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CoordinateSystem/CoordinateSystem.tex
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CoordinateSystem/CoordinateSystem.tex
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%
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% `coord.tex' -- describes the coordinate systems and transforms used
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% by the FG scenery management subsystem
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%
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% Written by Curtis Olson. Started July, 1997.
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%
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% $Id$
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\documentclass[12pt]{article}
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% \usepackage{times}
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% \usepackage{mathptm}
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\usepackage{anysize}
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\papersize{11in}{8.5in}
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\marginsize{1in}{1in}{1in}{1in}
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\usepackage{epsfig}
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\usepackage{setspace}
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\onehalfspacing
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\usepackage{url}
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\begin{document}
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\title{
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Flight Gear Internal Scenery Coordinate Systems and
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Representations.
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}
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\author{
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Curtis L. Olson\\
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(\texttt{curt@me.umn.edu})
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}
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\maketitle
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\section{Coordinate Systems}
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\subsection{Geocentric Coordinates}
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Geocentric coordinates are the polar coordinates centered at the
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center of the earth. Points are defined by the geocentric longitude,
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geocentric latitude, and distance from the \textit{center} of the
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earth. Note, due to the non-spherical nature of the earth, the
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geocentric latitude is not exactly the same as the traditional
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latitude you would see on a map.
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\subsection{Geodetic Coordinates}
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Geodetic coordinates are represented by longitude, latitude, and
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elevation above sea level. These are the coordinates you would read
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off a map, or see on your GPS. However, the geodetic latitude does
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not precisely correspond to the angle (in polar coordinates) from the
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center of the earth which the geocentric coordinate system reports.
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\subsection{Geocentric vs. Geodetic coordinates}
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The difference between geodetic and geocentric coordinates is subtle
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and must be understood. The problem arose because people started
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mapping the earth using latitude and longitude back when they thought
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the Earth was round (or a perfect sphere.) It's not though. It is
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more of an ellipse.
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Early map makers defined the standard \textit{geodetic} latitude as
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the angle between the local up vector and the equator. This is shown
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in figure \ref{fig:geodesy}. The point $\mathbf{B}$ marks our current
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position. The line $\mathbf{ABC}$ is tangent to the ellipse at point
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$\mathbf{B}$ and represents the local ``horizontal.'' The line
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$\mathbf{BF}$ represents the local ``up'' vector. Thus, in
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traditional map maker terms, the current latitude is the angle defined
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by $\angle \mathbf{DBF}$.
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However, as you can see from the figure, the line $\mathbf{BF}$ does
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not extend through the center of the earth. Instead, the line
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$\mathbf{BE}$ extends through the center of the earth. So in
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\textit{geocentric} coordinates, our latitude would be reported as the
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angle $\angle \mathbf{DBE}$.
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\begin{figure}[hbt]
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\centerline{
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\psfig{file=geodesy.eps}
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}
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\caption{Geocentric vs. Geodetic coordinates}
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\label{fig:geodesy}
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\end{figure}
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The LaRCsim flight model operates in geocentric coordinates
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internally, but reports the current position in both coordinate
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systems.
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\subsection{World Geodetic System 1984 (WGS 84)}
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The world is not a perfect sphere. WGS-84 defines a standard model
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for dealing with this. The LaRCsim flight model code already uses the
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WGS-84 standard in its calculations.
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For those that are interested here are a couple of URLS for more
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information:
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\noindent
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\url{http://acro.harvard.edu/SSA/BGA/wg84figs.html} \\
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\url{http://www.afmc.wpafb.af.mil/organizations/HQ-AFMC/IN/mist/dma/wgs1984.htm}
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\\
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\url{http://www.nima.mil/publications/guides/dtf/datums.html}
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To maintain simulation accuracy, the WGS-84 model should be used when
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translating geodetic coordinates (via geocentric coordinates) into the
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FG Cartesian coordinate system. The code to do this can probably be
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borrowed from the LaRCsim code. It is located in
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\texttt{ls\_geodesy.c}.
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\subsection{Cartesian Coordinates}
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Internally, all flight gear scenery is represented using a Cartesian
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coordinate system. The origin of this coordinate system is the center
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of the earth. The X axis runs along a line from the center of the
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earth out to the equator at the zero meridian. The Z axis runs along
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a line between the north and south poles with north being positive.
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The Y axis is parallel to a line running through the center of the
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earth out through the equator somewhere in the Indian Ocean. Figure
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\ref{fig:coords} shows the orientation of the X, Y, and Z axes in
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relationship to the earth.
