersects the segment being inserted. */ /* Their intersection point is inserted, splitting the shell edge. */ /* */ /* Otherwise, return zero. */ /* */ /*****************************************************************************/ int scoutsegment(searchtri, endpoint2, newmark) struct triedge *searchtri; point endpoint2; int newmark; { struct triedge crosstri; struct edge crossedge; point leftpoint, rightpoint; point endpoint1; enum finddirectionresult collinear; shelle sptr; /* Temporary variable used by tspivot(). */ collinear = finddirection(searchtri, endpoint2); dest(*searchtri, rightpoint); apex(*searchtri, leftpoint); if (((leftpoint[0] == endpoint2[0]) && (leftpoint[1] == endpoint2[1])) || ((rightpoint[0] == endpoint2[0]) && (rightpoint[1] == endpoint2[1]))) { /* The segment is already an edge in the mesh. */ if ((leftpoint[0] == endpoint2[0]) && (leftpoint[1] == endpoint2[1])) { lprevself(*searchtri); } /* Insert a shell edge, if there isn't already one there. */ insertshelle(searchtri, newmark); return 1; } else if (collinear == LEFTCOLLINEAR) { /* We've collided with a point between the segment's endpoints. */ /* Make the collinear point be the triangle's origin. */ lprevself(*searchtri); insertshelle(searchtri, newmark); /* Insert the remainder of the segment. */ return scoutsegment(searchtri, endpoint2, newmark); } else if (collinear == RIGHTCOLLINEAR) { /* We've collided with a point between the segment's endpoints. */ insertshelle(searchtri, newmark); /* Make the collinear point be the triangle's origin. */ lnextself(*searchtri); /* Insert the remainder of the segment. */ return scoutsegment(searchtri, endpoint2, newmark); } else { lnext(*searchtri, crosstri); tspivot(crosstri, crossedge); /* Check for a crossing segment. */ if (crossedge.sh == dummysh) { return 0; } else { org(*searchtri, endpoint1); /* Insert a point at the intersection. */ segmentintersection(&crosstri, &crossedge, endpoint2); triedgecopy(crosstri, *searchtri); insertshelle(searchtri, newmark); /* Insert the remainder of the segment. */ return scoutsegment(searchtri, endpoint2, newmark); } } } /*****************************************************************************/ /* */ /* conformingedge() Force a segment into a conforming Delaunay */ /* triangulation by inserting a point at its midpoint, */ /* and recursively forcing in the two half-segments if */ /* necessary. */ /* */ /* Generates a sequence of edges connecting `endpoint1' to `endpoint2'. */ /* `newmark' is the boundary marker of the segment, assigned to each new */ /* splitting point and shell edge. */ /* */ /* Note that conformingedge() does not always maintain the conforming */ /* Delaunay property. Once inserted, segments are locked into place; */ /* points inserted later (to force other segments in) may render these */ /* fixed segments non-Delaunay. The conforming Delaunay property will be */ /* restored by enforcequality() by splitting encroached segments. */ /* */ /*****************************************************************************/ #ifndef REDUCED #ifndef CDT_ONLY void conformingedge(endpoint1, endpoint2, newmark) point endpoint1; point endpoint2; int newmark; { struct triedge searchtri1, searchtri2; struct edge brokenshelle; point newpoint; point midpoint1, midpoint2; enum insertsiteresult success; int result1, result2; int i; shelle sptr; /* Temporary variable used by tspivot(). */ if (verbose > 2) { printf("Forcing segment into triangulation by recursive splitting:\n"); printf(" (%.12g, %.12g) (%.12g, %.12g)\n", endpoint1[0], endpoint1[1], endpoint2[0], endpoint2[1]); } /* Create a new point to insert in the middle of the segment. */ newpoint = (point) poolalloc(&points); /* Interpolate coordinates and attributes. */ for (i = 0; i < 2 + nextras; i++) { newpoint[i] = 0.5 * (endpoint1[i] + endpoint2[i]); } setpointmark(newpoint, newmark); /* Find a boundary triangle to search from. */ searchtri1.tri = (triangle *) NULL; /* Attempt to insert the new point. */ success = insertsite(newpoint, &searchtri1, (struct edge *) NULL, 0, 0); if (success == DUPLICATEPOINT) { if (verbose > 2) { printf(" Segment intersects existing point (%.12g, %.12g).\n", newpoint[0], newpoint[1]); } /* Use the point that's already there. */ pointdealloc(newpoint); org(searchtri1, newpoint); } else { if (success == VIOLATINGPOINT) { if (verbose > 2) { printf(" Two segments intersect at (%.12g, %.12g).\n", newpoint[0], newpoint[1]); } /* By fluke, we've landed right on another segment. Split it. */ tspivot(searchtri1, brokenshelle); success = insertsite(newpoint, &searchtri1, &brokenshelle, 0, 0); if (success != SUCCESSFULPOINT) { printf("Internal error in conformingedge():\n"); printf(" Failure to split a segment.\n"); internalerror(); } } /* The point has been inserted successfully. */ if (steinerleft > 0) { steinerleft--; } } triedgecopy(searchtri1, searchtri2); result1 = scoutsegment(&searchtri1, endpoint1, newmark); result2 = scoutsegment(&searchtri2, endpoint2, newmark); if (!result1) { /* The origin of searchtri1 may have changed if a collision with an */ /* intervening vertex on the segment occurred. */ org(searchtri1, midpoint1); conformingedge(midpoint1, endpoint1, newmark); } if (!result2) { /* The origin of searchtri2 may have changed if a collision with an */ /* intervening vertex on the segment occurred. */ org(searchtri2, midpoint2); conformingedge(midpoint2, endpoint2, newmark); } } #endif /* not CDT_ONLY */ #endif /* not REDUCED */ /*****************************************************************************/ /* */ /* delaunayfixup() Enforce the Delaunay condition at an edge, fanning out */ /* recursively from an existing point. Pay special */ /* attention to stacking inverted triangles. */ /* */ /* This is a support routine for inserting segments into a constrained */ /* Delaunay triangulation. */ /* */ /* The origin of fixuptri is treated as if it has just been inserted, and */ /* the local Delaunay condition needs to be enforced. It is only enforced */ /* in one sector, however, that being the angular range defined by */ /* fixuptri. */ /* */ /* This routine also needs to make decisions regarding the "stacking" of */ /* triangles. (Read the description of constrainededge() below before */ /* reading on here, so you understand the algorithm.) If the position of */ /* the new point (the origin of fixuptri) indicates that the vertex before */ /* it on the polygon is a reflex vertex, then "stack" the triangle by */ /* doing nothing. (fixuptri is an inverted triangle, which is how stacked */ /* triangles are identified.) */ /* */ /* Otherwise, check whether the vertex before that was a reflex vertex. */ /* If so, perform an edge flip, thereby eliminating an inverted triangle */ /* (popping it off the stack). The edge flip may result in the creation */ /* of a new inverted triangle, depending on whether or not the new vertex */ /* is visible to the vertex three edges behind on the polygon. */ /* */ /* If neither of the two vertices behind the new vertex are reflex */ /* vertices, fixuptri and fartri, the triangle opposite it, are not */ /* inverted; hence, ensure that the edge between them is locally Delaunay. */ /* */ /* `leftside' indicates whether or not fixuptri is to the left of the */ /* segment being inserted. (Imagine that the segment is pointing up from */ /* endpoint1 to endpoint2.) */ /* */ /*****************************************************************************/ void delaunayfixup(fixuptri, leftside) struct triedge *fixuptri; int leftside; { struct triedge neartri; struct triedge fartri; struct edge faredge; point nearpoint, leftpoint, rightpoint, farpoint; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ lnext(*fixuptri, neartri); sym(neartri, fartri); /* Check if the edge opposite the origin of fixuptri can be flipped. */ if (fartri.tri == dummytri) { return; } tspivot(neartri, faredge); if (faredge.sh != dummysh) { return; } /* Find all the relevant vertices. */ apex(neartri, nearpoint); org(neartri, leftpoint); dest(neartri, rightpoint); apex(fartri, farpoint); /* Check whether the previous polygon vertex is a reflex vertex. */ if (leftside) { if (counterclockwise(nearpoint, leftpoint, farpoint) <= 0.0) { /* leftpoint is a reflex vertex too. Nothing can */ /* be done until a convex section is found. */ return; } } else { if (counterclockwise(farpoint, rightpoint, nearpoint) <= 0.0) { /* rightpoint is a reflex vertex too. Nothing can */ /* be done until a convex section is found. */ return; } } if (counterclockwise(rightpoint, leftpoint, farpoint) > 0.0) { /* fartri is not an inverted triangle, and farpoint is not a reflex */ /* vertex. As there are no reflex vertices, fixuptri isn't an */ /* inverted triangle, either. Hence, test the edge between the */ /* triangles to ensure it is locally Delaunay. */ if (incircle(leftpoint, farpoint, rightpoint, nearpoint) <= 0.0) { return; } /* Not locally Delaunay; go on to an edge flip. */ } /* else fartri is inverted; remove it from the stack by flipping. */ flip(&neartri); lprevself(*fixuptri); /* Restore the origin of fixuptri after the flip. */ /* Recursively process the two triangles that result from the flip. */ delaunayfixup(fixuptri, leftside); delaunayfixup(&fartri, leftside); } /*****************************************************************************/ /* */ /* constrainededge() Force a segment into a constrained Delaunay */ /* triangulation by deleting the triangles it */ /* intersects, and triangulating the polygons that */ /* form on each side of it. */ /* */ /* Generates a single edge connecting `endpoint1' to `endpoint2'. The */ /* triangle `starttri' has `endpoint1' as its origin. `newmark' is the */ /* boundary marker of the segment. */ /* */ /* To insert a segment, every triangle whose interior intersects the */ /* segment is deleted. The union of these deleted triangles is a polygon */ /* (which is not necessarily monotone, but is close enough), which is */ /* divided into two polygons by the new segment. This routine's task is */ /* to generate the Delaunay triangulation of these two polygons. */ /* */ /* You might think of this routine's behavior as a two-step process. The */ /* first step is to walk from endpoint1 to endpoint2, flipping each edge */ /* encountered. This step creates a fan of edges connected to endpoint1, */ /* including the desired edge to endpoint2. The second step enforces the */ /* Delaunay condition on each side of the segment in an incremental manner: */ /* proceeding along the polygon from endpoint1 to endpoint2 (this is done */ /* independently on each side of the segment), each vertex is "enforced" */ /* as if it had just been inserted, but affecting only the previous */ /* vertices. The result is the same as if the vertices had been inserted */ /* in the order they appear on the polygon, so the result is Delaunay. */ /* */ /* In truth, constrainededge() interleaves these two steps. The procedure */ /* walks from endpoint1 to endpoint2, and each time an edge is encountered */ /* and flipped, the newly exposed vertex (at the far end of the flipped */ /* edge) is "enforced" upon the previously flipped edges, usually affecting */ /* only one side of the polygon (depending upon which side of the segment */ /* the vertex falls on). */ /* */ /* The algorithm is complicated by the need to handle polygons that are not */ /* convex. Although the polygon is not necessarily monotone, it can be */ /* triangulated in a manner similar to the stack-based algorithms for */ /* monotone polygons. For each reflex vertex (local concavity) of the */ /* polygon, there will be an inverted triangle formed by one of the edge */ /* flips. (An inverted triangle is one with negative area - that is, its */ /* vertices are arranged in clockwise order - and is best thought of as a */ /* wrinkle in the fabric of the mesh.) Each inverted triangle can be */ /* thought of as a reflex vertex pushed on the stack, waiting to be fixed */ /* later. */ /* */ /* A reflex vertex is popped from the stack when a vertex is inserted that */ /* is visible to the reflex vertex. (However, if the vertex behind the */ /* reflex vertex is not visible to the reflex vertex, a new inverted */ /* triangle will take its place on the stack.) These details are handled */ /* by the delaunayfixup() routine above. */ /* */ /*****************************************************************************/ void constrainededge(starttri, endpoint2, newmark) struct triedge *starttri; point endpoint2; int newmark; { struct triedge fixuptri, fixuptri2; struct edge fixupedge; point endpoint1; point farpoint; REAL area; int collision; int done; triangle ptr; /* Temporary variable used by sym() and oprev(). */ shelle sptr; /* Temporary variable used by tspivot(). */ org(*starttri, endpoint1); lnext(*starttri, fixuptri); flip(&fixuptri); /* `collision' indicates whether we have found a point directly */ /* between endpoint1 and endpoint2. */ collision = 0; done = 0; do { org(fixuptri, farpoint); /* `farpoint' is the extreme point of the polygon we are "digging" */ /* to get from endpoint1 to endpoint2. */ if ((farpoint[0] == endpoint2[0]) && (farpoint[1] == endpoint2[1])) { oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around endpoint2. */ delaunayfixup(&fixuptri, 0); delaunayfixup(&fixuptri2, 1); done = 1; } else { /* Check whether farpoint is to the left or right of the segment */ /* being inserted, to decide which edge of fixuptri to dig */ /* through next. */ area = counterclockwise(endpoint1, endpoint2, farpoint); if (area == 0.0) { /* We've collided with a point between endpoint1 and endpoint2. */ collision = 1; oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around farpoint. */ delaunayfixup(&fixuptri, 0); delaunayfixup(&fixuptri2, 1); done = 1; } else { if (area > 0.0) { /* farpoint is to the left of the segment. */ oprev(fixuptri, fixuptri2); /* Enforce the Delaunay condition around farpoint, on the */ /* left side of the segment only. */ delaunayfixup(&fixuptri2, 1); /* Flip the edge that crosses the segment. After the edge is */ /* flipped, one of its endpoints is the fan vertex, and the */ /* destination of fixuptri is the fan vertex. */ lprevself(fixuptri); } else { /* farpoint is to the right of the segment. */ delaunayfixup(&fixuptri, 0); /* Flip the edge that crosses the segment. After the edge is */ /* flipped, one of its endpoints is the fan vertex, and the */ /* destination of fixuptri is the fan vertex. */ oprevself(fixuptri); } /* Check for two intersecting segments. */ tspivot(fixuptri, fixupedge); if (fixupedge.sh == dummysh) { flip(&fixuptri); /* May create an inverted triangle on the left. */ } else { /* We've collided with a segment between endpoint1 and endpoint2. */ collision = 1; /* Insert a point at the intersection. */ segmentintersection(&fixuptri, &fixupedge, endpoint2); done = 1; } } } } while (!done); /* Insert a shell edge to make the segment permanent. */ insertshelle(&fixuptri, newmark); /* If there was a collision with an interceding vertex, install another */ /* segment connecting that vertex with endpoint2. */ if (collision) { /* Insert the remainder of the segment. */ if (!scoutsegment(&fixuptri, endpoint2, newmark)) { constrainededge(&fixuptri, endpoint2, newmark); } } } /*****************************************************************************/ /* */ /* insertsegment() Insert a PSLG segment into a triangulation. */ /* */ /*****************************************************************************/ void insertsegment(endpoint1, endpoint2, newmark) point endpoint1; point endpoint2; int newmark; { struct triedge searchtri1, searchtri2; triangle encodedtri; point checkpoint; triangle ptr; /* Temporary variable used by sym(). */ if (verbose > 1) { printf(" Connecting (%.12g, %.12g) to (%.12g, %.12g).