EVOLUTION-MANAGER

Edit File: geodesic.h

/**
 * \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 87, 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 |f| &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 |f| &lt; 1/50; for the
 * WGS84 ellipsoid, the errors are less than 15 nanometers. (Reasonably
 * accurate results are obtained for |f| &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 defined(__cplusplus)
extern "C" {
#endif

/**
 * The struct containing information about the ellipsoid. This must be
 * initialized by geod_init() before use.
 **********************************************************************/
 struct geod_geodesic {
 double a; /**< the equatorial radius */
 double f; /**< the flattening */
 /**< @cond SKIP */
 double f1, e2, ep2, n, b, c2, etol2;
 double A3x[6], C3x[15], C4x[21];
 /**< @endcond */
 };

/**
 * The struct containing information about a single geodesic. This must be
 * initialized by geod_lineinit(), geod_directline(), geod_gendirectline(),
 * or geod_inverseline() before use.
 **********************************************************************/
 struct geod_geodesicline {
 double lat1; /**< the starting latitude */
 double lon1; /**< the starting longitude */
 double azi1; /**< the starting azimuth */
 double a; /**< the equatorial radius */
 double f; /**< the flattening */
 double salp1; /**< sine of \e azi1 */
 double calp1; /**< cosine of \e azi1 */
 double a13; /**< arc length to reference point */
 double s13; /**< distance to reference point */
 /**< @cond SKIP */
 double b, c2, f1, salp0, calp0, k2,
 ssig1, csig1, dn1, stau1, ctau1, somg1, comg1,
 A1m1, A2m1, A3c, B11, B21, B31, A4, B41;
 double C1a[6+1], C1pa[6+1], C2a[6+1], C3a[6], C4a[6];
 /**< @endcond */
 unsigned caps; /**< the capabilities */
 };

/**
 * The struct for accumulating information about a geodesic polygon. This is
 * used for computing the perimeter and area of a polygon. This must be
 * initialized by geod_polygon_init() before use.
 **********************************************************************/
 struct geod_polygon {
 double lat; /**< the current latitude */
 double lon; /**< the current longitude */
 /**< @cond SKIP */
 double lat0;
 double lon0;
 double A[2];
 double P[2];
 int polyline;
 int crossings;
 /**< @endcond */
 unsigned num; /**< the number of points so far */
 };

/**
   * Initialize a geod_geodesic object.
   *
   * @param[out] g a pointer to the object to be initialized.
   * @param[in] a the equatorial radius (meters).
   * @param[in] f the flattening.
   **********************************************************************/
  void geod_init(struct geod_geodesic* g, double a, double f);

/**
   * Solve the direct geodesic problem.
   *
   * @param[in] g a pointer to the geod_geodesic object specifying the
   *   ellipsoid.
   * @param[in] lat1 latitude of point 1 (degrees).
   * @param[in] lon1 longitude of point 1 (degrees).
   * @param[in] azi1 azimuth at point 1 (degrees).
   * @param[in] s12 distance from point 1 to point 2 (meters); it can be
   *   negative.
   * @param[out] plat2 pointer to the latitude of point 2 (degrees).
   * @param[out] plon2 pointer to the longitude of point 2 (degrees).
   * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees).
   *
   * \e g must have been initialized with a call to geod_init().  \e lat1
   * should be in the range [&minus;90&deg;, 90&deg;].  The values of \e lon2
   * and \e azi2 returned are in the range [&minus;180&deg;, 180&deg;].  Any of
   * the "return" arguments \e plat2, etc., may be replaced by 0, if you do not
   * need some quantities computed.
   *
   * If either point is at a pole, the azimuth is defined by keeping the
   * longitude fixed, writing \e lat = &plusmn;(90&deg; &minus; &epsilon;), and
   * taking the limit &epsilon; &rarr; 0+.  An arc length greater that 180&deg;
   * signifies a geodesic which is not a shortest path.  (For a prolate
   * ellipsoid, an additional condition is necessary for a shortest path: the
   * longitudinal extent must not exceed of 180&deg;.)
   *
   * Example, determine the point 10000 km NE of JFK:
   @code{.c}
   struct geod_geodesic g;
   double lat, lon;
   geod_init(&g, 6378137, 1/298.257223563);
   geod_direct(&g, 40.64, -73.78, 45.0, 10e6, &lat, &lon, 0);
   printf("%.5f %.5f\n", lat, lon);
   @endcode
   **********************************************************************/
  void geod_direct(const struct geod_geodesic* g,
                   double lat1, double lon1, double azi1, double s12,
                   double* plat2, double* plon2, double* pazi2);

