-*- coding: utf-8; fill-column: 72; -*- Add-ons in FlightGear ===================== This document explains how add-ons work in FlightGear. The add-on feature was first added in FlightGear 2017.3. This document describes an evolution of the framework that appeared in FlightGear 2017.4. Contents -------- 1. Terminology 2. The addon-metadata.xml file 3. Add-ons and the Property Tree a) Add-on metadata b) Subtree reserved for add-on developers 4. Resources under the add-on directory 5. Persistent storage location for add-ons 6. Add-on-specific menus and dialogs a) Add-on-specific menus b) Add-on-specific dialogs 7. How to run code after an add-on is loaded 8. Overview of the C++ API 9. Nasal API 10. Add-on development; in-sim reload of Nasal code Introduction ------------ fgfs can be passed the --addon= option, where indicates an add-on directory. Such a directory, when used as the argument of --addon, receives special treatment : 1) The add-on directory is added to the list of aircraft paths. 2) The add-on directory must contain a PropertyList file called addon-metadata.xml that gives the name of the add-on, its identifier (id), its version and possibly a few other things (see details below). 3) The add-on directory may contain a PropertyList file called addon-config.xml, in which case it will be loaded into the Property Tree at FlightGear startup, as if it were passed to the --config fgfs option. 4) The add-on directory must contain a Nasal file called addon-main.nas. This file will be loaded at startup too, and its main() function run in the namespace __addon[ADDON_ID]__, where ADDON_ID is the add-on identifier specified in the addon-metadata.xml file. The main() function is passed one argument: the addons.Addon object (a Nasal ghost, see below) corresponding to the add-on being loaded. This operation is done by $FG_ROOT/Nasal/addons.nas at the time of this writing. Also, the Property Tree is populated (under /addons) with information about registered add-ons. More details will be given below. The --addon option can be specified zero or more times; each of the operations indicated above is carried out for every specified add-on in the order given by the --addon options used: that's what we call add-on registration order, or add-on load order. In other words, add-ons are registered and loaded in the order specified by the --addon options used. 1. Terminology ~~~~~~~~~~~ add-on base path Path to a directory containing all of the add-on files. This is the path passed to the --addon fgfs option, when one wants to load the add-on in question. add-on identifier (id) A string such as org.flightgear.addons.ATCChatter or user.joe.MyGreatAddon, used to uniquely identify an add-on. The add-on identifier is declared in /addon-metadata.xml, where is the add-on base path. add-on registration When a --addon option is processed, FlightGear ensures that the add-on identifier found in the corresponding addon-metadata.xml file isn't already used by an add-on from a previous --addon option on the same command line, and stores the add-on metadata inside dedicated C++ objects. This process is called add-on registration. add-on loading The following sequence of actions: a) loading an add-on's addon-main.nas file in the namespace __addon[ADDON_ID]__ b) calling its main() function is performed later (see $FG_ROOT/Nasal/addons.nas) and called add-on loading. 2. The addon-metadata.xml file ~~~~~~~~~~~~~~~~~~~~~~~~~~~ Every add-on must have in its base directory a file called 'addon-metadata.xml'. This section explains how to write this file. Sample addon-metadata.xml file ============================== Here is an example of an addon-metadata.xml file, for a hypothetical add-on called “Flying Turtle” distributed by Joe User: FlightGear add-on metadata 1 user.joe.FlyingTurtle Flying Turtle 1.0.0rc2 Joe User optional_address@example.com http://joe.example.com/foobar/ Jane Maintainer jane@example.com https://jane.example.com/ Jane Maintainer jane@example.com https://jane.example.com/ Allow flying with new foobar powers. This add-on enables something really great involving turtles... GNU GPL version 2 or later COPYING https://www.gnu.org/licenses/old-licenses/gpl-2.0.en.html 2017.4.0 none https://example.com/quux https://example.com/quux/download https://example.com/quux/support https://example.com/quux/code-repository first tag second tag etc. General rules ============= We use the terms “field” or “node” interchangeably here to refer to nodes of the addon-metadata.xml PropertyList file (technically, a field always has a value, possibly empty, therefore fields are all leaf nodes). Leading and trailing whitespace in each field of addon-metadata.xml is removed. All other whitespace is a priori preserved (this