GeodesicExact.hpp

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/**
 * \file GeodesicExact.hpp
 * \brief Header for GeographicLib::GeodesicExact class
 *
 * Copyright (c) Charles Karney (2012-2024) <karney@alum.mit.edu> and licensed
 * under the MIT/X11 License.  For more information, see
 * https://geographiclib.sourceforge.io/
 **********************************************************************/
 
#if !defined(GEOGRAPHICLIB_GEODESICEXACT_HPP)
#define GEOGRAPHICLIB_GEODESICEXACT_HPP 1
 
#include <GeographicLib/Constants.hpp>
#include <GeographicLib/EllipticFunction.hpp>
#include <GeographicLib/DST.hpp>
 
namespace GeographicLib {
 
  class GeodesicLineExact;
  // Visual Studio needs this forward declaration...
  class GEOGRAPHICLIB_EXPORT DST;
 
  /**
   * \brief Exact geodesic calculations
   *
   * The equations for geodesics on an ellipsoid can be expressed in terms of
   * incomplete elliptic integrals.  The Geodesic class expands these integrals
   * in a series in the flattening \e f and this provides an accurate solution
   * for \e f &isin; [-0.01, 0.01].  The GeodesicExact class computes the
   * ellitpic integrals directly and so provides a solution which is valid for
   * all \e f.  However, in practice, its use should be limited to about
   * <i>b</i>/\e a &isin; [0.01, 100] or \e f &isin; [&minus;99, 0.99].
   *
   * For the WGS84 ellipsoid, these classes are 2--3 times \e slower than the
   * series solution and 2--3 times \e less \e accurate (because it's less easy
   * to control round-off errors with the elliptic integral formulation); i.e.,
   * the error is about 40 nm (40 nanometers) instead of 15 nm.  However the
   * error in the series solution scales as <i>f</i><sup>7</sup> while the
   * error in the elliptic integral solution depends weakly on \e f.  If the
   * quarter meridian distance is 10000 km and the ratio <i>b</i>/\e a = 1
   * &minus; \e f is varied then the approximate maximum error (expressed as a
   * distance) is <pre>
   *       1 - f  error (nm)
   *       1/128     387
   *       1/64      345
   *       1/32      269
   *       1/16      210
   *       1/8       115
   *       1/4        69
   *       1/2        36
   *         1        15
   *         2        25
   *         4        96
   *         8       318
   *        16       985
   *        32      2352
   *        64      6008
   *       128     19024
   * </pre>
   *
   * The area in this classes is computing by finding an accurate approximation
   * to the area integrand using a discrete sine transform fitting \e N equally
   * spaced points in &sigma;.  \e N chosen to ensure full accuracy for
   * <i>b</i>/\e a &isin; [0.01, 100] or \e f &isin; [&minus;99, 0.99].
   *
   * The algorithms are described in
   * - C. F. F. Karney,
   *   <a href="https://doi.org/10.1007/s00190-023-01813-2">
   *   Geodesics on an arbitrary ellipsoid of revolution</a>,
   *   J. Geodesy <b>98</b>, 4:1--14 (2024);
   *   DOI: <a href="https://doi.org/10.1007/s00190-023-01813-2">
   *   10.1007/s00190-023-01813-2</a>.
   * .
   * See \ref geodellip for the formulation.  See the documentation on the
   * Geodesic class for additional information on the geodesic problems.
   *
   * Example of use:
   * \include example-GeodesicExact.cpp
   *
   * <a href="GeodSolve.1.html">GeodSolve</a> is a command-line utility
   * providing access to the functionality of GeodesicExact and
   * GeodesicLineExact (via the -E option).
   **********************************************************************/
 
  class GEOGRAPHICLIB_EXPORT GeodesicExact {
  private:
    typedef Math::real real;
    friend class GeodesicLineExact;
    friend class Geodesic;    // Allow Geodesic to call the default constructor
    // Private default constructor to support Geodesic(a, f, exact)
    GeodesicExact() {};         // Do nothing; used with exact = false.
 