\n", endpoint1[0], endpoint1[1], endpoint2[0], endpoint2[1]); } /* Find a triangle whose origin is the segment's first endpoint. */ checkpoint = (point) NULL; encodedtri = point2tri(endpoint1); if (encodedtri != (triangle) NULL) { decode(encodedtri, searchtri1); org(searchtri1, checkpoint); } if (checkpoint != endpoint1) { /* Find a boundary triangle to search from. */ searchtri1.tri = dummytri; searchtri1.orient = 0; symself(searchtri1); /* Search for the segment's first endpoint by point location. */ if (locate(endpoint1, &searchtri1) != ONVERTEX) { printf( "Internal error in insertsegment(): Unable to locate PSLG point\n"); printf(" (%.12g, %.12g) in triangulation.\n", endpoint1[0], endpoint1[1]); internalerror(); } } /* Remember this triangle to improve subsequent point location. */ triedgecopy(searchtri1, recenttri); /* Scout the beginnings of a path from the first endpoint */ /* toward the second. */ if (scoutsegment(&searchtri1, endpoint2, newmark)) { /* The segment was easily inserted. */ return; } /* The first endpoint may have changed if a collision with an intervening */ /* vertex on the segment occurred. */ org(searchtri1, endpoint1); /* Find a triangle whose origin is the segment's second endpoint. */ checkpoint = (point) NULL; encodedtri = point2tri(endpoint2); if (encodedtri != (triangle) NULL) { decode(encodedtri, searchtri2); org(searchtri2, checkpoint); } if (checkpoint != endpoint2) { /* Find a boundary triangle to search from. */ searchtri2.tri = dummytri; searchtri2.orient = 0; symself(searchtri2); /* Search for the segment's second endpoint by point location. */ if (locate(endpoint2, &searchtri2) != ONVERTEX) { printf( "Internal error in insertsegment(): Unable to locate PSLG point\n"); printf(" (%.12g, %.12g) in triangulation.\n", endpoint2[0], endpoint2[1]); internalerror(); } } /* Remember this triangle to improve subsequent point location. */ triedgecopy(searchtri2, recenttri); /* Scout the beginnings of a path from the second endpoint */ /* toward the first. */ if (scoutsegment(&searchtri2, endpoint1, newmark)) { /* The segment was easily inserted. */ return; } /* The second endpoint may have changed if a collision with an intervening */ /* vertex on the segment occurred. */ org(searchtri2, endpoint2); #ifndef REDUCED #ifndef CDT_ONLY if (splitseg) { /* Insert vertices to force the segment into the triangulation. */ conformingedge(endpoint1, endpoint2, newmark); } else { #endif /* not CDT_ONLY */ #endif /* not REDUCED */ /* Insert the segment directly into the triangulation. */ constrainededge(&searchtri1, endpoint2, newmark); #ifndef REDUCED #ifndef CDT_ONLY } #endif /* not CDT_ONLY */ #endif /* not REDUCED */ } /*****************************************************************************/ /* */ /* markhull() Cover the convex hull of a triangulation with shell edges. */ /* */ /*****************************************************************************/ void markhull() { struct triedge hulltri; struct triedge nexttri; struct triedge starttri; triangle ptr; /* Temporary variable used by sym() and oprev(). */ /* Find a triangle handle on the hull. */ hulltri.tri = dummytri; hulltri.orient = 0; symself(hulltri); /* Remember where we started so we know when to stop. */ triedgecopy(hulltri, starttri); /* Go once counterclockwise around the convex hull. */ do { /* Create a shell edge if there isn't already one here. */ insertshelle(&hulltri, 1); /* To find the next hull edge, go clockwise around the next vertex. */ lnextself(hulltri); oprev(hulltri, nexttri); while (nexttri.tri != dummytri) { triedgecopy(nexttri, hulltri); oprev(hulltri, nexttri); } } while (!triedgeequal(hulltri, starttri)); } /*****************************************************************************/ /* */ /* formskeleton() Create the shell edges of a triangulation, including */ /* PSLG edges and edges on the convex hull. */ /* */ /* The PSLG edges are read from a .poly file. The return value is the */ /* number of segments in the file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY int formskeleton(segmentlist, segmentmarkerlist, numberofsegments) int *segmentlist; int *segmentmarkerlist; int numberofsegments; #else /* not TRILIBRARY */ int formskeleton(polyfile, polyfilename) FILE *polyfile; char *polyfilename; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY char polyfilename[6]; int index; #else /* not TRILIBRARY */ char inputline[INPUTLINESIZE]; char *stringptr; #endif /* not TRILIBRARY */ point endpoint1, endpoint2; int segments; int segmentmarkers; int end1, end2; int boundmarker; int i; if (poly) { if (!quiet) { printf("Inserting segments into Delaunay triangulation.\n"); } #ifdef TRILIBRARY strcpy(polyfilename, "input"); segments = numberofsegments; segmentmarkers = segmentmarkerlist != (int *) NULL; index = 0; #else /* not TRILIBRARY */ /* Read the segments from a .poly file. */ /* Read number of segments and number of boundary markers. */ stringptr = readline(inputline, polyfile, polyfilename); segments = (int) strtol (stringptr, &stringptr, 0); stringptr = findfield(stringptr); if (*stringptr == '\0') { segmentmarkers = 0; } else { segmentmarkers = (int) strtol (stringptr, &stringptr, 0); } #endif /* not TRILIBRARY */ /* If segments are to be inserted, compute a mapping */ /* from points to triangles. */ if (segments > 0) { if (verbose) { printf(" Inserting PSLG segments.\n"); } makepointmap(); } boundmarker = 0; /* Read and insert the segments. */ for (i = 1; i <= segments; i++) { #ifdef TRILIBRARY end1 = segmentlist[index++]; end2 = segmentlist[index++]; if (segmentmarkers) { boundmarker = segmentmarkerlist[i - 1]; } #else /* not TRILIBRARY */ stringptr = readline(inputline, polyfile, inpolyfilename); stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Segment %d has no endpoints in %s.\n", i, polyfilename); exit(1); } else { end1 = (int) strtol (stringptr, &stringptr, 0); } stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Segment %d is missing its second endpoint in %s.\n", i, polyfilename); exit(1); } else { end2 = (int) strtol (stringptr, &stringptr, 0); } if (segmentmarkers) { stringptr = findfield(stringptr); if (*stringptr == '\0') { boundmarker = 0; } else { boundmarker = (int) strtol (stringptr, &stringptr, 0); } } #endif /* not TRILIBRARY */ if ((end1 < firstnumber) || (end1 >= firstnumber + inpoints)) { if (!quiet) { printf("Warning: Invalid first endpoint of segment %d in %s.\n", i, polyfilename); } } else if ((end2 < firstnumber) || (end2 >= firstnumber + inpoints)) { if (!quiet) { printf("Warning: Invalid second endpoint of segment %d in %s.\n", i, polyfilename); } } else { endpoint1 = getpoint(end1); endpoint2 = getpoint(end2); if ((endpoint1[0] == endpoint2[0]) && (endpoint1[1] == endpoint2[1])) { if (!quiet) { printf("Warning: Endpoints of segment %d are coincident in %s.\n", i, polyfilename); } } else { insertsegment(endpoint1, endpoint2, boundmarker); } } } } else { segments = 0; } if (convex || !poly) { /* Enclose the convex hull with shell edges. */ if (verbose) { printf(" Enclosing convex hull with segments.\n"); } markhull(); } return segments; } /** **/ /** **/ /********* Segment (shell edge) insertion ends here *********/ /********* Carving out holes and concavities begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* infecthull() Virally infect all of the triangles of the convex hull */ /* that are not protected by shell edges. Where there are */ /* shell edges, set boundary markers as appropriate. */ /* */ /*****************************************************************************/ void infecthull() { struct triedge hulltri; struct triedge nexttri; struct triedge starttri; struct edge hulledge; triangle **deadtri; point horg, hdest; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ if (verbose) { printf(" Marking concavities (external triangles) for elimination.\n"); } /* Find a triangle handle on the hull. */ hulltri.tri = dummytri; hulltri.orient = 0; symself(hulltri); /* Remember where we started so we know when to stop. */ triedgecopy(hulltri, starttri); /* Go once counterclockwise around the convex hull. */ do { /* Ignore triangles that are already infected. */ if (!infected(hulltri)) { /* Is the triangle protected by a shell edge? */ tspivot(hulltri, hulledge); if (hulledge.sh == dummysh) { /* The triangle is not protected; infect it. */ infect(hulltri); deadtri = (triangle **) poolalloc(&viri); *deadtri = hulltri.tri; } else { /* The triangle is protected; set boundary markers if appropriate. */ if (mark(hulledge) == 0) { setmark(hulledge, 1); org(hulltri, horg); dest(hulltri, hdest); if (pointmark(horg) == 0) { setpointmark(horg, 1); } if (pointmark(hdest) == 0) { setpointmark(hdest, 1); } } } } /* To find the next hull edge, go clockwise around the next vertex. */ lnextself(hulltri); oprev(hulltri, nexttri); while (nexttri.tri != dummytri) { triedgecopy(nexttri, hulltri); oprev(hulltri, nexttri); } } while (!triedgeequal(hulltri, starttri)); } /*****************************************************************************/ /* */ /* plague() Spread the virus from all infected triangles to any neighbors */ /* not protected by shell edges. Delete all infected triangles. */ /* */ /* This is the procedure that actually creates holes and concavities. */ /* */ /* This procedure operates in two phases. The first phase identifies all */ /* the triangles that will die, and marks them as infected. They are */ /* marked to ensure that each triangle is added to the virus pool only */ /* once, so the procedure will terminate. */ /* */ /* The second phase actually eliminates the infected triangles. It also */ /* eliminates orphaned points. */ /* */ /*****************************************************************************/ void plague() { struct triedge testtri; struct triedge neighbor; triangle **virusloop; triangle **deadtri; struct edge neighborshelle; point testpoint; point norg, ndest; point deadorg, deaddest, deadapex; int killorg; triangle ptr; /* Temporary variable used by sym() and onext(). */ shelle sptr; /* Temporary variable used by tspivot(). */ if (verbose) { printf(" Marking neighbors of marked triangles.\n"); } /* Loop through all the infected triangles, spreading the virus to */ /* their neighbors, then to their neighbors' neighbors. */ traversalinit(&viri); virusloop = (triangle **) traverse(&viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; /* A triangle is marked as infected by messing with one of its shell */ /* edges, setting it to an illegal value. Hence, we have to */ /* temporarily uninfect this triangle so that we can examine its */ /* adjacent shell edges. */ uninfect(testtri); if (verbose > 2) { /* Assign the triangle an orientation for convenience in */ /* checking its points. */ testtri.orient = 0; org(testtri, deadorg); dest(testtri, deaddest); apex(testtri, deadapex); printf(" Checking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", deadorg[0], deadorg[1], deaddest[0], deaddest[1], deadapex[0], deadapex[1]); } /* Check each of the triangle's three neighbors. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { /* Find the neighbor. */ sym(testtri, neighbor); /* Check for a shell between the triangle and its neighbor. */ tspivot(testtri, neighborshelle); /* Check if the neighbor is nonexistent or already infected. */ if ((neighbor.tri == dummytri) || infected(neighbor)) { if (neighborshelle.sh != dummysh) { /* There is a shell edge separating the triangle from its */ /* neighbor, but both triangles are dying, so the shell */ /* edge dies too. */ shelledealloc(neighborshelle.sh); if (neighbor.tri != dummytri) { /* Make sure the shell edge doesn't get deallocated again */ /* later when the infected neighbor is visited. */ uninfect(neighbor); tsdissolve(neighbor); infect(neighbor); } } } else { /* The neighbor exists and is not infected. */ if (neighborshelle.sh == dummysh) { /* There is no shell edge protecting the neighbor, so */ /* the neighbor becomes infected. */ if (verbose > 2) { org(neighbor, deadorg); dest(neighbor, deaddest); apex(neighbor, deadapex); printf( " Marking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", deadorg[0], deadorg[1], deaddest[0], deaddest[1], deadapex[0], deadapex[1]); } infect(neighbor); /* Ensure that the neighbor's neighbors will be infected. */ deadtri = (triangle **) poolalloc(&viri); *deadtri = neighbor.tri; } else { /* The neighbor is protected by a shell edge. */ /* Remove this triangle from the shell edge. */ stdissolve(neighborshelle); /* The shell edge becomes a boundary. Set markers accordingly. */ if (mark(neighborshelle) == 0) { setmark(neighborshelle, 1); } org(neighbor, norg); dest(neighbor, ndest); if (pointmark(norg) == 0) { setpointmark(norg, 1); } if (pointmark(ndest) == 0) { setpointmark(ndest, 1); } } } } /* Remark the triangle as infected, so it doesn't get added to the */ /* virus pool again. */ infect(testtri); virusloop = (triangle **) traverse(&viri); } if (verbose) { printf(" Deleting marked triangles.