could depend on the particular field, though). Most fields are optional. In most cases, omitting a field is the same as leaving it empty. But don't write empty tag fields, it is really too ugly. ;-) Name and id =========== Nodes: /addon/name and /addon/identifier The add-on name is the pretty form. It should not be overly long, but otherwise isn't constrained. On the other hand, the add-on identifier (id), which serves to uniquely identify an add-on: - must contain only ASCII letters (A-Z, a-z) and dots ('.'); - must be in reverse DNS style (even if the domain doesn't exist), e.g., org.flightgear.addons.ATCChatter for an add-on distributed in FGAddon, or user.joe.FlyingTurtle for Joe User's “Flying Turtle” add-on. Of course, if Joe User owns a domain name and uses it to distribute his add-on, he should put it here. Authors and maintainers ======================= Nodes: /addon/authors and /addon/maintainers Authors are people who contributed significantly to the add-on. Maintainers are people currently in charge of maintaining it. It is possible to declare any number of authors and any number of maintainers---the example above shows only one maintainer for shortness, but this is not a restriction. For each author and maintainer, you can give a name, an email address and a URL. The name must be non-empty, but the email address and URL need not be specified or may be left empty, which is equivalent. Obviously, if no email address nor URL is given for any maintainer, it is highly desirable that /addon/urls/support contains a usable URL for contacting the add-on maintainers. The data in children nodes of /addon/maintainers may refer either to real persons or to more abstract entities such as mailing-lists. In case of a real person, the corresponding URL, if specified, is expected to be the person's home page. On the other hand, if a declared “maintainer” is a mailing-list, a good use for the 'url' field is to indicate the address of a web page from which people can subscribe to the mailing-list. Short and long descriptions =========================== Nodes: /addon/short-description and /addon/long-description The short description should fit on one line (try not to exceed, say, 78 characters), and in general consist of only one sentence. The long description is essentially free-form, but only break lines when you do want a line break at this point. In other words, don't wrap lines manually in the XML file: this will be automatically done by the software displaying the add-on description, according to the particular line width it uses (which can depend on the user's screen or configuration, etc.). A single \n inside a paragraph (see footnote [1]) means a hard line break. Two \n in a row (i.e., a blank line) should be used to separate paragraphs. Example: This is a paragraph. This is the second line of the same paragraph. It can be very, very, very long and contain several sentences. This is a different paragraph. Again, don't break lines (i.e., don't press Enter) unless a particular formatting reason makes it necessary. For instance, it is okay to break lines in order to present a list of items, but not for line wrapping. Licensing terms =============== Nodes: /addon/license/designation /addon/license/file /addon/license/url The /add-on/license/designation node should describe the add-on licensing terms in a short but accurate way, if possible. If this is not practically doable, use the value “Custom”. If the add-on is distributed under several licenses, use the value “Multiple”. In all cases, make sure the licensing terms are clearly specified in other files of the add-on (typically, at least README.txt or COPYING). Values for /add-on/license/designation could be “GNU GPL version 2 or later”, “CC0 1.0 Universal”, “3-clause BSD”, etc. In most cases, the add-on should contain a file containing the full license text. Use the /add-on/license/file node to point to this file: it should contain a file path that is relative to the add-on base directory. This path must use slash separators ('/'), even if you use Windows. The /add-on/license/url node should contain a single URL if there is an official, stable URL for the license under which the add-on is distributed. The term “official” here is to be interpreted in the context of the particular license. For instance, for a GNU license (GPL2, LGPL2.1, etc.), the URL domain must be gnu.org; for a CC license (CC0 1.0 Universal, CC-BY-SA 4.0...), it must be creativecommons.org, etc. Minimum and maximum FlightGear versions ======================================= Nodes: /addon/min-FG-version and /addon/max-FG-version These two nodes are optional and may be omitted unless the add-on is known not to work with particular FlightGear