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\begin{figure}[hbt]
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\centerline{
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\psfig{file=coord.eps}
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}
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\caption{Flight Gear Coordinate System}
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\label{fig:coords}
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\end{figure}
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\newpage
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\subsection{Converting between coordinate systems}
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Different aspects of the simulation will need to deal with positions
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represented in the various coordinate systems. Typically map data is
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presented in the geodetic coordinate system. The LaRCsim code uses
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the geocentric coordinate system. FG will use a Cartesian coordinate
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system for representing scenery internally. Potential add on items
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such as GPS's will need to know positions in the geodetic coordinate
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system, etc.
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FG will need to be able to convert positions between any of these
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coordinate systems. LaRCsim comes with code to convert back and forth
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between geodetic and geocentric coordinates. So, we only need to
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convert between geocentric and cartesian coordinates to complete the
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picture. Converting from geocentric to cartesian coordinates is done
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by using the following formula: \\
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$x = cos(lon_\mathit{geocentric}) * cos(lat_\mathit{geocentric}) *
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radius_\mathit{geocentric}$ \\
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$y = sin(lon_\mathit{geocentric}) * cos(lat_\mathit{geocentric}) *
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radius_\mathit{geocentric}$ \\
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$z = sin(lat_\mathit{geocentric}) * radius_\mathit{geocentric}$
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So far I haven't needed to convert from the cartesian coordinate
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system back into any of the polar representations so I haven't derived
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that part yet. However, it should be pretty straightforward.
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\section{Scenery Representation}
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This section is a work in progress. I am hashing out ideas as I
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write, so please feel free to send me your comments and suggestions.
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\subsection{External Scenery Representation}
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This section should deal with the external file format(s) that FG can
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use for input.
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\subsection{Internal Scenery Representation}
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This section describes how FG represents, manipulates, and
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transforms scenery internally.
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Internal, all FG scenery is defined using the coordinate system shown
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in figure \ref{fig:coords}. This means that regardless of the
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external scenery representation, FG will convert all object to it's
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internal coordinate system when it loads the external file. Note,
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when a default external FG scenery representation is created, its
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representation should probably parallel the internal FG representation
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as close as possible. This way, most necessary conversions can then
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be done offline in advance.
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\subsection{Scenery Partitioning}
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For a first stab, scenery will be partitioned along longitude and
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latitude lines. This will form trapezium shaped chunks. I'd like to
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shoot for 10km x 10km chunks. So, as we move towards the poles, the
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width in degrees of these areas will have to increase. Figure
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\ref{fig:trap} shows an exaggerated scenery area.
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\begin{figure}[hbt]
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\centerline{
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\psfig{file=trap.eps}
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}
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\caption{Scenery Partitioning Scheme}
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\label{fig:trap}
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\end{figure}
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\subsection{Reference Points}
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Each scenery area will have a reference point at the center of its
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area. This reference point (for purposes of avoiding floating point
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precision problems) defines the origin of a local coordinate system
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which is oriented the same as the global coordinate system. The only
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difference is the origin is translated from the center of the earth to
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the center of the individual areas. Figure \ref{fig:reference}
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demonstrates this.
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\begin{figure}[hbt]
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\centerline{
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\psfig{file=ref.eps}
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}
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\caption{Reference Points and Translations}
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\label{fig:reference}
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\end{figure}
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All the objects for a specific scenery area will be defined based on
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this local coordinate system. For each scenery area we define a
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vector $\vec{\mathbf{a}}$ which represents the distance from the
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center of the earth to the local coordinate system.
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\subsection{Putting the pieces of scenery together}
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To render a scene, the scenery manager will need to load all the
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nearby areas. Also, we define a vector $\vec{\mathbf{v}}$ which
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represents the distance from the center of the earth to the current
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view point. Before rendering each scenery area we translate it by
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$\vec{\mathbf{a}} - \vec{\mathbf{v}}$. This moves all the current
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scenery areas near the origin, while maintaining the relative
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positions and orientations. All these transformations are inexpensive
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to calculate and can be done easily with standard OpenGL calls.
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It is straightforward to calculate the proper view point and up vector
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so that the scenery will appear right side up when it is rendered.