\n"); } traversalinit(&viri); virusloop = (triangle **) traverse(&viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; /* Check each of the three corners of the triangle for elimination. */ /* This is done by walking around each point, checking if it is */ /* still connected to at least one live triangle. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { org(testtri, testpoint); /* Check if the point has already been tested. */ if (testpoint != (point) NULL) { killorg = 1; /* Mark the corner of the triangle as having been tested. */ setorg(testtri, NULL); /* Walk counterclockwise about the point. */ onext(testtri, neighbor); /* Stop upon reaching a boundary or the starting triangle. */ while ((neighbor.tri != dummytri) && (!triedgeequal(neighbor, testtri))) { if (infected(neighbor)) { /* Mark the corner of this triangle as having been tested. */ setorg(neighbor, NULL); } else { /* A live triangle. The point survives. */ killorg = 0; } /* Walk counterclockwise about the point. */ onextself(neighbor); } /* If we reached a boundary, we must walk clockwise as well. */ if (neighbor.tri == dummytri) { /* Walk clockwise about the point. */ oprev(testtri, neighbor); /* Stop upon reaching a boundary. */ while (neighbor.tri != dummytri) { if (infected(neighbor)) { /* Mark the corner of this triangle as having been tested. */ setorg(neighbor, NULL); } else { /* A live triangle. The point survives. */ killorg = 0; } /* Walk clockwise about the point. */ oprevself(neighbor); } } if (killorg) { if (verbose > 1) { printf(" Deleting point (%.12g, %.12g)\n", testpoint[0], testpoint[1]); } pointdealloc(testpoint); } } } /* Record changes in the number of boundary edges, and disconnect */ /* dead triangles from their neighbors. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { sym(testtri, neighbor); if (neighbor.tri == dummytri) { /* There is no neighboring triangle on this edge, so this edge */ /* is a boundary edge. This triangle is being deleted, so this */ /* boundary edge is deleted. */ hullsize--; } else { /* Disconnect the triangle from its neighbor. */ dissolve(neighbor); /* There is a neighboring triangle on this edge, so this edge */ /* becomes a boundary edge when this triangle is deleted. */ hullsize++; } } /* Return the dead triangle to the pool of triangles. */ triangledealloc(testtri.tri); virusloop = (triangle **) traverse(&viri); } /* Empty the virus pool. */ poolrestart(&viri); } /*****************************************************************************/ /* */ /* regionplague() Spread regional attributes and/or area constraints */ /* (from a .poly file) throughout the mesh. */ /* */ /* This procedure operates in two phases. The first phase spreads an */ /* attribute and/or an area constraint through a (segment-bounded) region. */ /* The triangles are marked to ensure that each triangle is added to the */ /* virus pool only once, so the procedure will terminate. */ /* */ /* The second phase uninfects all infected triangles, returning them to */ /* normal. */ /* */ /*****************************************************************************/ void regionplague(attribute, area) REAL attribute; REAL area; { struct triedge testtri; struct triedge neighbor; triangle **virusloop; triangle **regiontri; struct edge neighborshelle; point regionorg, regiondest, regionapex; triangle ptr; /* Temporary variable used by sym() and onext(). */ shelle sptr; /* Temporary variable used by tspivot(). */ if (verbose > 1) { printf(" Marking neighbors of marked triangles.\n"); } /* Loop through all the infected triangles, spreading the attribute */ /* and/or area constraint to their neighbors, then to their neighbors' */ /* neighbors. */ traversalinit(&viri); virusloop = (triangle **) traverse(&viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; /* A triangle is marked as infected by messing with one of its shell */ /* edges, setting it to an illegal value. Hence, we have to */ /* temporarily uninfect this triangle so that we can examine its */ /* adjacent shell edges. */ uninfect(testtri); if (regionattrib) { /* Set an attribute. */ setelemattribute(testtri, eextras, attribute); } if (vararea) { /* Set an area constraint. */ setareabound(testtri, area); } if (verbose > 2) { /* Assign the triangle an orientation for convenience in */ /* checking its points. */ testtri.orient = 0; org(testtri, regionorg); dest(testtri, regiondest); apex(testtri, regionapex); printf(" Checking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", regionorg[0], regionorg[1], regiondest[0], regiondest[1], regionapex[0], regionapex[1]); } /* Check each of the triangle's three neighbors. */ for (testtri.orient = 0; testtri.orient < 3; testtri.orient++) { /* Find the neighbor. */ sym(testtri, neighbor); /* Check for a shell between the triangle and its neighbor. */ tspivot(testtri, neighborshelle); /* Make sure the neighbor exists, is not already infected, and */ /* isn't protected by a shell edge. */ if ((neighbor.tri != dummytri) && !infected(neighbor) && (neighborshelle.sh == dummysh)) { if (verbose > 2) { org(neighbor, regionorg); dest(neighbor, regiondest); apex(neighbor, regionapex); printf(" Marking (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", regionorg[0], regionorg[1], regiondest[0], regiondest[1], regionapex[0], regionapex[1]); } /* Infect the neighbor. */ infect(neighbor); /* Ensure that the neighbor's neighbors will be infected. */ regiontri = (triangle **) poolalloc(&viri); *regiontri = neighbor.tri; } } /* Remark the triangle as infected, so it doesn't get added to the */ /* virus pool again. */ infect(testtri); virusloop = (triangle **) traverse(&viri); } /* Uninfect all triangles. */ if (verbose > 1) { printf(" Unmarking marked triangles.\n"); } traversalinit(&viri); virusloop = (triangle **) traverse(&viri); while (virusloop != (triangle **) NULL) { testtri.tri = *virusloop; uninfect(testtri); virusloop = (triangle **) traverse(&viri); } /* Empty the virus pool. */ poolrestart(&viri); } /*****************************************************************************/ /* */ /* carveholes() Find the holes and infect them. Find the area */ /* constraints and infect them. Infect the convex hull. */ /* Spread the infection and kill triangles. Spread the */ /* area constraints. */ /* */ /* This routine mainly calls other routines to carry out all these */ /* functions. */ /* */ /*****************************************************************************/ void carveholes(holelist, holes, regionlist, regions) REAL *holelist; int holes; REAL *regionlist; int regions; { struct triedge searchtri; struct triedge triangleloop; struct triedge *regiontris; triangle **holetri; triangle **regiontri; point searchorg, searchdest; enum locateresult intersect; int i; triangle ptr; /* Temporary variable used by sym(). */ if (!(quiet || (noholes && convex))) { printf("Removing unwanted triangles.\n"); if (verbose && (holes > 0)) { printf(" Marking holes for elimination.\n"); } } if (regions > 0) { /* Allocate storage for the triangles in which region points fall. */ regiontris = (struct triedge *) malloc(regions * sizeof(struct triedge)); if (regiontris == (struct triedge *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } if (((holes > 0) && !noholes) || !convex || (regions > 0)) { /* Initialize a pool of viri to be used for holes, concavities, */ /* regional attributes, and/or regional area constraints. */ poolinit(&viri, sizeof(triangle *), VIRUSPERBLOCK, POINTER, 0); } if (!convex) { /* Mark as infected any unprotected triangles on the boundary. */ /* This is one way by which concavities are created. */ infecthull(); } if ((holes > 0) && !noholes) { /* Infect each triangle in which a hole lies. */ for (i = 0; i < 2 * holes; i += 2) { /* Ignore holes that aren't within the bounds of the mesh. */ if ((holelist[i] >= xmin) && (holelist[i] <= xmax) && (holelist[i + 1] >= ymin) && (holelist[i + 1] <= ymax)) { /* Start searching from some triangle on the outer boundary. */ searchtri.tri = dummytri; searchtri.orient = 0; symself(searchtri); /* Ensure that the hole is to the left of this boundary edge; */ /* otherwise, locate() will falsely report that the hole */ /* falls within the starting triangle. */ org(searchtri, searchorg); dest(searchtri, searchdest); if (counterclockwise(searchorg, searchdest, &holelist[i]) > 0.0) { /* Find a triangle that contains the hole. */ intersect = locate(&holelist[i], &searchtri); if ((intersect != OUTSIDE) && (!infected(searchtri))) { /* Infect the triangle. This is done by marking the triangle */ /* as infect and including the triangle in the virus pool. */ infect(searchtri); holetri = (triangle **) poolalloc(&viri); *holetri = searchtri.tri; } } } } } /* Now, we have to find all the regions BEFORE we carve the holes, because */ /* locate() won't work when the triangulation is no longer convex. */ /* (Incidentally, this is the reason why regional attributes and area */ /* constraints can't be used when refining a preexisting mesh, which */ /* might not be convex; they can only be used with a freshly */ /* triangulated PSLG.) */ if (regions > 0) { /* Find the starting triangle for each region. */ for (i = 0; i < regions; i++) { regiontris[i].tri = dummytri; /* Ignore region points that aren't within the bounds of the mesh. */ if ((regionlist[4 * i] >= xmin) && (regionlist[4 * i] <= xmax) && (regionlist[4 * i + 1] >= ymin) && (regionlist[4 * i + 1] <= ymax)) { /* Start searching from some triangle on the outer boundary. */ searchtri.tri = dummytri; searchtri.orient = 0; symself(searchtri); /* Ensure that the region point is to the left of this boundary */ /* edge; otherwise, locate() will falsely report that the */ /* region point falls within the starting triangle. */ org(searchtri, searchorg); dest(searchtri, searchdest); if (counterclockwise(searchorg, searchdest, ®ionlist[4 * i]) > 0.0) { /* Find a triangle that contains the region point. */ intersect = locate(®ionlist[4 * i], &searchtri); if ((intersect != OUTSIDE) && (!infected(searchtri))) { /* Record the triangle for processing after the */ /* holes have been carved. */ triedgecopy(searchtri, regiontris[i]); } } } } } if (viri.items > 0) { /* Carve the holes and concavities. */ plague(); } /* The virus pool should be empty now. */ if (regions > 0) { if (!quiet) { if (regionattrib) { if (vararea) { printf("Spreading regional attributes and area constraints.\n"); } else { printf("Spreading regional attributes.\n"); } } else { printf("Spreading regional area constraints.\n"); } } if (regionattrib && !refine) { /* Assign every triangle a regional attribute of zero. */ traversalinit(&triangles); triangleloop.orient = 0; triangleloop.tri = triangletraverse(); while (triangleloop.tri != (triangle *) NULL) { setelemattribute(triangleloop, eextras, 0.0); triangleloop.tri = triangletraverse(); } } for (i = 0; i < regions; i++) { if (regiontris[i].tri != dummytri) { /* Make sure the triangle under consideration still exists. */ /* It may have been eaten by the virus. */ if (regiontris[i].tri[3] != (triangle) NULL) { /* Put one triangle in the virus pool. */ infect(regiontris[i]); regiontri = (triangle **) poolalloc(&viri); *regiontri = regiontris[i].tri; /* Apply one region's attribute and/or area constraint. */ regionplague(regionlist[4 * i + 2], regionlist[4 * i + 3]); /* The virus pool should be empty now. */ } } } if (regionattrib && !refine) { /* Note the fact that each triangle has an additional attribute. */ eextras++; } } /* Free up memory. */ if (((holes > 0) && !noholes) || !convex || (regions > 0)) { pooldeinit(&viri); } if (regions > 0) { free(regiontris); } } /** **/ /** **/ /********* Carving out holes and concavities ends here *********/ /********* Mesh quality maintenance begins here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* tallyencs() Traverse the entire list of shell edges, check each edge */ /* to see if it is encroached. If so, add it to the list. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void tallyencs() { struct edge edgeloop; int dummy; traversalinit(&shelles); edgeloop.shorient = 0; edgeloop.sh = shelletraverse(); while (edgeloop.sh != (shelle *) NULL) { /* If the segment is encroached, add it to the list. */ dummy = checkedge4encroach(&edgeloop); edgeloop.sh = shelletraverse(); } } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* precisionerror() Print an error message for precision problems. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void precisionerror() { printf("Try increasing the area criterion and/or reducing the minimum\n"); printf(" allowable angle so that tiny triangles are not created.\n"); #ifdef SINGLE printf("Alternatively, try recompiling me with double precision\n"); printf(" arithmetic (by removing \"#define SINGLE\" from the\n"); printf(" source file or \"-DSINGLE\" from the makefile).\n"); #endif /* SINGLE */ } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* repairencs() Find and repair all the encroached segments. */ /* */ /* Encroached segments are repaired by splitting them by inserting a point */ /* at or near their centers. */ /* */ /* `flaws' is a flag that specifies whether one should take note of new */ /* encroached segments and bad triangles that result from inserting points */ /* to repair existing encroached segments. */ /* */ /* When a segment is split, the two resulting subsegments are always */ /* tested to see if they are encroached upon, regardless of the value */ /* of `flaws'. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void repairencs(flaws) int flaws; { struct triedge enctri; struct triedge testtri; struct edge *encloop; struct edge testsh; point eorg, edest; point newpoint; enum insertsiteresult success; REAL segmentlength, nearestpoweroftwo; REAL split; int acuteorg, acutedest; int dummy; int i; triangle ptr; /* Temporary variable used by stpivot(). */ shelle sptr; /* Temporary variable used by snext(). */ while ((badsegments.items > 0) && (steinerleft != 0)) { traversalinit(&badsegments); encloop = badsegmenttraverse(); while ((encloop != (struct edge *) NULL) && (steinerleft != 0)) { /* To decide where to split a segment, we need to know if the */ /* segment shares an endpoint with an adjacent segment. */ /* The concern is that, if we simply split every encroached */ /* segment in its center, two adjacent segments with a small */ /* angle between them might lead to an infinite loop; each */ /* point added to split one segment will encroach upon the */ /* other segment, which must then be split with a point that */ /* will encroach upon the first segment, and so on forever. */ /* To avoid this, imagine a set of concentric circles, whose */ /* radii are powers of two, about each segment endpoint. */ /* These concentric circles determine where the segment is */ /* split. (If both endpoints are shared with adjacent */ /* segments, split the segment in the middle, and apply the */ /* concentric shells for later splittings.) */ /* Is the origin shared with another segment? */ stpivot(*encloop, enctri); lnext(enctri, testtri); tspivot(testtri, testsh); acuteorg = testsh.sh != dummysh; /* Is the destination shared with another segment? */ lnextself(testtri); tspivot(testtri, testsh); acutedest = testsh.sh != dummysh; /* Now, check the other side of the segment, if there's a triangle */ /* there. */ sym(enctri, testtri); if (testtri.tri != dummytri) { /* Is the destination shared with another segment? */ lnextself(testtri); tspivot(testtri, testsh); acutedest = acutedest || (testsh.sh != dummysh); /* Is the origin shared with another segment? */ lnextself(testtri); tspivot(testtri, testsh); acuteorg = acuteorg || (testsh.sh != dummysh); } sorg(*encloop, eorg); sdest(*encloop, edest); /* Use the concentric circles if exactly one endpoint is shared */ /* with another adjacent segment. */ if (acuteorg ^ acutedest) { segmentlength = sqrt((edest[0] - eorg[0]) * (edest[0] - eorg[0]) + (edest[1] - eorg[1]) * (edest[1] - eorg[1])); /* Find the power of two nearest the segment's length. */ nearestpoweroftwo = 1.0; while (segmentlength > SQUAREROOTTWO * nearestpoweroftwo) { nearestpoweroftwo *= 2.0; } while (segmentlength < (0.5 * SQUAREROOTTWO) * nearestpoweroftwo) { nearestpoweroftwo *= 0.5; } /* Where do we split the segment? */ split = 0.5 * nearestpoweroftwo / segmentlength; if (acutedest) { split = 1.0 - split; } } else { /* If we're not worried about adjacent segments, split */ /* this segment in the middle. */ split = 0.5; } /* Create the new point. */ newpoint = (point) poolalloc(&points); /* Interpolate its coordinate and attributes. */ for (i = 0; i < 2 + nextras; i++) { newpoint[i] = (1.0 - split) * eorg[i] + split * edest[i]; } setpointmark(newpoint, mark(*encloop)); if (verbose > 1) { printf( " Splitting edge (%.12g, %.12g) (%.12g, %.12g) at (%.12g, %.12g).\n", eorg[0], eorg[1], edest[0], edest[1], newpoint[0], newpoint[1]); } /* Check whether the new point lies on an endpoint. */ if (((newpoint[0] == eorg[0]) && (newpoint[1] == eorg[1])) || ((newpoint[0] == edest[0]) && (newpoint[1] == edest[1]))) { printf("Error: Ran out of precision at (%.12g, %.12g).\n", newpoint[0], newpoint[1]); printf("I attempted to split a segment to a smaller size than can\n"); printf(" be accommodated by the finite precision of floating point\n" ); printf(" arithmetic.\n"); precisionerror(); exit(1); } /* Insert the splitting point. This should always succeed. */ success = insertsite(newpoint, &enctri, encloop, flaws, flaws); if ((success != SUCCESSFULPOINT) && (success != ENCROACHINGPOINT)) { printf("Internal error in repairencs():\n"); printf(" Failure to split a segment.\n"); internalerror(); } if (steinerleft > 0) { steinerleft--; } /* Check the two new subsegments to see if they're encroached. */ dummy = checkedge4encroach(encloop); snextself(*encloop); dummy = checkedge4encroach(encloop); badsegmentdealloc(encloop); encloop = badsegmenttraverse(); } } } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* tallyfaces() Test every triangle in the mesh for quality measures. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void tallyfaces() { struct triedge triangleloop; if (verbose) { printf(" Making a list of bad triangles.\n"); } traversalinit(&triangles); triangleloop.orient = 0; triangleloop.tri = triangletraverse(); while (triangleloop.tri != (triangle *) NULL) { /* If the triangle is bad, enqueue it. */ testtriangle(&triangleloop); triangleloop.tri = triangletraverse(); } } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* findcircumcenter() Find the circumcenter of a triangle. */ /* */ /* The result is returned both in terms of x-y coordinates and xi-eta */ /* coordinates. The xi-eta coordinate system is defined in terms of the */ /* triangle: the origin of the triangle is the origin of the coordinate */ /* system; the destination of the triangle is one unit along the xi axis; */ /* and the apex of the triangle is one unit along the eta axis. */ /* */ /* The return value indicates which edge of the triangle is shortest. */ /* */ /*****************************************************************************/ enum circumcenterresult findcircumcenter(torg, tdest, tapex, circumcenter, xi, eta) point torg; point tdest; point tapex; point circumcenter; REAL *xi; REAL *eta; { REAL xdo, ydo, xao, yao, xad, yad; REAL dodist, aodist, addist; REAL denominator; REAL dx, dy; circumcentercount++; /* Compute the circumcenter of the triangle. */ xdo = tdest[0] - torg[0]; ydo = tdest[1] - torg[1]; xao = tapex[0] - torg[0]; yao = tapex[1] - torg[1]; dodist = xdo * xdo + ydo * ydo; aodist = xao * xao + yao * yao; if (noexact) { denominator = 0.5 / (xdo * yao - xao * ydo); } else { /* Use the counterclockwise() routine to ensure a positive (and */ /* reasonably accurate) result, avoiding any possibility of */ /* division by zero. */ denominator = 0.5 / counterclockwise(tdest, tapex, torg); /* Don't count the above as an orientation test. */ counterclockcount--; } circumcenter[0] = torg[0] - (ydo * aodist - yao * dodist) * denominator; circumcenter[1] = torg[1] + (xdo * aodist - xao * dodist) * denominator; /* To interpolate point attributes for the new point inserted at */ /* the circumcenter, define a coordinate system with a xi-axis, */ /* directed from the triangle's origin to its destination, and */ /* an eta-axis, directed from its origin to its apex. */ /* Calculate the xi and eta coordinates of the circumcenter. */ dx = circumcenter[0] - torg[0]; dy = circumcenter[1] - torg[1]; *xi = (dx * yao - xao * dy) * (2.0 * denominator); *eta = (xdo * dy - dx * ydo) * (2.0 * denominator); xad = tapex[0] - tdest[0]; yad = tapex[1] - tdest[1]; addist = xad * xad + yad * yad; if ((addist < dodist) && (addist < aodist)) { return OPPOSITEORG; } else if (dodist < aodist) { return OPPOSITEAPEX; } else { return OPPOSITEDEST; } } /*****************************************************************************/ /* */ /* splittriangle() Inserts a point at the circumcenter of a triangle. */ /* Deletes the newly inserted point if it encroaches upon */ /* a segment. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void splittriangle(badtri) struct badface *badtri; { point borg, bdest, bapex; point newpoint; REAL xi, eta; enum insertsiteresult success; enum circumcenterresult shortedge; int errorflag; int i; org(badtri->badfacetri, borg); dest(badtri->badfacetri, bdest); apex(badtri->badfacetri, bapex); /* Make sure that this triangle is still the same triangle it was */ /* when it was tested and determined to be of bad quality. */ /* Subsequent transformations may have made it a different triangle. */ if ((borg == badtri->faceorg) && (bdest == badtri->facedest) && (bapex == badtri->faceapex)) { if (verbose > 1) { printf(" Splitting this triangle at its circumcenter:\n"); printf(" (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", borg[0], borg[1], bdest[0], bdest[1], bapex[0], bapex[1]); } errorflag = 0; /* Create a new point at the triangle's circumcenter. */ newpoint = (point) poolalloc(&points); shortedge = findcircumcenter(borg, bdest, bapex, newpoint, &xi, &eta); /* Check whether the new point lies on a triangle vertex. */ if (((newpoint[0] == borg[0]) && (newpoint[1] == borg[1])) || ((newpoint[0] == bdest[0]) && (newpoint[1] == bdest[1])) || ((newpoint[0] == bapex[0]) && (newpoint[1] == bapex[1]))) { if (!quiet) { printf("Warning: New point (%.12g, %.12g) falls on existing vertex.\n" , newpoint[0], newpoint[1]); errorflag = 1; } pointdealloc(newpoint); } else { for (i = 2; i < 2 + nextras; i++) { /* Interpolate the point attributes at the circumcenter. */ newpoint[i] = borg[i] + xi * (bdest[i] - borg[i]) + eta * (bapex[i] - borg[i]); } /* The new point must be in the interior, and have a marker of zero. */ setpointmark(newpoint, 0); /* Ensure that the handle `badtri->badfacetri' represents the shortest */ /* edge of the triangle. This ensures that the circumcenter must */ /* fall to the left of this edge, so point location will work. */ if (shortedge == OPPOSITEORG) { lnextself(badtri->badfacetri); } else if (shortedge == OPPOSITEDEST) { lprevself(badtri->badfacetri); } /* Insert the circumcenter, searching from the edge of the triangle, */ /* and maintain the Delaunay property of the triangulation. */ success = insertsite(newpoint, &(badtri->badfacetri), (struct edge *) NULL, 1, 1); if (success == SUCCESSFULPOINT) { if (steinerleft > 0) { steinerleft--; } } else if (success == ENCROACHINGPOINT) { /* If the newly inserted point encroaches upon a segment, delete it. */ deletesite(&(badtri->badfacetri)); } else if (success == VIOLATINGPOINT) { /* Failed to insert the new point, but some segment was */ /* marked as being encroached. */ pointdealloc(newpoint); } else { /* success == DUPLICATEPOINT */ /* Failed to insert the new point because a vertex is already there. */ if (!quiet) { printf( "Warning: New point (%.12g, %.12g) falls on existing vertex.\n" , newpoint[0], newpoint[1]); errorflag = 1; } pointdealloc(newpoint); } } if (errorflag) { if (verbose) { printf(" The new point is at the circumcenter of triangle\n"); printf(" (%.12g, %.12g) (%.12g, %.12g) (%.12g, %.12g)\n", borg[0], borg[1], bdest[0], bdest[1], bapex[0], bapex[1]); } printf("This probably means that I am trying to refine triangles\n"); printf(" to a smaller size than can be accommodated by the finite\n"); printf(" precision of floating point arithmetic. (You can be\n"); printf(" sure of this if I fail to terminate.)\n"); precisionerror(); } } /* Return the bad triangle to the pool. */ pooldealloc(&badtriangles, (VOID *) badtri); } #endif /* not CDT_ONLY */ /*****************************************************************************/ /* */ /* enforcequality() Remove all the encroached edges and bad triangles */ /* from the triangulation. */ /* */ /*****************************************************************************/ #ifndef CDT_ONLY void enforcequality() { int i; if (!quiet) { printf("Adding Steiner points to enforce quality.\n"); } /* Initialize the pool of encroached segments. */ poolinit(&badsegments, sizeof(struct edge), BADSEGMENTPERBLOCK, POINTER, 0); if (verbose) { printf(" Looking for encroached segments.\n"); } /* Test all segments to see if they're encroached. */ tallyencs(); if (verbose && (badsegments.items > 0)) { printf(" Splitting encroached segments.\n"); } /* Note that steinerleft == -1 if an unlimited number */ /* of Steiner points is allowed. */ while ((badsegments.items > 0) && (steinerleft != 0)) { /* Fix the segments without noting newly encroached segments or */ /* bad triangles. The reason we don't want to note newly */ /* encroached segments is because some encroached segments are */ /* likely to be noted multiple times, and would then be blindly */ /* split multiple times. I should fix that some time. */ repairencs(0); /* Now, find all the segments that became encroached while adding */ /* points to split encroached segments. */ tallyencs(); } /* At this point, if we haven't run out of Steiner points, the */ /* triangulation should be (conforming) Delaunay. */ /* Next, we worry about enforcing triangle quality. */ if ((minangle > 0.0) || vararea || fixedarea) { /* Initialize the pool of bad triangles. */ poolinit(&badtriangles, sizeof(struct badface), BADTRIPERBLOCK, POINTER, 0); /* Initialize the queues of bad triangles. */ for (i = 0; i < 64; i++) { queuefront[i] = (struct badface *) NULL; queuetail[i] = &queuefront[i]; } /* Test all triangles to see if they're bad. */ tallyfaces(); if (verbose) { printf(" Splitting bad triangles.\n"); } while ((badtriangles.items > 0) && (steinerleft != 0)) { /* Fix one bad triangle by inserting a point at its circumcenter. */ splittriangle(dequeuebadtri()); /* Fix any encroached segments that may have resulted. Record */ /* any new bad triangles or encroached segments that result. */ if (badsegments.items > 0) { repairencs(1); } } } /* At this point, if we haven't run out of Steiner points, the */ /* triangulation should be (conforming) Delaunay and have no */ /* low-quality triangles. */ /* Might we have run out of Steiner points too soon? */ if (!quiet && (badsegments.items > 0) && (steinerleft == 0)) { printf("\nWarning: I ran out of Steiner points, but the mesh has\n"); if (badsegments.items == 1) { printf(" an encroached segment, and therefore might not be truly\n"); } else { printf(" %ld encroached segments, and therefore might not be truly\n", badsegments.items); } printf(" Delaunay. If the Delaunay property is important to you,\n"); printf(" try increasing the number of Steiner points (controlled by\n"); printf(" the -S switch) slightly and try again.