/**
 * The general direct geodesic problem.
 *
 * @param[in] g a pointer to the geod_geodesic object specifying the
 * ellipsoid.
 * @param[in] lat1 latitude of point 1 (degrees).
 * @param[in] lon1 longitude of point 1 (degrees).
 * @param[in] azi1 azimuth at point 1 (degrees).
 * @param[in] flags bitor'ed combination of geod_flags(); \e flags &
 * GEOD_ARCMODE determines the meaning of \e s12_a12 and \e flags &
 * GEOD_LONG_UNROLL "unrolls" \e lon2.
 * @param[in] s12_a12 if \e flags & GEOD_ARCMODE is 0, this is the distance
 * from point 1 to point 2 (meters); otherwise it is the arc length
 * from point 1 to point 2 (degrees); it can be negative.
 * @param[out] plat2 pointer to the latitude of point 2 (degrees).
 * @param[out] plon2 pointer to the longitude of point 2 (degrees).
 * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees).
 * @param[out] ps12 pointer to the distance from point 1 to point 2
 * (meters).
 * @param[out] pm12 pointer to the reduced length of geodesic (meters).
 * @param[out] pM12 pointer to the geodesic scale of point 2 relative to
 * point 1 (dimensionless).
 * @param[out] pM21 pointer to the geodesic scale of point 1 relative to
 * point 2 (dimensionless).
 * @param[out] pS12 pointer to the area under the geodesic
 * (meters2).
 * @return \e a12 arc length from point 1 to point 2 (degrees).
 *
 * \e g must have been initialized with a call to geod_init(). \e lat1
 * should be in the range [&minus;90&deg;, 90&deg;]. The function value \e
 * a12 equals \e s12_a12 if \e flags & GEOD_ARCMODE. Any of the "return"
 * arguments, \e plat2, etc., may be replaced by 0, if you do not need some
 * quantities computed.
 *
 * With \e flags & GEOD_LONG_UNROLL bit set, the longitude is "unrolled" so
 * that the quantity \e lon2 &minus; \e lon1 indicates how many times and in
 * what sense the geodesic encircles the ellipsoid.
 **********************************************************************/
 double geod_gendirect(const struct geod_geodesic* g,
 double lat1, double lon1, double azi1,
 unsigned flags, double s12_a12,
 double* plat2, double* plon2, double* pazi2,
 double* ps12, double* pm12, double* pM12, double* pM21,
 double* pS12);

/**
   * Solve the inverse geodesic problem.
   *
   * @param[in] g a pointer to the geod_geodesic object specifying the
   *   ellipsoid.
   * @param[in] lat1 latitude of point 1 (degrees).
   * @param[in] lon1 longitude of point 1 (degrees).
   * @param[in] lat2 latitude of point 2 (degrees).
   * @param[in] lon2 longitude of point 2 (degrees).
   * @param[out] ps12 pointer to the distance from point 1 to point 2
   *   (meters).
   * @param[out] pazi1 pointer to the azimuth at point 1 (degrees).
   * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees).
   *
   * \e g must have been initialized with a call to geod_init().  \e lat1 and
   * \e lat2 should be in the range [&minus;90&deg;, 90&deg;].  The values of
   * \e azi1 and \e azi2 returned are in the range [&minus;180&deg;, 180&deg;].
   * Any of the "return" arguments, \e ps12, etc., may be replaced by 0, if you
   * do not need some quantities computed.
   *
   * If either point is at a pole, the azimuth is defined by keeping the
   * longitude fixed, writing \e lat = &plusmn;(90&deg; &minus; &epsilon;), and
   * taking the limit &epsilon; &rarr; 0+.
   *
   * The solution to the inverse problem is found using Newton's method.  If
   * this fails to converge (this is very unlikely in geodetic applications
   * but does occur for very eccentric ellipsoids), then the bisection method
   * is used to refine the solution.
   *
   * Example, determine the distance between JFK and Singapore Changi Airport:
   @code{.c}
   struct geod_geodesic g;
   double s12;
   geod_init(&g, 6378137, 1/298.257223563);
   geod_inverse(&g, 40.64, -73.78, 1.36, 103.99, &s12, 0, 0);
   printf("%.3f\n", s12);
   @endcode
   **********************************************************************/
  void geod_inverse(const struct geod_geodesic* g,
                    double lat1, double lon1, double lat2, double lon2,
                    double* ps12, double* pazi1, double* pazi2);