    static const unsigned maxit1_ = 20;
    unsigned maxit2_;
    real tiny_, tol0_, tol1_, tol2_, tolb_, xthresh_;
 
    static constexpr unsigned CAP_NONE = 0U;
    static constexpr unsigned CAP_E    = 1U<<0;
    // Skip 1U<<1 for compatibility with Geodesic (not required)
    static constexpr unsigned CAP_D    = 1U<<2;
    static constexpr unsigned CAP_H    = 1U<<3;
    static constexpr unsigned CAP_C4   = 1U<<4;
    static constexpr unsigned CAP_ALL  = 0x1FU;
    static constexpr unsigned CAP_MASK = CAP_ALL;
    static constexpr unsigned OUT_ALL  = 0x7F80U;
    static constexpr unsigned OUT_MASK = 0xFF80U;       // Includes LONG_UNROLL
 
    static real Astroid(real x, real y);
 
    real _a, _f, _f1, _e2, _ep2, _n, _b, _c2, _etol2;
    int _nC4;
    DST _fft;
 
    void Lengths(const EllipticFunction& E,
                 real sig12,
                 real ssig1, real csig1, real dn1,
                 real ssig2, real csig2, real dn2,
                 real cbet1, real cbet2, unsigned outmask,
                 real& s12s, real& m12a, real& m0,
                 real& M12, real& M21) const;
    real InverseStart(EllipticFunction& E,
                      real sbet1, real cbet1, real dn1,
                      real sbet2, real cbet2, real dn2,
                      real lam12, real slam12, real clam12,
                      real& salp1, real& calp1,
                      real& salp2, real& calp2, real& dnm) const;
    real Lambda12(real sbet1, real cbet1, real dn1,
                  real sbet2, real cbet2, real dn2,
                  real salp1, real calp1, real slam120, real clam120,
                  real& salp2, real& calp2, real& sig12,
                  real& ssig1, real& csig1, real& ssig2, real& csig2,
                  EllipticFunction& E,
                  real& domg12, bool diffp, real& dlam12) const;
    real GenInverse(real lat1, real lon1, real lat2, real lon2,
                    unsigned outmask, real& s12,
                    real& salp1, real& calp1, real& salp2, real& calp2,
                    real& m12, real& M12, real& M21, real& S12) const;
 
    class I4Integrand {
    private:
      real X, tX, tdX, sX, sX1, sXX1, asinhsX, _k2;
      static real asinhsqrt(real x);
      static real t(real x);
      static real td(real x);
      // static real Dt(real x, real y);
      real DtX(real y) const;
    public:
      I4Integrand(real ep2, real k2);
      real operator()(real sig) const;
    };
 
  public:
 