\n\n"); } } #endif /* not CDT_ONLY */ /** **/ /** **/ /********* Mesh quality maintenance ends here *********/ /*****************************************************************************/ /* */ /* highorder() Create extra nodes for quadratic subparametric elements. */ /* */ /*****************************************************************************/ void highorder() { struct triedge triangleloop, trisym; struct edge checkmark; point newpoint; point torg, tdest; int i; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ if (!quiet) { printf("Adding vertices for second-order triangles.\n"); } /* The following line ensures that dead items in the pool of nodes */ /* cannot be allocated for the extra nodes associated with high */ /* order elements. This ensures that the primary nodes (at the */ /* corners of elements) will occur earlier in the output files, and */ /* have lower indices, than the extra nodes. */ points.deaditemstack = (VOID *) NULL; traversalinit(&triangles); triangleloop.tri = triangletraverse(); /* To loop over the set of edges, loop over all triangles, and look at */ /* the three edges of each triangle. If there isn't another triangle */ /* adjacent to the edge, operate on the edge. If there is another */ /* adjacent triangle, operate on the edge only if the current triangle */ /* has a smaller pointer than its neighbor. This way, each edge is */ /* considered only once. */ while (triangleloop.tri != (triangle *) NULL) { for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { sym(triangleloop, trisym); if ((triangleloop.tri < trisym.tri) || (trisym.tri == dummytri)) { org(triangleloop, torg); dest(triangleloop, tdest); /* Create a new node in the middle of the edge. Interpolate */ /* its attributes. */ newpoint = (point) poolalloc(&points); for (i = 0; i < 2 + nextras; i++) { newpoint[i] = 0.5 * (torg[i] + tdest[i]); } /* Set the new node's marker to zero or one, depending on */ /* whether it lies on a boundary. */ setpointmark(newpoint, trisym.tri == dummytri); if (useshelles) { tspivot(triangleloop, checkmark); /* If this edge is a segment, transfer the marker to the new node. */ if (checkmark.sh != dummysh) { setpointmark(newpoint, mark(checkmark)); } } if (verbose > 1) { printf(" Creating (%.12g, %.12g).\n", newpoint[0], newpoint[1]); } /* Record the new node in the (one or two) adjacent elements. */ triangleloop.tri[highorderindex + triangleloop.orient] = (triangle) newpoint; if (trisym.tri != dummytri) { trisym.tri[highorderindex + trisym.orient] = (triangle) newpoint; } } } triangleloop.tri = triangletraverse(); } } /********* File I/O routines begin here *********/ /** **/ /** **/ /*****************************************************************************/ /* */ /* readline() Read a nonempty line from a file. */ /* */ /* A line is considered "nonempty" if it contains something that looks like */ /* a number. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY char *readline(string, infile, infilename) char *string; FILE *infile; char *infilename; { char *result; /* Search for something that looks like a number. */ do { result = fgets(string, INPUTLINESIZE, infile); if (result == (char *) NULL) { printf(" Error: Unexpected end of file in %s.\n", infilename); exit(1); } /* Skip anything that doesn't look like a number, a comment, */ /* or the end of a line. */ while ((*result != '\0') && (*result != '#') && (*result != '.') && (*result != '+') && (*result != '-') && ((*result < '0') || (*result > '9'))) { result++; } /* If it's a comment or end of line, read another line and try again. */ } while ((*result == '#') || (*result == '\0')); return result; } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* findfield() Find the next field of a string. */ /* */ /* Jumps past the current field by searching for whitespace, then jumps */ /* past the whitespace to find the next field. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY char *findfield(string) char *string; { char *result; result = string; /* Skip the current field. Stop upon reaching whitespace. */ while ((*result != '\0') && (*result != '#') && (*result != ' ') && (*result != '\t')) { result++; } /* Now skip the whitespace and anything else that doesn't look like a */ /* number, a comment, or the end of a line. */ while ((*result != '\0') && (*result != '#') && (*result != '.') && (*result != '+') && (*result != '-') && ((*result < '0') || (*result > '9'))) { result++; } /* Check for a comment (prefixed with `#'). */ if (*result == '#') { *result = '\0'; } return result; } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* readnodes() Read the points from a file, which may be a .node or .poly */ /* file. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void readnodes(nodefilename, polyfilename, polyfile) char *nodefilename; char *polyfilename; FILE **polyfile; { FILE *infile; point pointloop; char inputline[INPUTLINESIZE]; char *stringptr; char *infilename; REAL x, y; int firstnode; int nodemarkers; int currentmarker; int i, j; if (poly) { /* Read the points from a .poly file. */ if (!quiet) { printf("Opening %s.\n", polyfilename); } *polyfile = fopen(polyfilename, "r"); if (*polyfile == (FILE *) NULL) { printf(" Error: Cannot access file %s.\n", polyfilename); exit(1); } /* Read number of points, number of dimensions, number of point */ /* attributes, and number of boundary markers. */ stringptr = readline(inputline, *polyfile, polyfilename); inpoints = (int) strtol (stringptr, &stringptr, 0); stringptr = findfield(stringptr); if (*stringptr == '\0') { mesh_dim = 2; } else { mesh_dim = (int) strtol (stringptr, &stringptr, 0); } stringptr = findfield(stringptr); if (*stringptr == '\0') { nextras = 0; } else { nextras = (int) strtol (stringptr, &stringptr, 0); } stringptr = findfield(stringptr); if (*stringptr == '\0') { nodemarkers = 0; } else { nodemarkers = (int) strtol (stringptr, &stringptr, 0); } if (inpoints > 0) { infile = *polyfile; infilename = polyfilename; readnodefile = 0; } else { /* If the .poly file claims there are zero points, that means that */ /* the points should be read from a separate .node file. */ readnodefile = 1; infilename = innodefilename; } } else { readnodefile = 1; infilename = innodefilename; *polyfile = (FILE *) NULL; } if (readnodefile) { /* Read the points from a .node file. */ if (!quiet) { printf("Opening %s.\n", innodefilename); } infile = fopen(innodefilename, "r"); if (infile == (FILE *) NULL) { printf(" Error: Cannot access file %s.\n", innodefilename); exit(1); } /* Read number of points, number of dimensions, number of point */ /* attributes, and number of boundary markers. */ stringptr = readline(inputline, infile, innodefilename); inpoints = (int) strtol (stringptr, &stringptr, 0); stringptr = findfield(stringptr); if (*stringptr == '\0') { mesh_dim = 2; } else { mesh_dim = (int) strtol (stringptr, &stringptr, 0); } stringptr = findfield(stringptr); if (*stringptr == '\0') { nextras = 0; } else { nextras = (int) strtol (stringptr, &stringptr, 0); } stringptr = findfield(stringptr); if (*stringptr == '\0') { nodemarkers = 0; } else { nodemarkers = (int) strtol (stringptr, &stringptr, 0); } } if (inpoints < 3) { printf("Error: Input must have at least three input points.\n"); exit(1); } if (mesh_dim != 2) { printf("Error: Triangle only works with two-dimensional meshes.\n"); exit(1); } initializepointpool(); /* Read the points. */ for (i = 0; i < inpoints; i++) { pointloop = (point) poolalloc(&points); stringptr = readline(inputline, infile, infilename); if (i == 0) { firstnode = (int) strtol (stringptr, &stringptr, 0); if ((firstnode == 0) || (firstnode == 1)) { firstnumber = firstnode; } } stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Point %d has no x coordinate.\n", firstnumber + i); exit(1); } x = (REAL) strtod(stringptr, &stringptr); stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Point %d has no y coordinate.\n", firstnumber + i); exit(1); } y = (REAL) strtod(stringptr, &stringptr); pointloop[0] = x; pointloop[1] = y; /* Read the point attributes. */ for (j = 2; j < 2 + nextras; j++) { stringptr = findfield(stringptr); if (*stringptr == '\0') { pointloop[j] = 0.0; } else { pointloop[j] = (REAL) strtod(stringptr, &stringptr); } } if (nodemarkers) { /* Read a point marker. */ stringptr = findfield(stringptr); if (*stringptr == '\0') { setpointmark(pointloop, 0); } else { currentmarker = (int) strtol (stringptr, &stringptr, 0); setpointmark(pointloop, currentmarker); } } else { /* If no markers are specified in the file, they default to zero. */ setpointmark(pointloop, 0); } /* Determine the smallest and largest x and y coordinates. */ if (i == 0) { xmin = xmax = x; ymin = ymax = y; } else { xmin = (x < xmin) ? x : xmin; xmax = (x > xmax) ? x : xmax; ymin = (y < ymin) ? y : ymin; ymax = (y > ymax) ? y : ymax; } } if (readnodefile) { fclose(infile); } /* Nonexistent x value used as a flag to mark circle events in sweepline */ /* Delaunay algorithm. */ xminextreme = 10 * xmin - 9 * xmax; } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* transfernodes() Read the points from memory. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void transfernodes(pointlist, pointattriblist, pointmarkerlist, numberofpoints, numberofpointattribs) REAL *pointlist; REAL *pointattriblist; int *pointmarkerlist; int numberofpoints; int numberofpointattribs; { point pointloop; REAL x, y; int i, j; int coordindex; int attribindex; inpoints = numberofpoints; mesh_dim = 2; nextras = numberofpointattribs; readnodefile = 0; if (inpoints < 3) { printf("Error: Input must have at least three input points.\n"); exit(1); } initializepointpool(); /* Read the points. */ coordindex = 0; attribindex = 0; for (i = 0; i < inpoints; i++) { pointloop = (point) poolalloc(&points); /* Read the point coordinates. */ x = pointloop[0] = pointlist[coordindex++]; y = pointloop[1] = pointlist[coordindex++]; /* Read the point attributes. */ for (j = 0; j < numberofpointattribs; j++) { pointloop[2 + j] = pointattriblist[attribindex++]; } if (pointmarkerlist != (int *) NULL) { /* Read a point marker. */ setpointmark(pointloop, pointmarkerlist[i]); } else { /* If no markers are specified, they default to zero. */ setpointmark(pointloop, 0); } x = pointloop[0]; y = pointloop[1]; /* Determine the smallest and largest x and y coordinates. */ if (i == 0) { xmin = xmax = x; ymin = ymax = y; } else { xmin = (x < xmin) ? x : xmin; xmax = (x > xmax) ? x : xmax; ymin = (y < ymin) ? y : ymin; ymax = (y > ymax) ? y : ymax; } } /* Nonexistent x value used as a flag to mark circle events in sweepline */ /* Delaunay algorithm. */ xminextreme = 10 * xmin - 9 * xmax; } #endif /* TRILIBRARY */ /*****************************************************************************/ /* */ /* readholes() Read the holes, and possibly regional attributes and area */ /* constraints, from a .poly file. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void readholes(polyfile, polyfilename, hlist, holes, rlist, regions) FILE *polyfile; char *polyfilename; REAL **hlist; int *holes; REAL **rlist; int *regions; { REAL *holelist; REAL *regionlist; char inputline[INPUTLINESIZE]; char *stringptr; int index; int i; /* Read the holes. */ stringptr = readline(inputline, polyfile, polyfilename); *holes = (int) strtol (stringptr, &stringptr, 0); if (*holes > 0) { holelist = (REAL *) malloc(2 * *holes * sizeof(REAL)); *hlist = holelist; if (holelist == (REAL *) NULL) { printf("Error: Out of memory.\n"); exit(1); } for (i = 0; i < 2 * *holes; i += 2) { stringptr = readline(inputline, polyfile, polyfilename); stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Hole %d has no x coordinate.\n", firstnumber + (i >> 1)); exit(1); } else { holelist[i] = (REAL) strtod(stringptr, &stringptr); } stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Hole %d has no y coordinate.\n", firstnumber + (i >> 1)); exit(1); } else { holelist[i + 1] = (REAL) strtod(stringptr, &stringptr); } } } else { *hlist = (REAL *) NULL; } #ifndef CDT_ONLY if ((regionattrib || vararea) && !refine) { /* Read the area constraints. */ stringptr = readline(inputline, polyfile, polyfilename); *regions = (int) strtol (stringptr, &stringptr, 0); if (*regions > 0) { regionlist = (REAL *) malloc(4 * *regions * sizeof(REAL)); *rlist = regionlist; if (regionlist == (REAL *) NULL) { printf("Error: Out of memory.\n"); exit(1); } index = 0; for (i = 0; i < *regions; i++) { stringptr = readline(inputline, polyfile, polyfilename); stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Region %d has no x coordinate.\n", firstnumber + i); exit(1); } else { regionlist[index++] = (REAL) strtod(stringptr, &stringptr); } stringptr = findfield(stringptr); if (*stringptr == '\0') { printf("Error: Region %d has no y coordinate.\n", firstnumber + i); exit(1); } else { regionlist[index++] = (REAL) strtod(stringptr, &stringptr); } stringptr = findfield(stringptr); if (*stringptr == '\0') { printf( "Error: Region %d has no region attribute or area constraint.\n", firstnumber + i); exit(1); } else { regionlist[index++] = (REAL) strtod(stringptr, &stringptr); } stringptr = findfield(stringptr); if (*stringptr == '\0') { regionlist[index] = regionlist[index - 1]; } else { regionlist[index] = (REAL) strtod(stringptr, &stringptr); } index++; } } } else { /* Set `*regions' to zero to avoid an accidental free() later. */ *regions = 0; *rlist = (REAL *) NULL; } #endif /* not CDT_ONLY */ fclose(polyfile); } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* finishfile() Write the command line to the output file so the user */ /* can remember how the file was generated. Close the file. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void finishfile(outfile, argc, argv) FILE *outfile; int argc; char **argv; { int i; fprintf(outfile, "# Generated by"); for (i = 0; i < argc; i++) { fprintf(outfile, " "); fputs(argv[i], outfile); } fprintf(outfile, "\n"); fclose(outfile); } #endif /* not TRILIBRARY */ /*****************************************************************************/ /* */ /* writenodes() Number the points and write them to a .node file. */ /* */ /* To save memory, the point numbers are written over the shell markers */ /* after the points are written to a file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writenodes(pointlist, pointattriblist, pointmarkerlist) REAL **pointlist; REAL **pointattriblist; int **pointmarkerlist; #else /* not TRILIBRARY */ void writenodes(nodefilename, argc, argv) char *nodefilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY REAL *plist; REAL *palist; int *pmlist; int coordindex; int attribindex; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ point pointloop; int pointnumber; int i; #ifdef TRILIBRARY if (!quiet) { printf("Writing points.\n"); } /* Allocate memory for output points if necessary. */ if (*pointlist == (REAL *) NULL) { *pointlist = (REAL *) malloc(points.items * 2 * sizeof(REAL)); if (*pointlist == (REAL *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } /* Allocate memory for output point attributes if necessary. */ if ((nextras > 0) && (*pointattriblist == (REAL *) NULL)) { *pointattriblist = (REAL *) malloc(points.items * nextras * sizeof(REAL)); if (*pointattriblist == (REAL *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } /* Allocate memory for output point markers if necessary. */ if (!nobound && (*pointmarkerlist == (int *) NULL)) { *pointmarkerlist = (int *) malloc(points.items * sizeof(int)); if (*pointmarkerlist == (int *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } plist = *pointlist; palist = *pointattriblist; pmlist = *pointmarkerlist; coordindex = 0; attribindex = 0; #else /* not TRILIBRARY */ if (!quiet) { printf("Writing %s.\n", nodefilename); } outfile = fopen(nodefilename, "w"); if (outfile == (FILE *) NULL) { printf(" Error: Cannot create file %s.\n", nodefilename); exit(1); } /* Number of points, number of dimensions, number of point attributes, */ /* and number of boundary markers (zero or one). */ fprintf(outfile, "%ld %d %d %d\n", points.items, mesh_dim, nextras, 1 - nobound); #endif /* not TRILIBRARY */ traversalinit(&points); pointloop = pointtraverse(); pointnumber = firstnumber; while (pointloop != (point) NULL) { #ifdef TRILIBRARY /* X and y coordinates. */ plist[coordindex++] = pointloop[0]; plist[coordindex++] = pointloop[1]; /* Point attributes. */ for (i = 0; i < nextras; i++) { palist[attribindex++] = pointloop[2 + i]; } if (!nobound) { /* Copy the boundary marker. */ pmlist[pointnumber - firstnumber] = pointmark(pointloop); } #else /* not TRILIBRARY */ /* Point number, x and y coordinates. */ fprintf(outfile, "%4d %.17g %.17g", pointnumber, pointloop[0], pointloop[1]); for (i = 0; i < nextras; i++) { /* Write an attribute. */ fprintf(outfile, " %.17g", pointloop[i + 2]); } if (nobound) { fprintf(outfile, "\n"); } else { /* Write the boundary marker. */ fprintf(outfile, " %d\n", pointmark(pointloop)); } #endif /* not TRILIBRARY */ setpointmark(pointloop, pointnumber); pointloop = pointtraverse(); pointnumber++; } #ifndef TRILIBRARY finishfile(outfile, argc, argv); #endif /* not TRILIBRARY */ } /*****************************************************************************/ /* */ /* numbernodes() Number the points. */ /* */ /* Each point is assigned a marker equal to its number. */ /* */ /* Used when writenodes() is not called because no .node file is written. */ /* */ /*****************************************************************************/ void numbernodes() { point pointloop; int pointnumber; traversalinit(&points); pointloop = pointtraverse(); pointnumber = firstnumber; while (pointloop != (point) NULL) { setpointmark(pointloop, pointnumber); pointloop = pointtraverse(); pointnumber++; } } /*****************************************************************************/ /* */ /* writeelements() Write the triangles to an .ele file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writeelements(trianglelist, triangleattriblist) int **trianglelist; REAL **triangleattriblist; #else /* not TRILIBRARY */ void writeelements(elefilename, argc, argv) char *elefilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int *tlist; REAL *talist; int pointindex; int attribindex; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ struct triedge triangleloop; point p1, p2, p3; point mid1, mid2, mid3; int elementnumber; int i; #ifdef TRILIBRARY if (!quiet) { printf("Writing triangles.\n"); } /* Allocate memory for output triangles if necessary. */ if (*trianglelist == (int *) NULL) { *trianglelist = (int *) malloc(triangles.items * ((order + 1) * (order + 2) / 2) * sizeof(int)); if (*trianglelist == (int *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } /* Allocate memory for output triangle attributes if necessary. */ if ((eextras > 0) && (*triangleattriblist == (REAL *) NULL)) { *triangleattriblist = (REAL *) malloc(triangles.items * eextras * sizeof(REAL)); if (*triangleattriblist == (REAL *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } tlist = *trianglelist; talist = *triangleattriblist; pointindex = 0; attribindex = 0; #else /* not TRILIBRARY */ if (!quiet) { printf("Writing %s.\n", elefilename); } outfile = fopen(elefilename, "w"); if (outfile == (FILE *) NULL) { printf(" Error: Cannot create file %s.\n", elefilename); exit(1); } /* Number of triangles, points per triangle, attributes per triangle. */ fprintf(outfile, "%ld %d %d\n", triangles.items, (order + 1) * (order + 2) / 2, eextras); #endif /* not TRILIBRARY */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; elementnumber = firstnumber; while (triangleloop.tri != (triangle *) NULL) { org(triangleloop, p1); dest(triangleloop, p2); apex(triangleloop, p3); if (order == 1) { #ifdef TRILIBRARY tlist[pointindex++] = pointmark(p1); tlist[pointindex++] = pointmark(p2); tlist[pointindex++] = pointmark(p3); #else /* not TRILIBRARY */ /* Triangle number, indices for three points. */ fprintf(outfile, "%4d %4d %4d %4d", elementnumber, pointmark(p1), pointmark(p2), pointmark(p3)); #endif /* not TRILIBRARY */ } else { mid1 = (point) triangleloop.tri[highorderindex + 1]; mid2 = (point) triangleloop.tri[highorderindex + 2]; mid3 = (point) triangleloop.tri[highorderindex]; #ifdef TRILIBRARY tlist[pointindex++] = pointmark(p1); tlist[pointindex++] = pointmark(p2); tlist[pointindex++] = pointmark(p3); tlist[pointindex++] = pointmark(mid1); tlist[pointindex++] = pointmark(mid2); tlist[pointindex++] = pointmark(mid3); #else /* not TRILIBRARY */ /* Triangle number, indices for six points. */ fprintf(outfile, "%4d %4d %4d %4d %4d %4d %4d", elementnumber, pointmark(p1), pointmark(p2), pointmark(p3), pointmark(mid1), pointmark(mid2), pointmark(mid3)); #endif /* not TRILIBRARY */ } #ifdef TRILIBRARY for (i = 0; i < eextras; i++) { talist[attribindex++] = elemattribute(triangleloop, i); } #else /* not TRILIBRARY */ for (i = 0; i < eextras; i++) { fprintf(outfile, " %.17g", elemattribute(triangleloop, i)); } fprintf(outfile, "\n"); #endif /* not TRILIBRARY */ triangleloop.tri = triangletraverse(); elementnumber++; } #ifndef TRILIBRARY finishfile(outfile, argc, argv); #endif /* not TRILIBRARY */ } /*****************************************************************************/ /* */ /* writepoly() Write the segments and holes to a .poly file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writepoly(segmentlist, segmentmarkerlist) int **segmentlist; int **segmentmarkerlist; #else /* not TRILIBRARY */ void writepoly(polyfilename, holelist, holes, regionlist, regions, argc, argv) char *polyfilename; REAL *holelist; int holes; REAL *regionlist; int regions; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int *slist; int *smlist; int index; #else /* not TRILIBRARY */ FILE *outfile; int i; #endif /* not TRILIBRARY */ struct edge shelleloop; point endpoint1, endpoint2; int shellenumber; #ifdef TRILIBRARY if (!quiet) { printf("Writing segments.\n"); } /* Allocate memory for output segments if necessary. */ if (*segmentlist == (int *) NULL) { *segmentlist = (int *) malloc(shelles.items * 2 * sizeof(int)); if (*segmentlist == (int *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } /* Allocate memory for output segment markers if necessary. */ if (!nobound && (*segmentmarkerlist == (int *) NULL)) { *segmentmarkerlist = (int *) malloc(shelles.items * sizeof(int)); if (*segmentmarkerlist == (int *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } slist = *segmentlist; smlist = *segmentmarkerlist; index = 0; #else /* not TRILIBRARY */ if (!quiet) { printf("Writing %s.\n", polyfilename); } outfile = fopen(polyfilename, "w"); if (outfile == (FILE *) NULL) { printf(" Error: Cannot create file %s.\n", polyfilename); exit(1); } /* The zero indicates that the points are in a separate .node file. */ /* Followed by number of dimensions, number of point attributes, */ /* and number of boundary markers (zero or one). */ fprintf(outfile, "%d %d %d %d\n", 0, mesh_dim, nextras, 1 - nobound); /* Number of segments, number of boundary markers (zero or one). */ fprintf(outfile, "%ld %d\n", shelles.items, 1 - nobound); #endif /* not TRILIBRARY */ traversalinit(&shelles); shelleloop.sh = shelletraverse(); shelleloop.shorient = 0; shellenumber = firstnumber; while (shelleloop.sh != (shelle *) NULL) { sorg(shelleloop, endpoint1); sdest(shelleloop, endpoint2); #ifdef TRILIBRARY /* Copy indices of the segment's two endpoints. */ slist[index++] = pointmark(endpoint1); slist[index++] = pointmark(endpoint2); if (!nobound) { /* Copy the boundary marker. */ smlist[shellenumber - firstnumber] = mark(shelleloop); } #else /* not TRILIBRARY */ /* Segment number, indices of its two endpoints, and possibly a marker. */ if (nobound) { fprintf(outfile, "%4d %4d %4d\n", shellenumber, pointmark(endpoint1), pointmark(endpoint2)); } else { fprintf(outfile, "%4d %4d %4d %4d\n", shellenumber, pointmark(endpoint1), pointmark(endpoint2), mark(shelleloop)); } #endif /* not TRILIBRARY */ shelleloop.sh = shelletraverse(); shellenumber++; } #ifndef TRILIBRARY #ifndef CDT_ONLY fprintf(outfile, "%d\n", holes); if (holes > 0) { for (i = 0; i < holes; i++) { /* Hole number, x and y coordinates. */ fprintf(outfile, "%4d %.17g %.17g\n", firstnumber + i, holelist[2 * i], holelist[2 * i + 1]); } } if (regions > 0) { fprintf(outfile, "%d\n", regions); for (i = 0; i < regions; i++) { /* Region number, x and y coordinates, attribute, maximum area. */ fprintf(outfile, "%4d %.17g %.17g %.17g %.17g\n", firstnumber + i, regionlist[4 * i], regionlist[4 * i + 1], regionlist[4 * i + 2], regionlist[4 * i + 3]); } } #endif /* not CDT_ONLY */ finishfile(outfile, argc, argv); #endif /* not TRILIBRARY */ } /*****************************************************************************/ /* */ /* writeedges() Write the edges to a .edge file. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writeedges(edgelist, edgemarkerlist) int **edgelist; int **edgemarkerlist; #else /* not TRILIBRARY */ void writeedges(edgefilename, argc, argv) char *edgefilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int *elist; int *emlist; int index; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ struct triedge triangleloop, trisym; struct edge checkmark; point p1, p2; int edgenumber; triangle ptr; /* Temporary variable used by sym(). */ shelle sptr; /* Temporary variable used by tspivot(). */ #ifdef TRILIBRARY if (!quiet) { printf("Writing edges.\n"); } /* Allocate memory for edges if necessary. */ if (*edgelist == (int *) NULL) { *edgelist = (int *) malloc(edges * 2 * sizeof(int)); if (*edgelist == (int *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } /* Allocate memory for edge markers if necessary. */ if (!nobound && (*edgemarkerlist == (int *) NULL)) { *edgemarkerlist = (int *) malloc(edges * sizeof(int)); if (*edgemarkerlist == (int *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } elist = *edgelist; emlist = *edgemarkerlist; index = 0; #else /* not TRILIBRARY */ if (!quiet) { printf("Writing %s.\n", edgefilename); } outfile = fopen(edgefilename, "w"); if (outfile == (FILE *) NULL) { printf(" Error: Cannot create file %s.