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\end{document}
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25
CoordinateSystem/coord.fig
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CoordinateSystem/coord.fig
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#FIG 3.2
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Landscape
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Center
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Inches
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Letter
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100.00
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Single
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0
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1200 2
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5 1 0 1 0 7 0 0 -1 0.000 0 1 0 0 6000.000 2700.000 4200 5100 6000 5700 7800 5100
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5 1 1 1 0 7 0 0 -1 4.000 0 0 0 0 6000.000 7500.000 4200 5100 6000 4500 7800 5100
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1 3 0 1 0 7 0 0 -1 0.000 1 0.0000 6000 5100 1814 1814 6000 5100 7575 6000
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2 1 0 2 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 2.00 120.00 240.00
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6000 5100 8400 5100
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2 1 0 2 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 2.00 120.00 240.00
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6000 5100 6000 2700
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2 1 0 2 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 2.00 120.00 240.00
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6000 5100 6900 4200
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4 0 0 0 0 0 14 0.0000 4 195 1245 5925 2550 Z (North Pole)\001
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4 0 0 0 0 0 14 0.0000 4 195 1485 6975 4125 Y (Indian Ocean)\001
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4 0 0 0 0 0 14 0.0000 4 150 150 8550 5175 X\001
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4 0 0 0 0 0 14 0.0000 4 195 2025 6750 5475 (Zero Meridian, Africa)\001
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43
CoordinateSystem/geodesy.fig
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CoordinateSystem/geodesy.fig
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#FIG 3.2
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Landscape
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Center
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Inches
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Letter
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100.00
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Single
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0
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1200 2
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1 3 2 1 0 7 0 0 -1 3.000 1 0.0000 6300 4800 1800 1800 6300 4800 8100 4800
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1 1 0 2 0 7 0 0 -1 0.000 1 0.0000 6300 4800 2025 1500 6300 4800 8325 3300
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1 3 0 1 0 0 0 0 20 0.000 1 0.0000 6300 4800 44 44 6300 4800 6344 4795
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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7500 3600 8400 2700
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2 1 1 1 0 7 0 0 -1 4.000 0 0 -1 0 0 2
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6300 4800 7500 3600
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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7500 3600 8700 3600
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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7500 3600 8100 2400
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 1 2
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2 1 1.00 60.00 120.00
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2 1 1.00 60.00 120.00
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8700 4200 6000 2850
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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5625 4125 5400 3225
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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6000 5475 6300 6300
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2 1 1 1 0 7 0 0 -1 4.000 0 0 -1 0 0 2
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7500 3600 6750 5100
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4 0 0 0 0 0 14 0.0000 4 150 165 8550 4425 A\001
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4 0 0 0 0 0 14 0.0000 4 150 150 5775 2925 C\001
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4 0 0 0 0 0 14 0.0000 4 150 150 8550 3825 D\001
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4 0 0 0 0 0 14 0.0000 4 150 135 8325 3075 E\001
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4 0 0 0 0 0 14 0.0000 4 150 120 7800 2625 F\001
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4 0 0 0 0 0 12 0.0000 4 180 1125 5100 4350 Perfect Sphere\001
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4 0 0 0 0 0 12 0.0000 4 180 1980 5100 5400 True Earth ellipsoid shape\001
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4 0 0 0 0 0 14 0.0000 4 150 135 7500 3900 B\001
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27
CoordinateSystem/old-coord.fig
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27
CoordinateSystem/old-coord.fig
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@ -0,0 +1,27 @@
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#FIG 3.2
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Landscape
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Center
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Inches
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Letter
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100.00
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Single
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0
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1200 2
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5 1 0 1 0 7 0 0 -1 0.000 0 1 0 0 8962.500 6000.000 3600 5100 3525 6000 3600 6900