/**
 * The general inverse geodesic calculation.
 *
 * @param[in] g a pointer to the geod_geodesic object specifying the
 * ellipsoid.
 * @param[in] lat1 latitude of point 1 (degrees).
 * @param[in] lon1 longitude of point 1 (degrees).
 * @param[in] lat2 latitude of point 2 (degrees).
 * @param[in] lon2 longitude of point 2 (degrees).
 * @param[out] ps12 pointer to the distance from point 1 to point 2
 * (meters).
 * @param[out] pazi1 pointer to the azimuth at point 1 (degrees).
 * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees).
 * @param[out] pm12 pointer to the reduced length of geodesic (meters).
 * @param[out] pM12 pointer to the geodesic scale of point 2 relative to
 * point 1 (dimensionless).
 * @param[out] pM21 pointer to the geodesic scale of point 1 relative to
 * point 2 (dimensionless).
 * @param[out] pS12 pointer to the area under the geodesic
 * (meters2).
 * @return \e a12 arc length from point 1 to point 2 (degrees).
 *
 * \e g must have been initialized with a call to geod_init(). \e lat1 and
 * \e lat2 should be in the range [&minus;90&deg;, 90&deg;]. Any of the
 * "return" arguments \e ps12, etc., may be replaced by 0, if you do not need
 * some quantities computed.
 **********************************************************************/
 double geod_geninverse(const struct geod_geodesic* g,
 double lat1, double lon1, double lat2, double lon2,
 double* ps12, double* pazi1, double* pazi2,
 double* pm12, double* pM12, double* pM21,
 double* pS12);

/**
   * Initialize a geod_geodesicline object.
   *
   * @param[out] l a pointer to the object to be initialized.
   * @param[in] g a pointer to the geod_geodesic object specifying the
   *   ellipsoid.
   * @param[in] lat1 latitude of point 1 (degrees).
   * @param[in] lon1 longitude of point 1 (degrees).
   * @param[in] azi1 azimuth at point 1 (degrees).
   * @param[in] caps bitor'ed combination of geod_mask() values specifying the
   *   capabilities the geod_geodesicline object should possess, i.e., which
   *   quantities can be returned in calls to geod_position() and
   *   geod_genposition().
   *
   * \e g must have been initialized with a call to geod_init().  \e lat1
   * should be in the range [&minus;90&deg;, 90&deg;].
   *
   * The geod_mask values are [see geod_mask()]:
   * - \e caps |= GEOD_LATITUDE for the latitude \e lat2; this is
   *   added automatically,
   * - \e caps |= GEOD_LONGITUDE for the latitude \e lon2,
   * - \e caps |= GEOD_AZIMUTH for the latitude \e azi2; this is
   *   added automatically,
   * - \e caps |= GEOD_DISTANCE for the distance \e s12,
   * - \e caps |= GEOD_REDUCEDLENGTH for the reduced length \e m12,
   * - \e caps |= GEOD_GEODESICSCALE for the geodesic scales \e M12
   *   and \e M21,
   * - \e caps |= GEOD_AREA for the area \e S12,
   * - \e caps |= GEOD_DISTANCE_IN permits the length of the
   *   geodesic to be given in terms of \e s12; without this capability the
   *   length can only be specified in terms of arc length.
   * .
   * A value of \e caps = 0 is treated as GEOD_LATITUDE | GEOD_LONGITUDE |
   * GEOD_AZIMUTH | GEOD_DISTANCE_IN (to support the solution of the "standard"
   * direct problem).
   *
   * When initialized by this function, point 3 is undefined (l->s13 = l->a13 =
   * NaN).
   **********************************************************************/
  void geod_lineinit(struct geod_geodesicline* l,
                     const struct geod_geodesic* g,
                     double lat1, double lon1, double azi1, unsigned caps);

