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/** * \file geodesic.h * \brief API for the geodesic routines in C * * This an implementation in C of the geodesic algorithms described in * - C. F. F. Karney, * <a href="https://doi.org/10.1007/s00190-012-0578-z"> * Algorithms for geodesics</a>, * J. Geodesy <b>87</b>, 43--55 (2013); * DOI: <a href="https://doi.org/10.1007/s00190-012-0578-z"> * 10.1007/s00190-012-0578-z</a>; * addenda: <a href="https://geographiclib.sourceforge.io/geod-addenda.html"> * geod-addenda.html</a>. * . * The principal advantages of these algorithms over previous ones (e.g., * Vincenty, 1975) are * - accurate to round off for |<i>f</i>| &lt; 1/50; * - the solution of the inverse problem is always found; * - differential and integral properties of geodesics are computed. * * The shortest path between two points on the ellipsoid at (\e lat1, \e * lon1) and (\e lat2, \e lon2) is called the geodesic. Its length is * \e s12 and the geodesic from point 1 to point 2 has forward azimuths * \e azi1 and \e azi2 at the two end points. * * Traditionally two geodesic problems are considered: * - the direct problem -- given \e lat1, \e lon1, \e s12, and \e azi1, * determine \e lat2, \e lon2, and \e azi2. This is solved by the function * geod_direct(). * - the inverse problem -- given \e lat1, \e lon1, and \e lat2, \e lon2, * determine \e s12, \e azi1, and \e azi2. This is solved by the function * geod_inverse(). * * The ellipsoid is specified by its equatorial radius \e a (typically in * meters) and flattening \e f. The routines are accurate to round off with * double precision arithmetic provided that |<i>f</i>| &lt; 1/50; for the * WGS84 ellipsoid, the errors are less than 15 nanometers. (Reasonably * accurate results are obtained for |<i>f</i>| &lt; 1/5.) For a prolate * ellipsoid, specify \e f &lt; 0. * * The routines also calculate several other quantities of interest * - \e S12 is the area between the geodesic from point 1 to point 2 and the * equator; i.e., it is the area, measured counter-clockwise, of the * quadrilateral with corners (\e lat1,\e lon1), (0,\e lon1), (0,\e lon2), * and (\e lat2,\e lon2). * - \e m12, the reduced length of the geodesic is defined such that if * the initial azimuth is perturbed by \e dazi1 (radians) then the * second point is displaced by \e m12 \e dazi1 in the direction * perpendicular to the geodesic. On a curved surface the reduced * length obeys a symmetry relation, \e m12 + \e m21 = 0. On a flat * surface, we have \e m12 = \e s12. * - \e M12 and \e M21 are geodesic scales. If two geodesics are * parallel at point 1 and separated by a small distance \e dt, then * they are separated by a distance \e M12 \e dt at point 2. \e M21 * is defined similarly (with the geodesics being parallel to one * another at point 2). On a flat surface, we have \e M12 = \e M21 * = 1. * - \e a12 is the arc length on the auxiliary sphere. This is a * construct for converting the problem to one in spherical * trigonometry. \e a12 is measured in degrees. The spherical arc * length from one equator crossing to the next is always 180&deg;. * * If points 1, 2, and 3 lie on a single geodesic, then the following * addition rules hold: * - \e s13 = \e s12 + \e s23 * - \e a13 = \e a12 + \e a23 * - \e S13 = \e S12 + \e S23 * - \e m13 = \e m12 \e M23 + \e m23 \e M21 * - \e M13 = \e M12 \e M23 &minus; (1 &minus; \e M12 \e M21) \e * m23 / \e m12 * - \e M31 = \e M32 \e M21 &minus; (1 &minus; \e M23 \e M32) \e * m12 / \e m23 * * The shortest distance returned by the solution of the inverse problem is * (obviously) uniquely defined. However, in a few special cases there are * multiple azimuths which yield the same shortest distance. Here is a * catalog of those cases: * - \e lat1 = &minus;\e lat2 (with neither point at a pole). If \e azi1 = \e * azi2, the geodesic is unique. Otherwise