versions. /addon/min-FG-version defaults to 2017.4.0 and /addon/max-FG-version to the special value 'none' (only allowed for /addon/max-FG-version). Apart from this special case, every non-empty value present in one of these two fields must be a proper FlightGear version number usable with simgear::strutils::compare_versions(), for instance '2017.4.1'. Add-on version ============== Node: /addon/version The /addon/version node gives the version of the add-on and must obey a strict syntax[2], which is a subset of what is described in PEP 440: https://www.python.org/dev/peps/pep-0440/ Valid examples are, in increasing sort order: 1.2.5.dev1 # first development release of 1.2.5 1.2.5.dev4 # fourth development release of 1.2.5 1.2.5 1.2.9 1.2.10a1.dev2 # second dev release of the first alpha release of 1.2.10 1.2.10a1 # first alpha release of 1.2.10 1.2.10b5 # fifth beta release of 1.2.10 1.2.10rc12 # twelfth release candidate for 1.2.10 1.2.10 1.3.0 2017.4.12a2 2017.4.12b1 2017.4.12rc1 2017.4.12 .devN suffixes can of course be used on beta and release candidates too, just as with the 1.2.10a1.dev2 example given above for an alpha release. Note that a development release always sorts before the corresponding non-development release (e.g., 2017.2.1b5.dev4 comes before 2017.2.1b5). Other fields ============ The other nodes of 'addon-metadata.xml' should be self-explanatory. :-) 3. Add-ons and the Property Tree ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ a) Add-on metadata ^^^^^^^^^^^^^^^ The most important metadata for each registered add-on is made accessible in the Property Tree under /addons/by-id/ADDON_ID and the property /addons/by-id/ADDON_ID/loaded can be checked or listened to, in order to determine when a particular add-on is loaded. There is also a Nasal interface to access add-on metadata in a convenient way (see below). More precisely, when an add-on is registered, its name, id, base path, version (converted to a string), loaded status (boolean) and load sequence number (int) become available in the Property Tree as /addons/by-id/ADDON_ID/{name,id,path,version,loaded,load-seq-num}. The loaded status is initially false, and set to true when the add-on loading phase is complete. There are also /addons/addon[i]/path nodes where i is 0 for the first registered add-on, 1 for the second one, etc. b) Subtree reserved for add-on developers ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For each add-on, the subtree of the global Property Tree starting at /addons/by-id/ADDON_ID/addon-devel is reserved for the specific needs of the add-on, where ADDON_ID stands for the add-on identifier. For instance, developers of the add-on whose identifier is user.joe.FlyingTurtle can store whatever they want under /addons/by-id/user.joe.FlyingTurtle/addon-devel with the assurance that doing this won't clash with properties used by the add-on framework. Example: /addons/by-id/user.joe.FlyingTurtle/addon-devel/some/property and /addons/by-id/user.joe.FlyingTurtle/addon-devel/other/property could be two properties used for the specific needs of the add-on whose identifier is user.joe.FlyingTurtle. Add-on developers should *not* use other places in the /addons subtree of the Property Tree for their custom properties. 4. Resources under the add-on directory ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Many functions in FlightGear use files that are located using the SimGear ResourceManager class. This class allows one to point to files by relative path in aircraft source files and other places. The resource manager queries a set of providers, some of which look inside aircraft paths (starting with the current aircraft), others inside scenery paths, others under $FG_ROOT, etc. The first file that matches the specified resource path is used. One of these providers only handles resource paths with a very specific syntax, which is: [addon=ADDON_ID]path/relative/to/the/addon/directory (for instance, [addon=user.joe.FlyingTurtle]images/eject-button.png) When you use such a syntax in a place that is expected to contain a resource path, it will only find the specified file under the directory of the add-on whose identifier is ADDON_ID. This allows one to specifically target a particular file inside a particular add-on, instead of crossing fingers and hoping that the specified resource won't be found by coincidence in another place such as an aircraft directory, a scenery directory or inside $FG_ROOT (such mistakes can easily happen when unrelated places use files with rather generic names, such as button.png, system.xml, etc.). The [addon=ADDON_ID]relative/path syntax is useful where resources are specified inside non-Nasal files (e.g., in property-rule configuration files, which use the XML format). For the particular case of Nasal code, there is a better way that is explained below (see “Nasal API”): the resourcePath() method of addons.Addon objects returns a string like "[addon=ADDON_ID]relative/path" when you pass it the string "relative/path". This is a good thing to use, because then your Nasal code doesn't need to know about the particular syntax for add-on-specific resources and, more interestingly, doesn't have to hardcode the add-on identifier every time you need to access a resource inside the add-on directory. 