/**
   * Initialize a geod_geodesicline object in terms of the direct geodesic
   * problem.
   *
   * @param[out] l a pointer to the object to be initialized.
   * @param[in] g a pointer to the geod_geodesic object specifying the
   *   ellipsoid.
   * @param[in] lat1 latitude of point 1 (degrees).
   * @param[in] lon1 longitude of point 1 (degrees).
   * @param[in] azi1 azimuth at point 1 (degrees).
   * @param[in] s12 distance from point 1 to point 2 (meters); it can be
   *   negative.
   * @param[in] caps bitor'ed combination of geod_mask() values specifying the
   *   capabilities the geod_geodesicline object should possess, i.e., which
   *   quantities can be returned in calls to geod_position() and
   *   geod_genposition().
   *
   * This function sets point 3 of the geod_geodesicline to correspond to point
   * 2 of the direct geodesic problem.  See geod_lineinit() for more
   * information.
   **********************************************************************/
  void geod_directline(struct geod_geodesicline* l,
                       const struct geod_geodesic* g,
                       double lat1, double lon1, double azi1, double s12,
                       unsigned caps);

/**
   * Initialize a geod_geodesicline object in terms of the direct geodesic
   * problem spacified in terms of either distance or arc length.
   *
   * @param[out] l a pointer to the object to be initialized.
   * @param[in] g a pointer to the geod_geodesic object specifying the
   *   ellipsoid.
   * @param[in] lat1 latitude of point 1 (degrees).
   * @param[in] lon1 longitude of point 1 (degrees).
   * @param[in] azi1 azimuth at point 1 (degrees).
   * @param[in] flags either GEOD_NOFLAGS or GEOD_ARCMODE to determining the
   *   meaning of the \e s12_a12.
   * @param[in] s12_a12 if \e flags = GEOD_NOFLAGS, this is the distance
   *   from point 1 to point 2 (meters); if \e flags = GEOD_ARCMODE, it is
   *   the arc length from point 1 to point 2 (degrees); it can be
   *   negative.
   * @param[in] caps bitor'ed combination of geod_mask() values specifying the
   *   capabilities the geod_geodesicline object should possess, i.e., which
   *   quantities can be returned in calls to geod_position() and
   *   geod_genposition().
   *
   * This function sets point 3 of the geod_geodesicline to correspond to point
   * 2 of the direct geodesic problem.  See geod_lineinit() for more
   * information.
   **********************************************************************/
  void geod_gendirectline(struct geod_geodesicline* l,
                          const struct geod_geodesic* g,
                          double lat1, double lon1, double azi1,
                          unsigned flags, double s12_a12,
                          unsigned caps);

/**
   * Initialize a geod_geodesicline object in terms of the inverse geodesic
   * problem.
   *
   * @param[out] l a pointer to the object to be initialized.
   * @param[in] g a pointer to the geod_geodesic object specifying the
   *   ellipsoid.
   * @param[in] lat1 latitude of point 1 (degrees).
   * @param[in] lon1 longitude of point 1 (degrees).
   * @param[in] lat2 latitude of point 2 (degrees).
   * @param[in] lon2 longitude of point 2 (degrees).
   * @param[in] caps bitor'ed combination of geod_mask() values specifying the
   *   capabilities the geod_geodesicline object should possess, i.e., which
   *   quantities can be returned in calls to geod_position() and
   *   geod_genposition().
   *
   * This function sets point 3 of the geod_geodesicline to correspond to point
   * 2 of the inverse geodesic problem.  See geod_lineinit() for more
   * information.
   **********************************************************************/
  void geod_inverseline(struct geod_geodesicline* l,
                       const struct geod_geodesic* g,
                       double lat1, double lon1, double lat2, double lon2,
                       unsigned caps);