there are two geodesics and the * second one is obtained by setting [\e azi1, \e azi2] &rarr; [\e azi2, \e * azi1], [\e M12, \e M21] &rarr; [\e M21, \e M12], \e S12 &rarr; &minus;\e * S12. (This occurs when the longitude difference is near &plusmn;180&deg; * for oblate ellipsoids.) * - \e lon2 = \e lon1 &plusmn; 180&deg; (with neither point at a pole). If \e * azi1 = 0&deg; or &plusmn;180&deg;, the geodesic is unique. Otherwise * there are two geodesics and the second one is obtained by setting [\e * azi1, \e azi2] &rarr; [&minus;\e azi1, &minus;\e azi2], \e S12 &rarr; * &minus;\e S12. (This occurs when \e lat2 is near &minus;\e lat1 for * prolate ellipsoids.) * - Points 1 and 2 at opposite poles. There are infinitely many geodesics * which can be generated by setting [\e azi1, \e azi2] &rarr; [\e azi1, \e * azi2] + [\e d, &minus;\e d], for arbitrary \e d. (For spheres, this * prescription applies when points 1 and 2 are antipodal.) * - \e s12 = 0 (coincident points). There are infinitely many geodesics which * can be generated by setting [\e azi1, \e azi2] &rarr; [\e azi1, \e azi2] + * [\e d, \e d], for arbitrary \e d. * * These routines are a simple transcription of the corresponding C++ classes * in <a href="https://geographiclib.sourceforge.io"> GeographicLib</a>. The * "class data" is represented by the structs geod_geodesic, geod_geodesicline, * geod_polygon and pointers to these objects are passed as initial arguments * to the member functions. Most of the internal comments have been retained. * However, in the process of transcription some documentation has been lost * and the documentation for the C++ classes, GeographicLib::Geodesic, * GeographicLib::GeodesicLine, and GeographicLib::PolygonAreaT, should be * consulted. The C++ code remains the "reference implementation". Think * twice about restructuring the internals of the C code since this may make * porting fixes from the C++ code more difficult. * * Copyright (c) Charles Karney (2012-2018) <charles@karney.com> and licensed * under the MIT/X11 License. For more information, see * https://geographiclib.sourceforge.io/ * * This library was distributed with * <a href="../index.html">GeographicLib</a> 1.49. **********************************************************************/ #if !defined(GEODESIC_H) #define GEODESIC_H 1 /** * The major version of the geodesic library. (This tracks the version of * GeographicLib.) **********************************************************************/ #define GEODESIC_VERSION_MAJOR 1 /** * The minor version of the geodesic library. (This tracks the version of * GeographicLib.) **********************************************************************/ #define GEODESIC_VERSION_MINOR 49 /** * The patch level of the geodesic library. (This tracks the version of * GeographicLib.) **********************************************************************/ #define GEODESIC_VERSION_PATCH 3 /** * Pack the version components into a single integer. Users should not rely on * this particular packing of the components of the version number; see the * documentation for GEODESIC_VERSION, below. **********************************************************************/ #define GEODESIC_VERSION_NUM(a,b,c) ((((a) * 10000 + (b)) * 100) + (c)) /** * The version of the geodesic library as a single integer, packed as MMmmmmpp * where MM is the major version, mmmm is the minor version, and pp is the * patch level. Users should not rely on this particular packing of the * components of the version number. Instead they should use a test such as * @code{.c} #if GEODESIC_VERSION >= GEODESIC_VERSION_NUM(1,40,0) ... #endif * @endcode **********************************************************************/ #define GEODESIC_VERSION \ GEODESIC_VERSION_NUM(GEODESIC_VERSION_MAJOR, \ GEODESIC_VERSION_MINOR, \ GEODESIC_VERSION_PATCH) #if defin