    /**
     * Bit masks for what calculations to do.  These masks do double duty.
     * They signify to the GeodesicLineExact::GeodesicLineExact constructor and
     * to GeodesicExact::Line what capabilities should be included in the
     * GeodesicLineExact object.  They also specify which results to return in
     * the general routines GeodesicExact::GenDirect and
     * GeodesicExact::GenInverse routines.  GeodesicLineExact::mask is a
     * duplication of this enum.
     **********************************************************************/
    enum mask {
      /**
       * No capabilities, no output.
       * @hideinitializer
       **********************************************************************/
      NONE          = 0U,
      /**
       * Calculate latitude \e lat2.  (It's not necessary to include this as a
       * capability to GeodesicLineExact because this is included by default.)
       * @hideinitializer
       **********************************************************************/
      LATITUDE      = 1U<<7  | CAP_NONE,
      /**
       * Calculate longitude \e lon2.
       * @hideinitializer
       **********************************************************************/
      LONGITUDE     = 1U<<8  | CAP_H,
      /**
       * Calculate azimuths \e azi1 and \e azi2.  (It's not necessary to
       * include this as a capability to GeodesicLineExact because this is
       * included by default.)
       * @hideinitializer
       **********************************************************************/
      AZIMUTH       = 1U<<9  | CAP_NONE,
      /**
       * Calculate distance \e s12.
       * @hideinitializer
       **********************************************************************/
      DISTANCE      = 1U<<10 | CAP_E,
      /**
       * A combination of the common capabilities: GeodesicExact::LATITUDE,
       * GeodesicExact::LONGITUDE, GeodesicExact::AZIMUTH,
       * GeodesicExact::DISTANCE.
       * @hideinitializer
       **********************************************************************/
      STANDARD      = LATITUDE | LONGITUDE | AZIMUTH | DISTANCE,
      /**
       * Allow distance \e s12 to be used as input in the direct geodesic
       * problem.
       * @hideinitializer
       **********************************************************************/
      DISTANCE_IN   = 1U<<11 | CAP_E,
      /**
       * Calculate reduced length \e m12.
       * @hideinitializer
       **********************************************************************/
      REDUCEDLENGTH = 1U<<12 | CAP_D,
      /**
       * Calculate geodesic scales \e M12 and \e M21.
       * @hideinitializer
       **********************************************************************/
      GEODESICSCALE = 1U<<13 | CAP_D,
      /**
       * Calculate area \e S12.
       * @hideinitializer
       **********************************************************************/
      AREA          = 1U<<14 | CAP_C4,
      /**
       * Unroll \e lon2 in the direct calculation.
       * @hideinitializer
       **********************************************************************/
      LONG_UNROLL   = 1U<<15,
      /**
       * All capabilities, calculate everything.  (GeodesicExact::LONG_UNROLL
       * is not included in this mask.)
       * @hideinitializer
       **********************************************************************/
      ALL           = OUT_ALL| CAP_ALL,
    };
 
    /** \name Constructor
     **********************************************************************/
    ///@{
    /**
     * Constructor for an ellipsoid with
     *
     * @param[in] a equatorial radius (meters).
     * @param[in] f flattening of ellipsoid.  Setting \e f = 0 gives a sphere.
     *   Negative \e f gives a prolate ellipsoid.
     * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
     *   positive.
     **********************************************************************/
    GeodesicExact(real a, real f);
    ///@}
 
    /** \name Direct geodesic problem specified in terms of distance.
     **********************************************************************/
    ///@{
    /**
     * Perform the direct geodesic calculation where the length of the geodesic
     * is specified in terms of distance.
     *
     * @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
     *   signed.
     * @param[out] lat2 latitude of point 2 (degrees).
     * @param[out] lon2 longitude of point 2 (degrees).
     * @param[out] azi2 (forward) azimuth at point 2 (degrees).
     * @param[out] m12 reduced length of geodesic (meters).
     * @param[out] M12 geodesic scale of point 2 relative to point 1
     *   (dimensionless).
     * @param[out] M21 geodesic scale of point 1 relative to point 2
     *   (dimensionless).
     * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
     * @return \e a12 arc length of between point 1 and point 2 (degrees).
     *
     * \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;].
     *
     * 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;.)
     *
     * The following functions are overloaded versions of GeodesicExact::Direct
     * which omit some of the output parameters.  Note, however, that the arc
     * length is always computed and returned as the function value.
     **********************************************************************/
    Math::real Direct(real lat1, real lon1, real azi1, real s12,
                      real& lat2, real& lon2, real& azi2,
                      real& m12, real& M12, real& M21, real& S12)
      const {
      real t;
      return GenDirect(lat1, lon1, azi1, false, s12,
                       LATITUDE | LONGITUDE | AZIMUTH |
                       REDUCEDLENGTH | GEODESICSCALE | AREA,
                       lat2, lon2, azi2, t, m12, M12, M21, S12);
    }
 