/**
 * Compute the position along a geod_geodesicline.
 *
 * @param[in] l a pointer to the geod_geodesicline object specifying the
 * geodesic line.
 * @param[in] s12 distance from point 1 to point 2 (meters); it can be
 * negative.
 * @param[out] plat2 pointer to the latitude of point 2 (degrees).
 * @param[out] plon2 pointer to the longitude of point 2 (degrees); requires
 * that \e l was initialized with \e caps |= GEOD_LONGITUDE.
 * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees).
 *
 * \e l must have been initialized with a call, e.g., to geod_lineinit(),
 * with \e caps |= GEOD_DISTANCE_IN. The values of \e lon2 and \e azi2
 * returned are in the range [&minus;180&deg;, 180&deg;]. Any of the
 * "return" arguments \e plat2, etc., may be replaced by 0, if you do not
 * need some quantities computed.
 *
 * Example, compute way points between JFK and Singapore Changi Airport
 * the "obvious" way using geod_direct():
 @code{.c}
 struct geod_geodesic g;
 double s12, azi1, lat[101],lon[101];
 int i;
 geod_init(&g, 6378137, 1/298.257223563);
 geod_inverse(&g, 40.64, -73.78, 1.36, 103.99, &s12, &azi1, 0);
 for (i = 0; i < 101; ++i) {
 geod_direct(&g, 40.64, -73.78, azi1, i * s12 * 0.01, lat + i, lon + i, 0);
 printf("%.5f %.5f\n", lat[i], lon[i]);
 }
 @endcode
 * A faster way using geod_position():
 @code{.c}
 struct geod_geodesic g;
 struct geod_geodesicline l;
 double lat[101],lon[101];
 int i;
 geod_init(&g, 6378137, 1/298.257223563);
 geod_inverseline(&l, &g, 40.64, -73.78, 1.36, 103.99, 0);
 for (i = 0; i <= 100; ++i) {
 geod_position(&l, i * l.s13 * 0.01, lat + i, lon + i, 0);
 printf("%.5f %.5f\n", lat[i], lon[i]);
 }
 @endcode
 **********************************************************************/
 void geod_position(const struct geod_geodesicline* l, double s12,
 double* plat2, double* plon2, double* pazi2);

/**
 * The general position function.
 *
 * @param[in] l a pointer to the geod_geodesicline object specifying the
 * geodesic line.
 * @param[in] flags bitor'ed combination of geod_flags(); \e flags &
 * GEOD_ARCMODE determines the meaning of \e s12_a12 and \e flags &
 * GEOD_LONG_UNROLL "unrolls" \e lon2; if \e flags & GEOD_ARCMODE is 0,
 * then \e l must have been initialized with \e caps |= GEOD_DISTANCE_IN.
 * @param[in] s12_a12 if \e flags & GEOD_ARCMODE is 0, this is the
 * distance from point 1 to point 2 (meters); otherwise it is the
 * arc length from point 1 to point 2 (degrees); it can be
 * negative.
 * @param[out] plat2 pointer to the latitude of point 2 (degrees).
 * @param[out] plon2 pointer to the longitude of point 2 (degrees); requires
 * that \e l was initialized with \e caps |= GEOD_LONGITUDE.
 * @param[out] pazi2 pointer to the (forward) azimuth at point 2 (degrees).
 * @param[out] ps12 pointer to the distance from point 1 to point 2
 * (meters); requires that \e l was initialized with \e caps |=
 * GEOD_DISTANCE.
 * @param[out] pm12 pointer to the reduced length of geodesic (meters);
 * requires that \e l was initialized with \e caps |= GEOD_REDUCEDLENGTH.
 * @param[out] pM12 pointer to the geodesic scale of point 2 relative to
 * point 1 (dimensionless); requires that \e l was initialized with \e caps
 * |= GEOD_GEODESICSCALE.
 * @param[out] pM21 pointer to the geodesic scale of point 1 relative to
 * point 2 (dimensionless); requires that \e l was initialized with \e caps
 * |= GEOD_GEODESICSCALE.
 * @param[out] pS12 pointer to the area under the geodesic
 * (meters2); requires that \e l was initialized with \e caps |=
 * GEOD_AREA.
 * @return \e a12 arc length from point 1 to point 2 (degrees).
 *
 * \e l must have been initialized with a call to geod_lineinit() with \e
 * caps |= GEOD_DISTANCE_IN. The value \e azi2 returned is in the range
 * [&minus;180&deg;, 180&deg;]. Any of the "return" arguments \e plat2,
 * etc., may be replaced by 0, if you do not need some quantities
 * computed. Requesting a value which \e l is not capable of computing
 * is not an error; the corresponding argument will not be altered.
 *
 * With \e flags & GEOD_LONG_UNROLL bit set, the longitude is "unrolled" so
 * that the quantity \e lon2 &minus; \e lon1 indicates how many times and in
 * what sense the geodesic encircles the ellipsoid.
 *
 * Example, compute way points between JFK and Singapore Changi Airport using
 * geod_genposition(). In this example, the points are evenly space in arc
 * length (and so only approximately equally spaced in distance). This is
 * faster than using geod_position() and would be appropriate if drawing the
 * path on a map.
 @code{.c}
 struct geod_geodesic g;
 struct geod_geodesicline l;
 double lat[101], lon[101];
 int i;
 geod_init(&g, 6378137, 1/298.257223563);
 geod_inverseline(&l, &g, 40.64, -73.78, 1.36, 103.99,
 GEOD_LATITUDE | GEOD_LONGITUDE);
 for (i = 0; i <= 100; ++i) {
 geod_genposition(&l, GEOD_ARCMODE, i * l.a13 * 0.01,
 lat + i, lon + i, 0, 0, 0, 0, 0, 0);
 printf("%.5f %.5f\n", lat[i], lon[i]);
 }
 @endcode
 **********************************************************************/
 double geod_genposition(const struct geod_geodesicline* l,
 unsigned flags, double s12_a12,
 double* plat2, double* plon2, double* pazi2,
 double* ps12, double* pm12,
 double* pM12, double* pM21,
 double* pS12);