\n", edgefilename); exit(1); } /* Number of edges, number of boundary markers (zero or one). */ fprintf(outfile, "%ld %d\n", edges, 1 - nobound); #endif /* not TRILIBRARY */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); edgenumber = firstnumber; /* To loop over the set of edges, loop over all triangles, and look at */ /* the three edges of each triangle. If there isn't another triangle */ /* adjacent to the edge, operate on the edge. If there is another */ /* adjacent triangle, operate on the edge only if the current triangle */ /* has a smaller pointer than its neighbor. This way, each edge is */ /* considered only once. */ while (triangleloop.tri != (triangle *) NULL) { for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { sym(triangleloop, trisym); if ((triangleloop.tri < trisym.tri) || (trisym.tri == dummytri)) { org(triangleloop, p1); dest(triangleloop, p2); #ifdef TRILIBRARY elist[index++] = pointmark(p1); elist[index++] = pointmark(p2); #endif /* TRILIBRARY */ if (nobound) { #ifndef TRILIBRARY /* Edge number, indices of two endpoints. */ fprintf(outfile, "%4d %d %d\n", edgenumber, pointmark(p1), pointmark(p2)); #endif /* not TRILIBRARY */ } else { /* Edge number, indices of two endpoints, and a boundary marker. */ /* If there's no shell edge, the boundary marker is zero. */ if (useshelles) { tspivot(triangleloop, checkmark); if (checkmark.sh == dummysh) { #ifdef TRILIBRARY emlist[edgenumber - firstnumber] = 0; #else /* not TRILIBRARY */ fprintf(outfile, "%4d %d %d %d\n", edgenumber, pointmark(p1), pointmark(p2), 0); #endif /* not TRILIBRARY */ } else { #ifdef TRILIBRARY emlist[edgenumber - firstnumber] = mark(checkmark); #else /* not TRILIBRARY */ fprintf(outfile, "%4d %d %d %d\n", edgenumber, pointmark(p1), pointmark(p2), mark(checkmark)); #endif /* not TRILIBRARY */ } } else { #ifdef TRILIBRARY emlist[edgenumber - firstnumber] = trisym.tri == dummytri; #else /* not TRILIBRARY */ fprintf(outfile, "%4d %d %d %d\n", edgenumber, pointmark(p1), pointmark(p2), trisym.tri == dummytri); #endif /* not TRILIBRARY */ } } edgenumber++; } } triangleloop.tri = triangletraverse(); } #ifndef TRILIBRARY finishfile(outfile, argc, argv); #endif /* not TRILIBRARY */ } /*****************************************************************************/ /* */ /* writevoronoi() Write the Voronoi diagram to a .v.node and .v.edge */ /* file. */ /* */ /* The Voronoi diagram is the geometric dual of the Delaunay triangulation. */ /* Hence, the Voronoi vertices are listed by traversing the Delaunay */ /* triangles, and the Voronoi edges are listed by traversing the Delaunay */ /* edges. */ /* */ /* WARNING: In order to assign numbers to the Voronoi vertices, this */ /* procedure messes up the shell edges or the extra nodes of every */ /* element. Hence, you should call this procedure last. */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void writevoronoi(vpointlist, vpointattriblist, vpointmarkerlist, vedgelist, vedgemarkerlist, vnormlist) REAL **vpointlist; REAL **vpointattriblist; int **vpointmarkerlist; int **vedgelist; int **vedgemarkerlist; REAL **vnormlist; #else /* not TRILIBRARY */ void writevoronoi(vnodefilename, vedgefilename, argc, argv) char *vnodefilename; char *vedgefilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY REAL *plist; REAL *palist; int *elist; REAL *normlist; int coordindex; int attribindex; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ struct triedge triangleloop, trisym; point torg, tdest, tapex; REAL circumcenter[2]; REAL xi, eta; int vnodenumber, vedgenumber; int p1, p2; int i; triangle ptr; /* Temporary variable used by sym(). */ #ifdef TRILIBRARY if (!quiet) { printf("Writing Voronoi vertices.\n"); } /* Allocate memory for Voronoi vertices if necessary. */ if (*vpointlist == (REAL *) NULL) { *vpointlist = (REAL *) malloc(triangles.items * 2 * sizeof(REAL)); if (*vpointlist == (REAL *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } /* Allocate memory for Voronoi vertex attributes if necessary. */ if (*vpointattriblist == (REAL *) NULL) { *vpointattriblist = (REAL *) malloc(triangles.items * nextras * sizeof(REAL)); if (*vpointattriblist == (REAL *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } *vpointmarkerlist = (int *) NULL; plist = *vpointlist; palist = *vpointattriblist; coordindex = 0; attribindex = 0; #else /* not TRILIBRARY */ if (!quiet) { printf("Writing %s.\n", vnodefilename); } outfile = fopen(vnodefilename, "w"); if (outfile == (FILE *) NULL) { printf(" Error: Cannot create file %s.\n", vnodefilename); exit(1); } /* Number of triangles, two dimensions, number of point attributes, */ /* zero markers. */ fprintf(outfile, "%ld %d %d %d\n", triangles.items, 2, nextras, 0); #endif /* not TRILIBRARY */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; vnodenumber = firstnumber; while (triangleloop.tri != (triangle *) NULL) { org(triangleloop, torg); dest(triangleloop, tdest); apex(triangleloop, tapex); findcircumcenter(torg, tdest, tapex, circumcenter, &xi, &eta); #ifdef TRILIBRARY /* X and y coordinates. */ plist[coordindex++] = circumcenter[0]; plist[coordindex++] = circumcenter[1]; for (i = 2; i < 2 + nextras; i++) { /* Interpolate the point attributes at the circumcenter. */ palist[attribindex++] = torg[i] + xi * (tdest[i] - torg[i]) + eta * (tapex[i] - torg[i]); } #else /* not TRILIBRARY */ /* Voronoi vertex number, x and y coordinates. */ fprintf(outfile, "%4d %.17g %.17g", vnodenumber, circumcenter[0], circumcenter[1]); for (i = 2; i < 2 + nextras; i++) { /* Interpolate the point attributes at the circumcenter. */ fprintf(outfile, " %.17g", torg[i] + xi * (tdest[i] - torg[i]) + eta * (tapex[i] - torg[i])); } fprintf(outfile, "\n"); #endif /* not TRILIBRARY */ * (int *) (triangleloop.tri + 6) = vnodenumber; triangleloop.tri = triangletraverse(); vnodenumber++; } #ifndef TRILIBRARY finishfile(outfile, argc, argv); #endif /* not TRILIBRARY */ #ifdef TRILIBRARY if (!quiet) { printf("Writing Voronoi edges.\n"); } /* Allocate memory for output Voronoi edges if necessary. */ if (*vedgelist == (int *) NULL) { *vedgelist = (int *) malloc(edges * 2 * sizeof(int)); if (*vedgelist == (int *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } *vedgemarkerlist = (int *) NULL; /* Allocate memory for output Voronoi norms if necessary. */ if (*vnormlist == (REAL *) NULL) { *vnormlist = (REAL *) malloc(edges * 2 * sizeof(REAL)); if (*vnormlist == (REAL *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } elist = *vedgelist; normlist = *vnormlist; coordindex = 0; #else /* not TRILIBRARY */ if (!quiet) { printf("Writing %s.\n", vedgefilename); } outfile = fopen(vedgefilename, "w"); if (outfile == (FILE *) NULL) { printf(" Error: Cannot create file %s.\n", vedgefilename); exit(1); } /* Number of edges, zero boundary markers. */ fprintf(outfile, "%ld %d\n", edges, 0); #endif /* not TRILIBRARY */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); vedgenumber = firstnumber; /* To loop over the set of edges, loop over all triangles, and look at */ /* the three edges of each triangle. If there isn't another triangle */ /* adjacent to the edge, operate on the edge. If there is another */ /* adjacent triangle, operate on the edge only if the current triangle */ /* has a smaller pointer than its neighbor. This way, each edge is */ /* considered only once. */ while (triangleloop.tri != (triangle *) NULL) { for (triangleloop.orient = 0; triangleloop.orient < 3; triangleloop.orient++) { sym(triangleloop, trisym); if ((triangleloop.tri < trisym.tri) || (trisym.tri == dummytri)) { /* Find the number of this triangle (and Voronoi vertex). */ p1 = * (int *) (triangleloop.tri + 6); if (trisym.tri == dummytri) { org(triangleloop, torg); dest(triangleloop, tdest); #ifdef TRILIBRARY /* Copy an infinite ray. Index of one endpoint, and -1. */ elist[coordindex] = p1; normlist[coordindex++] = tdest[1] - torg[1]; elist[coordindex] = -1; normlist[coordindex++] = torg[0] - tdest[0]; #else /* not TRILIBRARY */ /* Write an infinite ray. Edge number, index of one endpoint, -1, */ /* and x and y coordinates of a vector representing the */ /* direction of the ray. */ fprintf(outfile, "%4d %d %d %.17g %.17g\n", vedgenumber, p1, -1, tdest[1] - torg[1], torg[0] - tdest[0]); #endif /* not TRILIBRARY */ } else { /* Find the number of the adjacent triangle (and Voronoi vertex). */ p2 = * (int *) (trisym.tri + 6); /* Finite edge. Write indices of two endpoints. */ #ifdef TRILIBRARY elist[coordindex] = p1; normlist[coordindex++] = 0.0; elist[coordindex] = p2; normlist[coordindex++] = 0.0; #else /* not TRILIBRARY */ fprintf(outfile, "%4d %d %d\n", vedgenumber, p1, p2); #endif /* not TRILIBRARY */ } vedgenumber++; } } triangleloop.tri = triangletraverse(); } #ifndef TRILIBRARY finishfile(outfile, argc, argv); #endif /* not TRILIBRARY */ } #ifdef TRILIBRARY void writeneighbors(neighborlist) int **neighborlist; #else /* not TRILIBRARY */ void writeneighbors(neighborfilename, argc, argv) char *neighborfilename; int argc; char **argv; #endif /* not TRILIBRARY */ { #ifdef TRILIBRARY int *nlist; int index; #else /* not TRILIBRARY */ FILE *outfile; #endif /* not TRILIBRARY */ struct triedge triangleloop, trisym; int elementnumber; int neighbor1, neighbor2, neighbor3; triangle ptr; /* Temporary variable used by sym(). */ #ifdef TRILIBRARY if (!quiet) { printf("Writing neighbors.\n"); } /* Allocate memory for neighbors if necessary. */ if (*neighborlist == (int *) NULL) { *neighborlist = (int *) malloc(triangles.items * 3 * sizeof(int)); if (*neighborlist == (int *) NULL) { printf("Error: Out of memory.\n"); exit(1); } } nlist = *neighborlist; index = 0; #else /* not TRILIBRARY */ if (!quiet) { printf("Writing %s.\n", neighborfilename); } outfile = fopen(neighborfilename, "w"); if (outfile == (FILE *) NULL) { printf(" Error: Cannot create file %s.\n", neighborfilename); exit(1); } /* Number of triangles, three edges per triangle. */ fprintf(outfile, "%ld %d\n", triangles.items, 3); #endif /* not TRILIBRARY */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; elementnumber = firstnumber; while (triangleloop.tri != (triangle *) NULL) { * (int *) (triangleloop.tri + 6) = elementnumber; triangleloop.tri = triangletraverse(); elementnumber++; } * (int *) (dummytri + 6) = -1; traversalinit(&triangles); triangleloop.tri = triangletraverse(); elementnumber = firstnumber; while (triangleloop.tri != (triangle *) NULL) { triangleloop.orient = 1; sym(triangleloop, trisym); neighbor1 = * (int *) (trisym.tri + 6); triangleloop.orient = 2; sym(triangleloop, trisym); neighbor2 = * (int *) (trisym.tri + 6); triangleloop.orient = 0; sym(triangleloop, trisym); neighbor3 = * (int *) (trisym.tri + 6); #ifdef TRILIBRARY nlist[index++] = neighbor1; nlist[index++] = neighbor2; nlist[index++] = neighbor3; #else /* not TRILIBRARY */ /* Triangle number, neighboring triangle numbers. */ fprintf(outfile, "%4d %d %d %d\n", elementnumber, neighbor1, neighbor2, neighbor3); #endif /* not TRILIBRARY */ triangleloop.tri = triangletraverse(); elementnumber++; } #ifndef TRILIBRARY finishfile(outfile, argc, argv); #endif /* TRILIBRARY */ } /*****************************************************************************/ /* */ /* writeoff() Write the triangulation to an .off file. */ /* */ /* OFF stands for the Object File Format, a format used by the Geometry */ /* Center's Geomview package. */ /* */ /*****************************************************************************/ #ifndef TRILIBRARY void writeoff(offfilename, argc, argv) char *offfilename; int argc; char **argv; { FILE *outfile; struct triedge triangleloop; point pointloop; point p1, p2, p3; if (!quiet) { printf("Writing %s.\n", offfilename); } outfile = fopen(offfilename, "w"); if (outfile == (FILE *) NULL) { printf(" Error: Cannot create file %s.\n", offfilename); exit(1); } /* Number of points, triangles, and edges. */ fprintf(outfile, "OFF\n%ld %ld %ld\n", points.items, triangles.items, edges); /* Write the points. */ traversalinit(&points); pointloop = pointtraverse(); while (pointloop != (point) NULL) { /* The "0.0" is here because the OFF format uses 3D coordinates. */ fprintf(outfile, " %.17g %.17g %.17g\n", pointloop[0], pointloop[1], 0.0); pointloop = pointtraverse(); } /* Write the triangles. */ traversalinit(&triangles); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; while (triangleloop.tri != (triangle *) NULL) { org(triangleloop, p1); dest(triangleloop, p2); apex(triangleloop, p3); /* The "3" means a three-vertex polygon. */ fprintf(outfile, " 3 %4d %4d %4d\n", pointmark(p1) - 1, pointmark(p2) - 1, pointmark(p3) - 1); triangleloop.tri = triangletraverse(); } finishfile(outfile, argc, argv); } #endif /* not TRILIBRARY */ /** **/ /** **/ /********* File I/O routines end here *********/ /*****************************************************************************/ /* */ /* quality_statistics() Print statistics about the quality of the mesh. */ /* */ /*****************************************************************************/ void quality_statistics() { struct triedge triangleloop; point p[3]; REAL cossquaretable[8]; REAL ratiotable[16]; REAL dx[3], dy[3]; REAL edgelength[3]; REAL dotproduct; REAL cossquare; REAL triarea; REAL shortest, longest; REAL trilongest2; REAL smallestarea, biggestarea; REAL triminaltitude2; REAL minaltitude; REAL triaspect2; REAL worstaspect; REAL smallestangle, biggestangle; REAL radconst, degconst; int angletable[18]; int aspecttable[16]; int aspectindex; int tendegree; int acutebiggest; int i, ii, j, k; printf("Mesh quality statistics:\n\n"); radconst = PI / 18.0; degconst = 180.0 / PI; for (i = 0; i < 8; i++) { cossquaretable[i] = cos(radconst * (REAL) (i + 1)); cossquaretable[i] = cossquaretable[i] * cossquaretable[i]; } for (i = 0; i < 18; i++) { angletable[i] = 0; } ratiotable[0] = 1.5; ratiotable[1] = 2.0; ratiotable[2] = 2.5; ratiotable[3] = 3.0; ratiotable[4] = 4.0; ratiotable[5] = 6.0; ratiotable[6] = 10.0; ratiotable[7] = 15.0; ratiotable[8] = 25.0; ratiotable[9] = 50.0; ratiotable[10] = 100.0; ratiotable[11] = 300.0; ratiotable[12] = 1000.0; ratiotable[13] = 10000.0; ratiotable[14] = 100000.0; ratiotable[15] = 0.0; for (i = 0; i < 16; i++) { aspecttable[i] = 0; } worstaspect = 0.0; minaltitude = xmax - xmin + ymax - ymin; minaltitude = minaltitude * minaltitude; shortest = minaltitude; longest = 0.0; smallestarea = minaltitude; biggestarea = 0.0; worstaspect = 0.0; smallestangle = 0.0; biggestangle = 2.0; acutebiggest = 1; traversalinit(&triangles); triangleloop.tri = triangletraverse(); triangleloop.orient = 0; while (triangleloop.tri != (triangle *) NULL) { org(triangleloop, p[0]); dest(triangleloop, p[1]); apex(triangleloop, p[2]); trilongest2 = 0.0; for (i = 0; i < 3; i++) { j = plus1mod3[i]; k = minus1mod3[i]; dx[i] = p[j][0] - p[k][0]; dy[i] = p[j][1] - p[k][1]; edgelength[i] = dx[i] * dx[i] + dy[i] * dy[i]; if (edgelength[i] > trilongest2) { trilongest2 = edgelength[i]; } if (edgelength[i] > longest) { longest = edgelength[i]; } if (edgelength[i] < shortest) { shortest = edgelength[i]; } } triarea = counterclockwise(p[0], p[1], p[2]); if (triarea < smallestarea) { smallestarea = triarea; } if (triarea > biggestarea) { biggestarea = triarea; } triminaltitude2 = triarea * triarea / trilongest2; if (triminaltitude2 < minaltitude) { minaltitude = triminaltitude2; } triaspect2 = trilongest2 / triminaltitude2; if (triaspect2 > worstaspect) { worstaspect = triaspect2; } aspectindex = 0; while ((triaspect2 > ratiotable[aspectindex] * ratiotable[aspectindex]) && (aspectindex < 15)) { aspectindex++; } aspecttable[aspectindex]++; for (i = 0; i < 3; i++) { j = plus1mod3[i]; k = minus1mod3[i]; dotproduct = dx[j] * dx[k] + dy[j] * dy[k]; cossquare = dotproduct * dotproduct / (edgelength[j] * edgelength[k]); tendegree = 8; for (ii = 7; ii >= 0; ii--) { if (cossquare > cossquaretable[ii]) { tendegree = ii; } } if (dotproduct <= 0.0) { angletable[tendegree]++; if (cossquare > smallestangle) { smallestangle = cossquare; } if (acutebiggest && (cossquare < biggestangle)) { biggestangle = cossquare; } } else { angletable[17 - tendegree]++; if (acutebiggest || (cossquare > biggestangle)) { biggestangle = cossquare; acutebiggest = 0; } } } triangleloop.tri = triangletraverse(); } shortest = sqrt(shortest); longest = sqrt(longest); minaltitude = sqrt(minaltitude); worstaspect = sqrt(worstaspect); smallestarea *= 2.0; biggestarea *= 2.0; if (smallestangle >= 1.0) { smallestangle = 0.0; } else { smallestangle = degconst * acos(sqrt(smallestangle)); } if (biggestangle >= 1.0) { biggestangle = 180.0; } else { if (acutebiggest) { biggestangle = degconst * acos(sqrt(biggestangle)); } else { biggestangle = 180.0 - degconst * acos(sqrt(biggestangle)); } } printf(" Smallest area: %16.5g | Largest area: %16.5g\n", smallestarea, biggestarea); printf(" Shortest edge: %16.5g | Longest edge: %16.5g\n", shortest, longest); printf(" Shortest altitude: %12.5g | Largest aspect ratio: %8.5g\n\n", minaltitude, worstaspect); printf(" Aspect ratio histogram:\n"); printf(" 1.1547 - %-6.6g : %8d | %6.6g - %-6.6g : %8d\n", ratiotable[0], aspecttable[0], ratiotable[7], ratiotable[8], aspecttable[8]); for (i = 1; i < 7; i++) { printf(" %6.6g - %-6.6g : %8d | %6.6g - %-6.6g : %8d\n", ratiotable[i - 1], ratiotable[i], aspecttable[i], ratiotable[i + 7], ratiotable[i + 8], aspecttable[i + 8]); } printf(" %6.6g - %-6.6g : %8d | %6.6g - : %8d\n", ratiotable[6], ratiotable[7], aspecttable[7], ratiotable[14], aspecttable[15]); printf( " (Triangle aspect ratio is longest edge divided by shortest altitude)\n\n"); printf(" Smallest angle: %15.5g | Largest angle: %15.5g\n\n", smallestangle, biggestangle); printf(" Angle histogram:\n"); for (i = 0; i < 9; i++) { printf(" %3d - %3d degrees: %8d | %3d - %3d degrees: %8d\n", i * 10, i * 10 + 10, angletable[i], i * 10 + 90, i * 10 + 100, angletable[i + 9]); } printf("\n"); } /*****************************************************************************/ /* */ /* statistics() Print all sorts of cool facts. */ /* */ /*****************************************************************************/ void statistics() { printf("\nStatistics:\n\n"); printf(" Input points: %d\n", inpoints); if (refine) { printf(" Input triangles: %d\n", inelements); } if (poly) { printf(" Input segments: %d\n", insegments); if (!refine) { printf(" Input holes: %d\n", holes); } } printf("\n Mesh points: %ld\n", points.items); printf(" Mesh triangles: %ld\n", triangles.items); printf(" Mesh edges: %ld\n", edges); if (poly || refine) { printf(" Mesh boundary edges: %ld\n", hullsize); printf(" Mesh segments: %ld\n\n", shelles.items); } else { printf(" Mesh convex hull edges: %ld\n\n", hullsize); } if (verbose) { quality_statistics(); printf("Memory allocation statistics:\n\n"); printf(" Maximum number of points: %ld\n", points.maxitems); printf(" Maximum number of triangles: %ld\n", triangles.maxitems); if (shelles.maxitems > 0) { printf(" Maximum number of segments: %ld\n", shelles.maxitems); } if (viri.maxitems > 0) { printf(" Maximum number of viri: %ld\n", viri.maxitems); } if (badsegments.maxitems > 0) { printf(" Maximum number of encroached segments: %ld\n", badsegments.maxitems); } if (badtriangles.maxitems > 0) { printf(" Maximum number of bad triangles: %ld\n", badtriangles.maxitems); } if (splaynodes.maxitems > 0) { printf(" Maximum number of splay tree nodes: %ld\n", splaynodes.maxitems); } printf(" Approximate heap memory use (bytes): %ld\n\n", points.maxitems * points.itembytes + triangles.maxitems * triangles.itembytes + shelles.maxitems * shelles.itembytes + viri.maxitems * viri.itembytes + badsegments.maxitems * badsegments.itembytes + badtriangles.maxitems * badtriangles.itembytes + splaynodes.maxitems * splaynodes.itembytes); printf("Algorithmic statistics:\n\n"); printf(" Number of incircle tests: %ld\n", incirclecount); printf(" Number of orientation tests: %ld\n", counterclockcount); if (hyperbolacount > 0) { printf(" Number of right-of-hyperbola tests: %ld\n", hyperbolacount); } if (circumcentercount > 0) { printf(" Number of circumcenter computations: %ld\n", circumcentercount); } if (circletopcount > 0) { printf(" Number of circle top computations: %ld\n", circletopcount); } printf("\n"); } } /*****************************************************************************/ /* */ /* main() or triangulate() Gosh, do everything. */ /* */ /* The sequence is roughly as follows. Many of these steps can be skipped, */ /* depending on the command line switches. */ /* */ /* - Initialize constants and parse the command line. */ /* - Read the points from a file and either */ /* - triangulate them (no -r), or */ /* - read an old mesh from files and reconstruct it (-r). */ /* - Insert the PSLG segments (-p), and possibly segments on the convex */ /* hull (-c). */ /* - Read the holes (-p), regional attributes (-pA), and regional area */ /* constraints (-pa). Carve the holes and concavities, and spread the */ /* regional attributes and area constraints. */ /* - Enforce the constraints on minimum angle (-q) and maximum area (-a). */ /* Also enforce the conforming Delaunay property (-q and -a). */ /* - Compute the number of edges in the resulting mesh. */ /* - Promote the mesh's linear triangles to higher order elements (-o). */ /* - Write the output files and print the statistics. */ /* - Check the consistency and Delaunay property of the mesh (-C). */ /* */ /*****************************************************************************/ #ifdef TRILIBRARY void triangulate(triswitches, in, out, vorout) char *triswitches; struct triangulateio *in; struct triangulateio *out; struct triangulateio *vorout; #else /* not TRILIBRARY */ int main(argc, argv) int argc; char **argv; #endif /* not TRILIBRARY */ { REAL *holearray; /* Array of holes. */ REAL *regionarray; /* Array of regional attributes and area constraints. */ #ifndef TRILIBRARY FILE *polyfile; #endif /* not TRILIBRARY */ #ifndef NO_TIMER /* Variables for timing the performance of Triangle. The types are */ /* defined in sys/time.h. */ struct timeval tv0, tv1, tv2, tv3, tv4, tv5, tv6; struct timezone tz; #endif /* NO_TIMER */ #ifndef NO_TIMER gettimeofday(&tv0, &tz); #endif /* NO_TIMER */ triangleinit(); #ifdef TRILIBRARY parsecommandline(1, &triswitches); #else /* not TRILIBRARY */ parsecommandline(argc, argv); #endif /* not TRILIBRARY */ #ifdef TRILIBRARY transfernodes(in->pointlist, in->pointattributelist, in->pointmarkerlist, in->numberofpoints, in->numberofpointattributes); #else /* not TRILIBRARY */ readnodes(innodefilename, inpolyfilename, &polyfile); #endif /* not TRILIBRARY */ #ifndef NO_TIMER if (!quiet) { gettimeofday(&tv1, &tz); } #endif /* NO_TIMER */ #ifdef CDT_ONLY hullsize = delaunay(); /* Triangulate the points. */ #else /* not CDT_ONLY */ if (refine) { /* Read and reconstruct a mesh. */ #ifdef TRILIBRARY hullsize = reconstruct(in->trianglelist, in->triangleattributelist, in->trianglearealist, in->numberoftriangles, in->numberofcorners, in->numberoftriangleattributes, in->segmentlist, in->segmentmarkerlist, in->numberofsegments); #else /* not TRILIBRARY */ hullsize = reconstruct(inelefilename, areafilename, inpolyfilename, polyfile); #endif /* not TRILIBRARY */ } else { hullsize = delaunay(); /* Triangulate the points. */ } #endif /* not CDT_ONLY */ #ifndef NO_TIMER if (!quiet) { gettimeofday(&tv2, &tz); if (refine) { printf("Mesh reconstruction"); } else { printf("Delaunay"); } printf(" milliseconds: %ld\n", 1000l * (tv2.tv_sec - tv1.tv_sec) + (tv2.tv_usec - tv1.tv_usec) / 1000l); } #endif /* NO_TIMER */ /* Ensure that no point can be mistaken for a triangular bounding */ /* box point in insertsite(). */ infpoint1 = (point) NULL; infpoint2 = (point) NULL; infpoint3 = (point) NULL; if (useshelles) { checksegments = 1; /* Segments will be introduced next. */ if (!refine) { /* Insert PSLG segments and/or convex hull segments. */ #ifdef TRILIBRARY insegments = formskeleton(in->segmentlist, in->segmentmarkerlist, in->numberofsegments); #else /* not TRILIBRARY */ insegments = formskeleton(polyfile, inpolyfilename); #endif /* not TRILIBRARY */ } } #ifndef NO_TIMER if (!quiet) { gettimeofday(&tv3, &tz); if (useshelles && !refine) { printf("Segment milliseconds: %ld\n", 1000l * (tv3.tv_sec - tv2.tv_sec) + (tv3.tv_usec - tv2.tv_usec) / 1000l); } } #endif /* NO_TIMER */ if (poly) { #ifdef TRILIBRARY holearray = in->holelist; holes = in->numberofholes; regionarray = in->regionlist; regions = in->numberofregions; #else /* not TRILIBRARY */ readholes(polyfile, inpolyfilename, &holearray, &holes, ®ionarray, ®ions); #endif /* not TRILIBRARY */ if (!refine) { /* Carve out holes and concavities. */ carveholes(holearray, holes, regionarray, regions); } } else { /* Without a PSLG, there can be no holes or regional attributes */ /* or area constraints. The following are set to zero to avoid */ /* an accidental free() later. */ holes = 0; regions = 0; } #ifndef NO_TIMER if (!quiet) { gettimeofday(&tv4, &tz); if (poly && !refine) { printf("Hole milliseconds: %ld\n", 1000l * (tv4.tv_sec - tv3.tv_sec) + (tv4.tv_usec - tv3.tv_usec) / 1000l); } } #endif /* NO_TIMER */ #ifndef CDT_ONLY if (quality) { enforcequality(); /* Enforce angle and area constraints. */ } #endif /* not CDT_ONLY */ #ifndef NO_TIMER if (!quiet) { gettimeofday(&tv5, &tz); #ifndef CDT_ONLY if (quality) { printf("Quality milliseconds: %ld\n", 1000l * (tv5.tv_sec - tv4.tv_sec) + (tv5.tv_usec - tv4.tv_usec) / 1000l); } #endif /* not CDT_ONLY */ } #endif /* NO_TIMER */ /* Compute the number of edges. */ edges = (3l * triangles.items + hullsize) / 2l; if (order > 1) { highorder(); /* Promote elements to higher polynomial order. */ } if (!quiet) { printf("\n"); } #ifdef TRILIBRARY out->numberofpoints = points.items; out->numberofpointattributes = nextras; out->numberoftriangles = triangles.items; out->numberofcorners = (order + 1) * (order + 2) / 2; out->numberoftriangleattributes = eextras; out->numberofedges = edges; if (useshelles) { out->numberofsegments = shelles.items; } else { out->numberofsegments = hullsize; } if (vorout != (struct triangulateio *) NULL) { vorout->numberofpoints = triangles.items; vorout->numberofpointattributes = nextras; vorout->numberofedges = edges; } #endif /* TRILIBRARY */ /* If not using iteration numbers, don't write a .node file if one was */ /* read, because the original one would be overwritten! */ if (nonodewritten || (noiterationnum && readnodefile)) { if (!quiet) { #ifdef TRILIBRARY printf("NOT writing points.\n"); #else /* not TRILIBRARY */ printf("NOT writing a .node file.\n"); #endif /* not TRILIBRARY */ } numbernodes(); /* We must remember to number the points. */ } else { #ifdef TRILIBRARY writenodes(&out->pointlist, &out->pointattributelist, &out->pointmarkerlist); #else /* not TRILIBRARY */ writenodes(outnodefilename, argc, argv); /* Numbers the points too. */ #endif /* TRILIBRARY */ } if (noelewritten) { if (!quiet) { #ifdef TRILIBRARY printf("NOT writing triangles.\n"); #else /* not TRILIBRARY */ printf("NOT writing an .ele file.\n"); #endif /* not TRILIBRARY */ } } else { #ifdef TRILIBRARY writeelements(&out->trianglelist, &out->triangleattributelist); #else /* not TRILIBRARY */ writeelements(outelefilename, argc, argv); #endif /* not TRILIBRARY */ } /* The -c switch (convex switch) causes a PSLG to be written */ /* even if none was read. */ if (poly || convex) { /* If not using iteration numbers, don't overwrite the .poly file. */ if (nopolywritten || noiterationnum) { if (!quiet) { #ifdef TRILIBRARY printf("NOT writing segments.\n"); #else /* not TRILIBRARY */ printf("NOT writing a .poly file.\n"); #endif /* not TRILIBRARY */ } } else { #ifdef TRILIBRARY writepoly(&out->segmentlist, &out->segmentmarkerlist); out->numberofholes = holes; out->numberofregions = regions; if (poly) { out->holelist = in->holelist; out->regionlist = in->regionlist; } else { out->holelist = (REAL *) NULL; out->regionlist = (REAL *) NULL; } #else /* not TRILIBRARY */ writepoly(outpolyfilename, holearray, holes, regionarray, regions, argc, argv); #endif /* not TRILIBRARY */ } } #ifndef TRILIBRARY #ifndef CDT_ONLY if (regions > 0) { free(regionarray); } #endif /* not CDT_ONLY */ if (holes > 0) { free(holearray); } if (geomview) { writeoff(offfilename, argc, argv); } #endif /* not TRILIBRARY */ if (edgesout) { #ifdef TRILIBRARY writeedges(&out->edgelist, &out->edgemarkerlist); #else /* not TRILIBRARY */ writeedges(edgefilename, argc, argv); #endif /* not TRILIBRARY */ } if (voronoi) { #ifdef TRILIBRARY writevoronoi(&vorout->pointlist, &vorout->pointattributelist, &vorout->pointmarkerlist, &vorout->edgelist, &vorout->edgemarkerlist, &vorout->normlist); #else /* not TRILIBRARY */ writevoronoi(vnodefilename, vedgefilename, argc, argv); #endif /* not TRILIBRARY */ } if (neighbors) { #ifdef TRILIBRARY writeneighbors(&out->neighborlist); #else /* not TRILIBRARY */ writeneighbors(neighborfilename, argc, argv); #endif /* not TRILIBRARY */ } if (!quiet) { #ifndef NO_TIMER gettimeofday(&tv6, &tz); printf("\nOutput milliseconds: %ld\n", 1000l * (tv6.tv_sec - tv5.tv_sec) + (tv6.tv_usec - tv5.tv_usec) / 1000l); printf("Total running milliseconds: %ld\n", 1000l * (tv6.tv_sec - tv0.tv_sec) + (tv6.tv_usec - tv0.tv_usec) / 1000l); #endif /* NO_TIMER */ statistics(); } #ifndef REDUCED if (docheck) { checkmesh(); checkdelaunay(); } #endif /* not REDUCED */ triangledeinit(); #ifndef TRILIBRARY return 0; #endif /* not TRILIBRARY */ }