    /**
     * See the documentation for GeodesicExact::Direct.
     **********************************************************************/
    Math::real Direct(real lat1, real lon1, real azi1, real s12,
                      real& lat2, real& lon2)
      const {
      real t;
      return GenDirect(lat1, lon1, azi1, false, s12,
                       LATITUDE | LONGITUDE,
                       lat2, lon2, t, t, t, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Direct.
     **********************************************************************/
    Math::real Direct(real lat1, real lon1, real azi1, real s12,
                      real& lat2, real& lon2, real& azi2)
      const {
      real t;
      return GenDirect(lat1, lon1, azi1, false, s12,
                       LATITUDE | LONGITUDE | AZIMUTH,
                       lat2, lon2, azi2, t, t, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Direct.
     **********************************************************************/
    Math::real Direct(real lat1, real lon1, real azi1, real s12,
                      real& lat2, real& lon2, real& azi2, real& m12)
      const {
      real t;
      return GenDirect(lat1, lon1, azi1, false, s12,
                       LATITUDE | LONGITUDE | AZIMUTH | REDUCEDLENGTH,
                       lat2, lon2, azi2, t, m12, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Direct.
     **********************************************************************/
    Math::real Direct(real lat1, real lon1, real azi1, real s12,
                      real& lat2, real& lon2, real& azi2,
                      real& M12, real& M21)
      const {
      real t;
      return GenDirect(lat1, lon1, azi1, false, s12,
                       LATITUDE | LONGITUDE | AZIMUTH | GEODESICSCALE,
                       lat2, lon2, azi2, t, t, M12, M21, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Direct.
     **********************************************************************/
    Math::real Direct(real lat1, real lon1, real azi1, real s12,
                      real& lat2, real& lon2, real& azi2,
                      real& m12, real& M12, real& M21)
      const {
      real t;
      return GenDirect(lat1, lon1, azi1, false, s12,
                       LATITUDE | LONGITUDE | AZIMUTH |
                       REDUCEDLENGTH | GEODESICSCALE,
                       lat2, lon2, azi2, t, m12, M12, M21, t);
    }
    ///@}
 
    /** \name Direct geodesic problem specified in terms of arc length.
     **********************************************************************/
    ///@{
    /**
     * Perform the direct geodesic calculation where the length of the geodesic
     * is specified in terms of arc length.
     *
     * @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] a12 arc length between point 1 and point 2 (degrees); it can
     *   be signed.
     * @param[out] lat2 latitude of point 2 (degrees).
     * @param[out] lon2 longitude of point 2 (degrees).
     * @param[out] azi2 (forward) azimuth at point 2 (degrees).
     * @param[out] s12 distance between point 1 and point 2 (meters).
     * @param[out] m12 reduced length of geodesic (meters).
     * @param[out] M12 geodesic scale of point 2 relative to point 1
     *   (dimensionless).
     * @param[out] M21 geodesic scale of point 1 relative to point 2
     *   (dimensionless).
     * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
     *
     * \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;].
     *
     * 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;.)
     *
     * The following functions are overloaded versions of GeodesicExact::Direct
     * which omit some of the output parameters.
     **********************************************************************/
    void ArcDirect(real lat1, real lon1, real azi1, real a12,
                   real& lat2, real& lon2, real& azi2, real& s12,
                   real& m12, real& M12, real& M21, real& S12)
      const {
      GenDirect(lat1, lon1, azi1, true, a12,
                LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
                REDUCEDLENGTH | GEODESICSCALE | AREA,
                lat2, lon2, azi2, s12, m12, M12, M21, S12);
    }
 