/**
   * Specify position of point 3 in terms of distance.
   *
   * @param[in,out] l a pointer to the geod_geodesicline object.
   * @param[in] s13 the distance from point 1 to point 3 (meters); it
   *   can be negative.
   *
   * This is only useful if the geod_geodesicline object has been constructed
   * with \e caps |= GEOD_DISTANCE_IN.
   **********************************************************************/
  void geod_setdistance(struct geod_geodesicline* l, double s13);

/**
   * Specify position of point 3 in terms of either distance or arc length.
   *
   * @param[in,out] l a pointer to the geod_geodesicline object.
   * @param[in] flags either GEOD_NOFLAGS or GEOD_ARCMODE to determining the
   *   meaning of the \e s13_a13.
   * @param[in] s13_a13 if \e flags = GEOD_NOFLAGS, this is the distance
   *   from point 1 to point 3 (meters); if \e flags = GEOD_ARCMODE, it is
   *   the arc length from point 1 to point 3 (degrees); it can be
   *   negative.
   *
   * If flags = GEOD_NOFLAGS, this calls geod_setdistance().  If flags =
   * GEOD_ARCMODE, the \e s13 is only set if the geod_geodesicline object has
   * been constructed with \e caps |= GEOD_DISTANCE.
   **********************************************************************/
  void geod_gensetdistance(struct geod_geodesicline* l,
                           unsigned flags, double s13_a13);

/**
   * Initialize a geod_polygon object.
   *
   * @param[out] p a pointer to the object to be initialized.
   * @param[in] polylinep non-zero if a polyline instead of a polygon.
   *
   * If \e polylinep is zero, then the sequence of vertices and edges added by
   * geod_polygon_addpoint() and geod_polygon_addedge() define a polygon and
   * the perimeter and area are returned by geod_polygon_compute().  If \e
   * polylinep is non-zero, then the vertices and edges define a polyline and
   * only the perimeter is returned by geod_polygon_compute().
   *
   * The area and perimeter are accumulated at two times the standard floating
   * point precision to guard against the loss of accuracy with many-sided
   * polygons.  At any point you can ask for the perimeter and area so far.
   *
   * An example of the use of this function is given in the documentation for
   * geod_polygon_compute().
   **********************************************************************/
  void geod_polygon_init(struct geod_polygon* p, int polylinep);

/**
   * Clear the polygon, allowing a new polygon to be started.
   *
   * @param[in,out] p a pointer to the object to be cleared.
   **********************************************************************/
  void geod_polygon_clear(struct geod_polygon* p);

/**
   * Add a point to the polygon or polyline.
   *
   * @param[in] g a pointer to the geod_geodesic object specifying the
   *   ellipsoid.
   * @param[in,out] p a pointer to the geod_polygon object specifying the
   *   polygon.
   * @param[in] lat the latitude of the point (degrees).
   * @param[in] lon the longitude of the point (degrees).
   *
   * \e g and \e p must have been initialized with calls to geod_init() and
   * geod_polygon_init(), respectively.  The same \e g must be used for all the
   * points and edges in a polygon.  \e lat should be in the range
   * [&minus;90&deg;, 90&deg;].
   *
   * An example of the use of this function is given in the documentation for
   * geod_polygon_compute().
   **********************************************************************/
  void geod_polygon_addpoint(const struct geod_geodesic* g,
                             struct geod_polygon* p,
                             double lat, double lon);

