path: root/src/geodesic.h

/**
 * \file geodesic.h
 * \brief Header for the geodesic routines in C
 *
 * This an implementation in C of the geodesic algorithms described in
 * - C. F. F. Karney,
 *   <a href="http://dx.doi.org/10.1007/s00190-012-0578-z">
 *   Algorithms for geodesics</a>,
 *   J. Geodesy <b>87</b>, 43--55 (2013);
 *   DOI: <a href="http://dx.doi.org/10.1007/s00190-012-0578-z">
 *   10.1007/s00190-012-0578-z</a>;
 *   addenda: <a href="http://geographiclib.sf.net/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, \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 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] = [\e
 *   azi2, \e azi1], [\e M12, \e M21] = [\e M21, \e M12], \e S12 =
 *   &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 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] = [&minus;\e azi1, &minus;\e azi2], \e
 *   S12 = &minus;\e S12.  (This occurs when the \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] =
 *   [\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] = [\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="http://geographiclib.sf.net"> GeographicLib</a>.  The
 * "class data" is represented by the structs geod_geodesic and
 * geod_geodesicline and these are passed as initial arguments to the member
 * functions.  (But note that the PolygonArea class has been replaced by a
 * simple function interface and the area computation does use the accurate
 * summing providing by the Accumulator class.)  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::PolygonArea, 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 (2012-2013) <charles@karney.com> and licensed
 * under the MIT/X11 License.  For more information, see
 * http://geographiclib.sourceforge.net/
 *
 * This library was distributed with
 * <a href="../index.html">GeographicLib</a> 1.30.
 **********************************************************************/

#if !defined(GEODESIC_H)
#define GEODESIC_H 1

#if defined(__cplusplus)
extern "C" {
#endif

  /**
   * The struct containing information about the ellipsoid.
   **********************************************************************/
  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.
   **********************************************************************/
  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 */
    /**< @cond SKIP */
    double b, c2, f1, salp0, calp0, k2,
      salp1, calp1, 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 */
  };

  /**
   * Initialize a geod_geodesic object
   *
   * @param[out] g 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);

  /**
   * Initialize a geod_geodesicline object
   *
   * @param[out] l the object to be initialized.
   * @param[in] g 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;]; \e azi1 should be in the
   * range [&minus;540&deg;, 540&deg;).
   *
   * The geod_mask values are
   * - \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).
   **********************************************************************/
  void geod_lineinit(struct geod_geodesicline* l,
                     const struct geod_geodesic* g,
                     double lat1, double lon1, double azi1, unsigned caps);

  /**
   * Solve the direct geodesic problem.
   *
   * @param[in] g 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 between point 1 and 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;]; \e lon1 and \e azi1
   * should be in the range [&minus;540&deg;, 540&deg;).  The values of \e lon2
   * and \e azi2 returned are in the range [&minus;180&deg;, 180&deg;).  Any of
   * the "return" arguments 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 and 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
   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);

  /**
   * Solve the inverse geodesic problem.
   *
   * @param[in] g 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 between point 1 and 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;]; \e lon1 and
   * \e lon2 should be in the range [&minus;540&deg;, 540&deg;).  The values of
   * \e azi1 and \e azi2 returned are in the range [&minus;180&deg;, 180&deg;).
   * Any of the "return" arguments 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 and 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
   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);

  /**
   * Compute the position along a geod_geodesicline.
   *
   * @param[in] l the geod_geodesicline object specifying the geodesic line.
   * @param[in] s12 distance between point 1 and 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 the \e 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 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
   * 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
   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
   struct geod_geodesic g;
   struct geod_geodesicline l;
   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);
   geod_lineinit(&l, &g, 40.64, -73.78, azi1, 0);
   for (i = 0; i < 101; ++i) {
     geod_position(&l, i * s12 * 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 direct geodesic problem.
   *
   * @param[in] g 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] arcmode flag determining the meaning of the \e
   *   s12_a12.
   * @param[in] s12_a12 if \e arcmode is 0, this is the distance between
   *   point 1 and point 2 (meters); otherwise it is the arc length between
   *   point 1 and 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 between point 1 and 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
   *   (meters<sup>2</sup>).
   * @return \e a12 arc length of between point 1 and 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;]; \e lon1 and \e azi1
   * should be in the range [&minus;540&deg;, 540&deg;).  The function value \e
   * a12 equals \e s12_a12 is \e arcmode is non-zero.  Any of the "return"
   * arguments plat2, etc., may be replaced by 0, if you do not need some
   * quantities computed.
   **********************************************************************/
  double geod_gendirect(const struct geod_geodesic* g,
                        double lat1, double lon1, double azi1,
                        int arcmode, double s12_a12,
                        double* plat2, double* plon2, double* pazi2,
                        double* ps12, double* pm12, double* pM12, double* pM21,
                        double* pS12);

  /**
   * The general inverse geodesic calculation.
   *
   * @param[in] g 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 between point 1 and 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
   *   (meters<sup>2</sup>).
   * @return \e a12 arc length of between point 1 and 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;]; \e lon1 and
   * \e lon2 should be in the range [&minus;540&deg;, 540&deg;).  Any of the
   * "return" arguments 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);

  /**
   * The general position function.
   *
   * @param[in] l the geod_geodesicline object specifying the geodesic line.
   * @param[in] arcmode flag determining the meaning of the second parameter;
   *   if arcmode is 0, then \e l must have been initialized with \e caps |=
   *   GEOD_DISTANCE_IN.
   * @param[in] s12_a12 if \e arcmode is 0, this is the distance between
   *   point 1 and point 2 (meters); otherwise it is the arc length between
   *   point 1 and 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 between point 1 and 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
   *   (meters<sup>2</sup>); requires that \e l was initialized with \e caps |=
   *   GEOD_AREA.

   * @return \e a12 arc length of between point 1 and point 2 (degrees).
   *
   * \e l must have been initialized with a call 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
   * 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.
   *
   * Example, computer 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 space in distance).  This is
   * faster than using geod_position() would be appropriate if drawing the path
   * on a map.
   @code
   struct geod_geodesic g;
   struct geod_geodesicline l;
   double a12, azi1, lat[101],lon[101];
   int i;
   geod_init(&g, 6378137, 1/298.257223563);
   a12 = geod_geninverse(&g, 40.64, -73.78, 1.36, 103.99,
                         0, &azi1, 0, 0, 0, 0, 0);
   geod_lineinit(&l, &g, 40.64, -73.78, azi1, GEOD_LATITUDE | GEOD_LONGITUDE);
   for (i = 0; i < 101; ++i) {
     geod_genposition(&l, 1, i * a12 * 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,
                          int arcmode, double s12_a12,
                          double* plat2, double* plon2, double* pazi2,
                          double* ps12, double* pm12,
                          double* pM12, double* pM21,
                          double* pS12);

  /**
   * The area of a geodesic polygon.
   *
   * @param[in] g 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 (meters<sup>2</sup>).
   * @param[out] pP pointer to the perimeter of the polygon (meters).
   *
   * \e lats should be in the range [&minus;90&deg;, 90&deg;]; \e lons should
   * be in the range [&minus;540&deg;, 540&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 Antarctic:
   @code
   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 geod_geodesicline capabilities.
   **********************************************************************/
  enum geod_mask {
    GEOD_NONE         = 0U,
    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 */
  };

#if defined(__cplusplus)
}
#endif

#endif