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5 1 0 1 0 7 0 0 -1 0.000 0 1 0 0 10762.500 6000.000 5400 5100 5325 6000 5400 6900
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5 1 0 1 0 7 0 0 -1 0.000 0 1 0 0 4500.000 -262.500 3600 5100 4500 5175 5400 5100
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5 1 0 1 0 7 0 0 -1 0.000 0 0 0 0 4500.000 1537.500 5400 6900 4500 6975 3600 6900
|
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5 1 2 1 0 7 0 0 -1 3.000 0 1 0 0 9825.000 6075.000 4500 5175 4425 6000 4500 6975
|
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5 1 2 1 0 7 0 0 -1 3.000 0 0 0 0 4425.000 637.500 5325 6000 4425 6075 3525 6000
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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6000 5175 4425 6075
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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6000 5175 8550 5175
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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6000 5175 6000 2625
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4 0 0 0 0 0 18 0.0000 4 195 180 8475 5475 X\001
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4 0 0 0 0 0 18 0.0000 4 195 180 5700 2700 Y\001
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4 0 0 0 0 0 18 0.0000 4 195 180 4500 5850 Z\001
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47
CoordinateSystem/ref.fig
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47
CoordinateSystem/ref.fig
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@ -0,0 +1,47 @@
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#FIG 3.2
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Landscape
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Center
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Inches
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Letter
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100.00
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Single
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0
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1200 2
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5 1 0 1 0 7 0 0 -1 0.000 0 1 0 0 6900.000 3304.688 5100 5700 6975 6300 8700 5700
|
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5 1 1 1 0 7 0 0 -1 4.000 0 0 0 0 6900.000 8100.000 5100 5700 6900 5100 8700 5700
|
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5 1 0 1 0 7 0 0 -1 4.000 0 0 0 0 6900.000 5175.000 5925 5100 6075 4650 6375 4350
|
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5 1 0 1 0 7 0 0 -1 4.000 0 0 0 0 8887.500 5062.500 6600 5325 6600 4800 6675 4425
|
||||
5 1 0 1 0 7 0 0 -1 4.000 0 1 0 0 6675.000 3975.000 5925 5100 6225 5250 6600 5325
|
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5 1 0 1 0 7 0 0 -1 4.000 0 1 0 0 6600.000 4087.500 6375 4350 6525 4425 6675 4425
|
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1 3 0 1 0 7 0 0 -1 0.000 1 0.0000 6900 5700 1802 1802 6900 5700 8700 5775
|
||||
2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
|
||||
2 1 1.00 60.00 120.00
|
||||
6300 4800 6300 3600
|
||||
2 1 0 2 0 7 0 0 -1 0.000 0 0 -1 1 0 2
|
||||
2 1 2.00 120.00 240.00
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6900 5700 9300 5700
|
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2 1 0 2 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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||||
2 1 2.00 120.00 240.00
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6900 5700 7800 4800
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2 1 0 2 0 7 0 0 -1 0.000 0 0 -1 1 0 2
|
||||
2 1 2.00 120.00 240.00
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||||
6900 5700 6900 3300
|
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
|
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6300 4800 7200 4800
|
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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2 1 1.00 60.00 120.00
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6300 4800 6750 4350
|
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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0 0 1.00 60.00 120.00
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6900 5700 6300 4800
|
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2 1 0 1 0 7 0 0 -1 0.000 0 0 -1 1 0 2
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0 0 1.00 60.00 120.00
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6075 4950 6300 4950
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4 0 0 0 0 0 14 0.0000 4 150 120 6825 3150 Z\001
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||||
4 0 0 0 0 0 14 0.0000 4 150 135 7875 4800 Y\001
|
||||
4 0 0 0 0 0 14 0.0000 4 150 150 9450 5775 X\001
|
||||
4 0 0 0 0 0 12 0.0000 4 135 135 7275 4875 X\001
|
||||
4 0 0 0 0 0 12 0.0000 4 135 135 6675 4275 Y\001
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||||
4 0 0 0 0 0 12 0.0000 4 135 120 6300 3525 Z\001
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||||
4 0 0 0 0 0 12 0.0000 4 90 90 6150 5100 a\001
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16
CoordinateSystem/trap.fig
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16
CoordinateSystem/trap.fig
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|
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@ -0,0 +1,16 @@
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#FIG 3.2
|
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Landscape
|
||||
Center
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||||
Inches
|
||||
Letter
|
||||
100.00
|
||||
Single
|
||||
0
|
||||
1200 2
|
||||
5 1 0 1 0 7 0 0 -1 0.000 0 1 0 0 6900.000 3304.688 5100 5700 6975 6300 8700 5700
|
||||
5 1 1 1 0 7 0 0 -1 4.000 0 0 0 0 6900.000 8100.000 5100 5700 6900 5100 8700 5700
|
||||
5 1 0 1 0 7 0 0 -1 4.000 0 0 0 0 6900.000 5175.000 5925 5100 6075 4650 6375 4350
|
||||
5 1 0 1 0 7 0 0 -1 4.000 0 0 0 0 8887.500 5062.500 6600 5325 6600 4800 6675 4425
|
||||
5 1 0 1 0 7 0 0 -1 4.000 0 1 0 0 6675.000 3975.000 5925 5100 6225 5250 6600 5325
|
||||
5 1 0 1 0 7 0 0 -1 4.000 0 1 0 0 6600.000 4087.500 6375 4350 6525 4425 6675 4425
|
||||
1 3 0 1 0 7 0 0 -1 0.000 1 0.0000 6900 5700 1802 1802 6900 5700 8700 5775
|
||||
Loading…
Reference in a new issue