/**
   * Add an edge to the polygon or polyline.
   *
   * @param[in] g a pointer to the geod_geodesic object specifying the
   *   ellipsoid.
   * @param[in,out] p a pointer to the geod_polygon object specifying the
   *   polygon.
   * @param[in] azi azimuth at current point (degrees).
   * @param[in] s distance from current point to next point (meters).
   *
   * \e g and \e p must have been initialized with calls to geod_init() and
   * geod_polygon_init(), respectively.  The same \e g must be used for all the
   * points and edges in a polygon.  This does nothing if no points have been
   * added yet.  The \e lat and \e lon fields of \e p give the location of the
   * new vertex.
   **********************************************************************/
  void geod_polygon_addedge(const struct geod_geodesic* g,
                            struct geod_polygon* p,
                            double azi, double s);

/**
 * Return the results for a polygon.
 *
 * @param[in] g a pointer to the geod_geodesic object specifying the
 * ellipsoid.
 * @param[in] p a pointer to the geod_polygon object specifying the polygon.
 * @param[in] reverse if non-zero then clockwise (instead of
 * counter-clockwise) traversal counts as a positive area.
 * @param[in] sign if non-zero then return a signed result for the area if
 * the polygon is traversed in the "wrong" direction instead of returning
 * the area for the rest of the earth.
 * @param[out] pA pointer to the area of the polygon (meters2);
 * only set if \e polyline is non-zero in the call to geod_polygon_init().
 * @param[out] pP pointer to the perimeter of the polygon or length of the
 * polyline (meters).
 * @return the number of points.
 *
 * The area and perimeter are accumulated at two times the standard floating
 * point precision to guard against the loss of accuracy with many-sided
 * polygons. Only simple polygons (which are not self-intersecting) are
 * allowed. There's no need to "close" the polygon by repeating the first
 * vertex. Set \e pA or \e pP to zero, if you do not want the corresponding
 * quantity returned.
 *
 * More points can be added to the polygon after this call.
 *
 * Example, compute the perimeter and area of the geodesic triangle with
 * vertices (0&deg;N,0&deg;E), (0&deg;N,90&deg;E), (90&deg;N,0&deg;E).
 @code{.c}
 double A, P;
 int n;
 struct geod_geodesic g;
 struct geod_polygon p;
 geod_init(&g, 6378137, 1/298.257223563);
 geod_polygon_init(&p, 0);

geod_polygon_addpoint(&g, &p,  0,  0);
   geod_polygon_addpoint(&g, &p,  0, 90);
   geod_polygon_addpoint(&g, &p, 90,  0);
   n = geod_polygon_compute(&g, &p, 0, 1, &A, &P);
   printf("%d %.8f %.3f\n", n, P, A);
   @endcode
   **********************************************************************/
  unsigned geod_polygon_compute(const struct geod_geodesic* g,
                                const struct geod_polygon* p,
                                int reverse, int sign,
                                double* pA, double* pP);

/**
 * Return the results assuming a tentative final test point is added;
 * however, the data for the test point is not saved. This lets you report a
 * running result for the perimeter and area as the user moves the mouse
 * cursor. Ordinary floating point arithmetic is used to accumulate the data
 * for the test point; thus the area and perimeter returned are less accurate
 * than if geod_polygon_addpoint() and geod_polygon_compute() are used.
 *
 * @param[in] g a pointer to the geod_geodesic object specifying the
 * ellipsoid.
 * @param[in] p a pointer to the geod_polygon object specifying the polygon.
 * @param[in] lat the latitude of the test point (degrees).
 * @param[in] lon the longitude of the test point (degrees).
 * @param[in] reverse if non-zero then clockwise (instead of
 * counter-clockwise) traversal counts as a positive area.
 * @param[in] sign if non-zero then return a signed result for the area if
 * the polygon is traversed in the "wrong" direction instead of returning
 * the area for the rest of the earth.
 * @param[out] pA pointer to the area of the polygon (meters2);
 * only set if \e polyline is non-zero in the call to geod_polygon_init().
 * @param[out] pP pointer to the perimeter of the polygon or length of the
 * polyline (meters).
 * @return the number of points.
 *
 * \e lat should be in the range [&minus;90&deg;, 90&deg;].
 **********************************************************************/
 unsigned geod_polygon_testpoint(const struct geod_geodesic* g,
 const struct geod_polygon* p,
 double lat, double lon,
 int reverse, int sign,
 double* pA, double* pP);