    /**
     * See the documentation for GeodesicExact::ArcDirect.
     **********************************************************************/
    void ArcDirect(real lat1, real lon1, real azi1, real a12,
                   real& lat2, real& lon2) const {
      real t;
      GenDirect(lat1, lon1, azi1, true, a12,
                LATITUDE | LONGITUDE,
                lat2, lon2, t, t, t, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::ArcDirect.
     **********************************************************************/
    void ArcDirect(real lat1, real lon1, real azi1, real a12,
                   real& lat2, real& lon2, real& azi2) const {
      real t;
      GenDirect(lat1, lon1, azi1, true, a12,
                LATITUDE | LONGITUDE | AZIMUTH,
                lat2, lon2, azi2, t, t, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::ArcDirect.
     **********************************************************************/
    void ArcDirect(real lat1, real lon1, real azi1, real a12,
                   real& lat2, real& lon2, real& azi2, real& s12)
      const {
      real t;
      GenDirect(lat1, lon1, azi1, true, a12,
                LATITUDE | LONGITUDE | AZIMUTH | DISTANCE,
                lat2, lon2, azi2, s12, t, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::ArcDirect.
     **********************************************************************/
    void ArcDirect(real lat1, real lon1, real azi1, real a12,
                   real& lat2, real& lon2, real& azi2,
                   real& s12, real& m12) const {
      real t;
      GenDirect(lat1, lon1, azi1, true, a12,
                LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
                REDUCEDLENGTH,
                lat2, lon2, azi2, s12, m12, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::ArcDirect.
     **********************************************************************/
    void ArcDirect(real lat1, real lon1, real azi1, real a12,
                   real& lat2, real& lon2, real& azi2, real& s12,
                   real& M12, real& M21) const {
      real t;
      GenDirect(lat1, lon1, azi1, true, a12,
                LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
                GEODESICSCALE,
                lat2, lon2, azi2, s12, t, M12, M21, t);
    }
 
    /**
     * See the documentation for GeodesicExact::ArcDirect.
     **********************************************************************/
    void ArcDirect(real lat1, real lon1, real azi1, real a12,
                   real& lat2, real& lon2, real& azi2, real& s12,
                   real& m12, real& M12, real& M21) const {
      real t;
      GenDirect(lat1, lon1, azi1, true, a12,
                LATITUDE | LONGITUDE | AZIMUTH | DISTANCE |
                REDUCEDLENGTH | GEODESICSCALE,
                lat2, lon2, azi2, s12, m12, M12, M21, t);
    }
    ///@}
 
    /** \name General version of the direct geodesic solution.
     **********************************************************************/
    ///@{
 
    /**
     * The general direct geodesic calculation.  GeodesicExact::Direct and
     * GeodesicExact::ArcDirect are defined in terms of this function.
     *
     * @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 boolean flag determining the meaning of the second
     *   parameter.
     * @param[in] s12_a12 if \e arcmode is false, 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 signed.
     * @param[in] outmask a bitor'ed combination of GeodesicExact::mask values
     *   specifying which of the following parameters should be set.
     * @param[out] lat2 latitude of point 2 (degrees).
     * @param[out] lon2 longitude of point 2 (degrees).
     * @param[out] azi2 (forward) azimuth at point 2 (degrees).
     * @param[out] s12 distance between point 1 and point 2 (meters).
     * @param[out] m12 reduced length of geodesic (meters).
     * @param[out] M12 geodesic scale of point 2 relative to point 1
     *   (dimensionless).
     * @param[out] M21 geodesic scale of point 1 relative to point 2
     *   (dimensionless).
     * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
     * @return \e a12 arc length of between point 1 and point 2 (degrees).
     *
     * The GeodesicExact::mask values possible for \e outmask are
     * - \e outmask |= GeodesicExact::LATITUDE for the latitude \e lat2;
     * - \e outmask |= GeodesicExact::LONGITUDE for the latitude \e lon2;
     * - \e outmask |= GeodesicExact::AZIMUTH for the latitude \e azi2;
     * - \e outmask |= GeodesicExact::DISTANCE for the distance \e s12;
     * - \e outmask |= GeodesicExact::REDUCEDLENGTH for the reduced length \e
     *   m12;
     * - \e outmask |= GeodesicExact::GEODESICSCALE for the geodesic scales \e
     *   M12 and \e M21;
     * - \e outmask |= GeodesicExact::AREA for the area \e S12;
     * - \e outmask |= GeodesicExact::ALL for all of the above;
     * - \e outmask |= GeodesicExact::LONG_UNROLL to unroll \e lon2 instead of
     *   wrapping it into the range [&minus;180&deg;, 180&deg;].
     * .
     * The function value \e a12 is always computed and returned and this
     * equals \e s12_a12 is \e arcmode is true.  If \e outmask includes
     * GeodesicExact::DISTANCE and \e arcmode is false, then \e s12 = \e
     * s12_a12.  It is not necessary to include GeodesicExact::DISTANCE_IN in
     * \e outmask; this is automatically included is \e arcmode is false.
     *
     * With the GeodesicExact::LONG_UNROLL bit set, the quantity \e lon2
     * &minus; \e lon1 indicates how many times and in what sense the geodesic
     * encircles the ellipsoid.
     **********************************************************************/
    Math::real GenDirect(real lat1, real lon1, real azi1,
                         bool arcmode, real s12_a12, unsigned outmask,
                         real& lat2, real& lon2, real& azi2,
                         real& s12, real& m12, real& M12, real& M21,
                         real& S12) const;
    ///@}
 