5. Persistent storage location for add-ons ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If an add-on needs to store data that persists across FlightGear sessions, it can use a specific directory tree whose path is obtained with addon.storagePath, where 'addon' is an addons.Addon instance. This corresponds to $FG_HOME/Export/Addons/ADDON_ID, however it is simpler and better to use addon.storagePath instead of hardcoding and manually assembling this path in each add-on. Since the directory is likely not to exist until the add-on creates it, the recommended usage pattern is the following: 1) Create the add-on-specific storage directory if it doesn't already exist, and optionally get its path at the same time: storageDir = addon.createStorageDir(); Typically, you'll run this in the add-on main() function (at least, early enough) if your add-on uses the storage directory. Note that there is no need to check yourself whether the directory already exists: addon.createStorageDir() does that for you. 2) At any time, you can get a path to the add-on-specific storage directory with: storageDir = addon.storagePath Accessing addon.storagePath doesn't check for the existence nor the type of $FG_HOME/Export/Addons/ADDON_ID, thus it is a bit faster than addon.createStorageDir(). Use addon.storagePath whenever you know that the directory has already been created. The features described in this section were added in FlightGear 2018.2. 6. Add-on-specific menus and dialogs ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ a) Add-on-specific menus ^^^^^^^^^^^^^^^^^^^^^ Add-ons can easily provide their own menus. If an add-on is loaded that has a file named 'addon-menubar-items.xml' in its base directory, the menus described in this file are added to the FlightGear menu bar. The file should look like this: FlightGear add-on menu bar items 1 ... ... ... In this file, each element must be a valid menu description for the FlightGear menu system (the FlightGear standard menubar in $FG_ROOT/gui/menubar.xml provides good examples). Here is an example that adds one menu with an entry for running some Nasal code and another entry for opening a custom dialog (see below for add-on-specific dialogs): true nasal dialog-show my-foobar-dialog This feature was added in FlightGear 2018.2. For older versions, one can add menus via addon-config.xml, but it's a bit hackish because of the menu index problem. b) Add-on-specific dialogs ^^^^^^^^^^^^^^^^^^^^^^^ As is the case for aircraft, add-ons can provide their own dialogs by shipping the corresponding XML files in the subfolder gui/dialogs of the add-on base directory. In other words, with a file like /gui/dialogs/my-foobar-dialog.xml starting with: my-foobar-dialog ... the following element inside 'addon-menubar-items.xml' (see above) describes a valid menu entry for showing the custom dialog. dialog-show my-foobar-dialog See $FG_ROOT/gui/dialogs to get inspiration from FlightGear's standard dialogs. This feature was added in FlightGear 2018.2. 7. How to run code after an add-on is loaded ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ You may want to set up Nasal code to be run after an add-on is loaded; here is how to do that: var addonId = "user.joe.FlyingTurtle"; var loadedFlagNode = props.globals.getNode("/addons") .getChild("by-id", 0, 1) .getChild(addonId, 0, 1) .getChild("loaded", 0, 1); if (loadedFlagNode.getBoolValue()) { logprint(5, addonId ~ " is already loaded"); } else { # Define a function to be called after the add-on is loaded var id = setlistener( loadedFlagNode, func(changedNode, listenedNode) { if (listenedNode.getBoolValue()) { removelistener(id); logprint(5, addonId ~ " is loaded"); }; }, 0, 0); } 8. Overview of the C++ API ~~~~~~~~~~~~~~~~~~~~~~~ The add-on C++ infrastructure mainly relies on the following classes: AddonManager, Addon and AddonVersion. AddonManager is used to register add-ons, which later leads to their loading. AddonManager relies on an std::map, where keys are add-on identifiers and AddonRef is SGSharedPtr at the time of this writing (changing it to another kind of smart pointer should be a mere one-line change). This map holds the metadata of each registered add-on. Accessor methods are available for: - retrieving the lists of registered and loaded add-ons; - checking if a particular add-on has already been registered or loaded; - for each add-on, obtaining an Addon instance which can be queried for its identifier, its name, identifier, version, base path, the minimum and maximum FlightGear versions it requires, its base node in the Property Tree, its order in the load sequence... The AddonVersion class handles everything about add-on version numbers: - initialization from the individual components or from a string; - conversion to a string and output to an std::ostream; - access to every component; - comparisons using the standard operators: ==, !