/**
 * Return the results assuming a tentative final test point is added via an
 * azimuth and distance; however, the data for the test point is not saved.
 * This lets you report a running result for the perimeter and area as the
 * user moves the mouse cursor. Ordinary floating point arithmetic is used
 * to accumulate the data for the test point; thus the area and perimeter
 * returned are less accurate than if geod_polygon_addedge() and
 * geod_polygon_compute() are used.
 *
 * @param[in] g a pointer to the geod_geodesic object specifying the
 * ellipsoid.
 * @param[in] p a pointer to the geod_polygon object specifying the polygon.
 * @param[in] azi azimuth at current point (degrees).
 * @param[in] s distance from current point to final test point (meters).
 * @param[in] reverse if non-zero then clockwise (instead of
 * counter-clockwise) traversal counts as a positive area.
 * @param[in] sign if non-zero then return a signed result for the area if
 * the polygon is traversed in the "wrong" direction instead of returning
 * the area for the rest of the earth.
 * @param[out] pA pointer to the area of the polygon (meters2);
 * only set if \e polyline is non-zero in the call to geod_polygon_init().
 * @param[out] pP pointer to the perimeter of the polygon or length of the
 * polyline (meters).
 * @return the number of points.
 **********************************************************************/
 unsigned geod_polygon_testedge(const struct geod_geodesic* g,
 const struct geod_polygon* p,
 double azi, double s,
 int reverse, int sign,
 double* pA, double* pP);

/**
 * A simple interface for computing the area of a geodesic polygon.
 *
 * @param[in] g a pointer to the geod_geodesic object specifying the
 * ellipsoid.
 * @param[in] lats an array of latitudes of the polygon vertices (degrees).
 * @param[in] lons an array of longitudes of the polygon vertices (degrees).
 * @param[in] n the number of vertices.
 * @param[out] pA pointer to the area of the polygon (meters2).
 * @param[out] pP pointer to the perimeter of the polygon (meters).
 *
 * \e lats should be in the range [&minus;90&deg;, 90&deg;].
 *
 * Only simple polygons (which are not self-intersecting) are allowed.
 * There's no need to "close" the polygon by repeating the first vertex. The
 * area returned is signed with counter-clockwise traversal being treated as
 * positive.
 *
 * Example, compute the area of Antarctica:
 @code{.c}
 double
 lats[] = {-72.9, -71.9, -74.9, -74.3, -77.5, -77.4, -71.7, -65.9, -65.7,
 -66.6, -66.9, -69.8, -70.0, -71.0, -77.3, -77.9, -74.7},
 lons[] = {-74, -102, -102, -131, -163, 163, 172, 140, 113,
 88, 59, 25, -4, -14, -33, -46, -61};
 struct geod_geodesic g;
 double A, P;
 geod_init(&g, 6378137, 1/298.257223563);
 geod_polygonarea(&g, lats, lons, (sizeof lats) / (sizeof lats[0]), &A, &P);
 printf("%.0f %.2f\n", A, P);
 @endcode
 **********************************************************************/
 void geod_polygonarea(const struct geod_geodesic* g,
 double lats[], double lons[], int n,
 double* pA, double* pP);

/**
 * mask values for the \e caps argument to geod_lineinit().
 **********************************************************************/
 enum geod_mask {
 GEOD_NONE = 0U, /**< Calculate nothing */
 GEOD_LATITUDE = 1U<<7 | 0U, /**< Calculate latitude */
 GEOD_LONGITUDE = 1U<<8 | 1U<<3, /**< Calculate longitude */
 GEOD_AZIMUTH = 1U<<9 | 0U, /**< Calculate azimuth */
 GEOD_DISTANCE = 1U<<10 | 1U<<0, /**< Calculate distance */
 GEOD_DISTANCE_IN = 1U<<11 | 1U<<0 | 1U<<1,/**< Allow distance as input */
 GEOD_REDUCEDLENGTH= 1U<<12 | 1U<<0 | 1U<<2,/**< Calculate reduced length */
 GEOD_GEODESICSCALE= 1U<<13 | 1U<<0 | 1U<<2,/**< Calculate geodesic scale */
 GEOD_AREA = 1U<<14 | 1U<<4, /**< Calculate reduced length */
 GEOD_ALL = 0x7F80U| 0x1FU /**< Calculate everything */
 };

/**
 * flag values for the \e flags argument to geod_gendirect() and
 * geod_genposition()
 **********************************************************************/
 enum geod_flags {
 GEOD_NOFLAGS = 0U, /**< No flags */
 GEOD_ARCMODE = 1U<<0, /**< Position given in terms of arc distance */
 GEOD_LONG_UNROLL = 1U<<15 /**< Unroll the longitude */
 };

#if defined(__cplusplus)
}
#endif

#endif