=, <, <=, >, >=. Registering an add-on using AddonManager::registerAddon() ensures uniqueness of the add-on identifier and makes its name, identifier, base path, version (converted to a string), loaded status (boolean) and load sequence number (int) available in the Property Tree as /addons/by-id/ADDON_ID/{name,id,path,version,loaded,load-seq-num}. Note: if C++ code needs to use the add-on base path, better use AddonManager::addonBasePath() or Addon::getBasePath(), whose return values can't be tampered with by Nasal code. AddonManager::registerAddon() fails with a specific exception if the running FlightGear instance doesn't match the min-FG-version and max-FG-version requirements declared in the addon-metadata.xml file, as well as in the obvious other cases (required files such as addon-metadata.xml not found, invalid syntax in such files, etc.). The code in options.cxx (fgOptAddon()) catches such exceptions and displays the appropriate error message with SG_LOG() and fatalMessageBoxThenExit(). 9. Nasal API ~~~~~~~~~ The Nasal add-on API all lives in the 'addons' namespace. It gives Nasal code easy access to add-on metadata, for instance like this: var myAddon = addons.getAddon("user.joe.FlyingTurtle"); print(myAddon.id); print(myAddon.name); print(myAddon.version.str()); foreach (var author; myAddon.authors) { print(author.name, " ", author.email, " ", author.url); } foreach (var maintainer; myAddon.maintainers) { print(maintainer.name, " ", maintainer.email, " ", maintainer.url); } print(myAddon.shortDescription); print(myAddon.longDescription); print(myAddon.licenseDesignation); print(myAddon.licenseFile); print(myAddon.licenseUrl); print(myAddon.basePath); print(myAddon.minFGVersionRequired); print(myAddon.maxFGVersionRequired); print(myAddon.homePage); print(myAddon.downloadUrl); print(myAddon.supportUrl); print(myAddon.codeRepositoryUrl); foreach (var tag; myAddon.tags) { print(tag); } print(myAddon.loadSequenceNumber); # myAddon.node is a props.Node object for /addons/by-id/ADDON_ID print(myAddon.node.getPath()); Among other things, the Nasal add-on API allows one to get the version of any registered add-on as a ghost and reliably compare it to another instance of addons.AddonVersion: var myAddon = addons.getAddon("user.joe.FlyingTurtle"); var firstVersionOK = addons.AddonVersion.new("2.12.5rc1"); # Or alternatively: # var firstVersionOK = addons.AddonVersion.new(2, 12, 5, "rc1"); if (myAddon.version.lowerThan(firstVersionOK)) { ... Here follows the complete Nasal add-on API, at the time of this writing. All strings are encoded in UTF-8. Queries to the AddonManager: addons.isAddonRegistered(string addonId) -> bool (1 or 0) addons.registeredAddons() -> vector (in registration/load order) addons.isAddonLoaded(string addonId) -> bool (1 or 0) addons.loadedAddons() -> vector (in lexicographic order) addons.getAddon(string addonId) -> addons.Addon instance (ghost) Read-only data members (attributes) of addons.Addon objects: id the add-on identifier, in reverse DNS style (string) name the add-on “pretty name” (string) version the add-on version (instance of addons.AddonVersion, ghost) authors the add-on authors (vector of addons.Author ghosts) maintainers the add-on maintainers (vector of addons.Maintainer ghosts) shortDescription the add-on short description (string) longDescription the add-on long description (string) licenseDesignation licensing terms: "GNU GPL version 2 or later", "CC0 1.0 Universal", etc. (string) licenseFile relative, slash-separated path to a file under the add-on base directory containing the license text (string) licenseUrl stable, official URL for the add-on license text (string) basePath path to the add-on base directory (string) storagePath path to the add-on storage directory (string) This is $FG_HOME/Export/Addons/ADDON_ID. [added in FlightGear 2018.2] minFGVersionRequired minimum required FG version for the add-on (string) maxFGVersionRequired max. required FG version... or "none" (string) homePage add-on home page (string) downloadUrl add-on download URL (string) supportUrl add-on support URL (string) codeRepositoryUrl URL pointing to the development repository of the add-on (Git, Subversion, etc.; string) tags vector containing the add-on tags used to help users find add-ons (vector of strings) node base node for the add-on in the Property Tree: /addons/by-id/ADDON_ID (props.Node object) loadSequenceNumber 0 for the first registered add-on, 1 for the second one, etc. (integer) Member functions (methods) of addons.Addon objects: createStorageDir() -> string Create the add-on storage directory if it doesn't already exist (that is, $FG_HOME/Export/Addons/ADDON_ID). Return its path as a string. [added in FlightGear 2018.2] resourcePath(string relPath) -> string Return a resource path suitable for use with the simgear::ResourceManager. 'relPath' must be relative to the add-on base directory, and mustn't start with a '/'. You can use this method for instance to specify an image file for display in a Canvas widget. In you want a full path to the resource file (e.g., for troubleshooting), call resolvepath() with the return value of addons.Addon.resourcePath(). Read-only data members (attributes) of addons.AddonVersion objects: majorNumber non-negative integer minorNumber non-negative integer patchLevel non-negative integer suffix string such as "", "a1", "b2.dev45", "rc12"... Member functions (methods) of addons.AddonVersion objects: new(string version) | construct from string new(int major, int minor=0, int patchLevel=0, | construct string suffix="") | from components str() | string representation equal(addons.AddonVersion other) | nonEqual(addons.AddonVersion other) | compare to another lowerThan(addons.AddonVersion other) | addons.AddonVersion lowerThanOrEqual(addons.AddonVersion other) | instance greaterThan(addons.AddonVersion other) | greaterThanOrEqual(addons.AddonVersion other) | Read-only data members (attributes) of addons.Author objects: name author name (non-empty string) email email address of the author (string) url home page of the author (string) Read-only data members (attributes) of addons.Maintainer objects: name maintainer name (non-empty string) email email address of the maintainer (string) url home page of the maintainer, if a person; if the maintainer is a mailing-list, the URL can point to a web page from which people can subscribe to that mailing-list (string) 10. Add-on development; in-sim reload of Nasal code --------------------------------------------------- !!! WARNING: !!! The reload feature is meant for developers only, it should not be made !!! visible to end users. Unexpected side effects may occur due to reload, !!! if not implemented correctly. !!! We really don't want users to send bug reports due to reload going wrong. To make development of add-ons less time consuming, you can reload the Nasal part of your add-on without having to restart FlightGear. When an add-on is loaded, setlistener() and maketimer() wrappers are installed in the add-on's own namespace; these wrappers shadow and call themselves the standard setlistener() and maketimer() functions. The setlistener() and maketimer() wrapper functions keep track of every listener and timer they create. When the add-on is removed (e.g., as part of its reload sequence), removelistener() is called for each of these listeners, and each timer has its stop() method called. _ For the time being, you have to track any other resources outside the namespace of your add-on by yourself and clean them up in the unload() function, e.g. delete canvas or close any files you opened. You can define this unload() function in the addon-main.nas file. When your add-on is reloaded, its unload() function, if defined, will be called with one argument: the addons.Addon object (a Nasal ghost) corresponding to your add-on. unload() is run in the add-on's own namespace. In FlightGear Versions before 2020.1 the reload is triggered by setting the property /addon/by-id/ADDON_ID/reload to true (replace ADDON_ID with your particular add-on identifier). !!! Since version 2020.1 reload should be done like this: !!! fgcommand("addon-reload", props.Node.new({'id': 'ADDON_ID'})); You can add a menu item to trigger the reload easily, but it should be removed before publishing your add-on to endusers (see the above warning). Please have a look at the skeleton add-on at https://sourceforge.net/p/flightgear/fgaddon/HEAD/tree/trunk/Addons/Skeleton/ Footnotes --------- [1] \n represents end-of-line in string literals of languages such as C, C++, Python and many others. We use this convention here to represent the end-of-line character sequence in the XML data. [2] MAJOR.MINOR.PATCHLEVEL[{a|b|rc}N1][.devN2] where MAJOR, MINOR and PATCHLEVEL are non-negative integers, and N1 and N2 are positive integers.