    /** \name Inverse geodesic problem.
     **********************************************************************/
    ///@{
    /**
     * Perform the inverse geodesic calculation.
     *
     * @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] s12 distance between point 1 and point 2 (meters).
     * @param[out] azi1 azimuth at point 1 (degrees).
     * @param[out] azi2 (forward) azimuth at point 2 (degrees).
     * @param[out] m12 reduced length of geodesic (meters).
     * @param[out] M12 geodesic scale of point 2 relative to point 1
     *   (dimensionless).
     * @param[out] M21 geodesic scale of point 1 relative to point 2
     *   (dimensionless).
     * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
     * @return \e a12 arc length of between point 1 and point 2 (degrees).
     *
     * \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;].
     *
     * 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 following functions are overloaded versions of
     * GeodesicExact::Inverse which omit some of the output parameters.  Note,
     * however, that the arc length is always computed and returned as the
     * function value.
     **********************************************************************/
    Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
                       real& s12, real& azi1, real& azi2, real& m12,
                       real& M12, real& M21, real& S12) const {
      return GenInverse(lat1, lon1, lat2, lon2,
                        DISTANCE | AZIMUTH |
                        REDUCEDLENGTH | GEODESICSCALE | AREA,
                        s12, azi1, azi2, m12, M12, M21, S12);
    }
 
    /**
     * See the documentation for GeodesicExact::Inverse.
     **********************************************************************/
    Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
                       real& s12) const {
      real t;
      return GenInverse(lat1, lon1, lat2, lon2,
                        DISTANCE,
                        s12, t, t, t, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Inverse.
     **********************************************************************/
    Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
                       real& azi1, real& azi2) const {
      real t;
      return GenInverse(lat1, lon1, lat2, lon2,
                        AZIMUTH,
                        t, azi1, azi2, t, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Inverse.
     **********************************************************************/
    Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
                       real& s12, real& azi1, real& azi2)
      const {
      real t;
      return GenInverse(lat1, lon1, lat2, lon2,
                        DISTANCE | AZIMUTH,
                        s12, azi1, azi2, t, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Inverse.
     **********************************************************************/
    Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
                       real& s12, real& azi1, real& azi2, real& m12)
      const {
      real t;
      return GenInverse(lat1, lon1, lat2, lon2,
                        DISTANCE | AZIMUTH | REDUCEDLENGTH,
                        s12, azi1, azi2, m12, t, t, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Inverse.
     **********************************************************************/
    Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
                       real& s12, real& azi1, real& azi2,
                       real& M12, real& M21) const {
      real t;
      return GenInverse(lat1, lon1, lat2, lon2,
                        DISTANCE | AZIMUTH | GEODESICSCALE,
                        s12, azi1, azi2, t, M12, M21, t);
    }
 
    /**
     * See the documentation for GeodesicExact::Inverse.
     **********************************************************************/
    Math::real Inverse(real lat1, real lon1, real lat2, real lon2,
                       real& s12, real& azi1, real& azi2, real& m12,
                       real& M12, real& M21) const {
      real t;
      return GenInverse(lat1, lon1, lat2, lon2,
                        DISTANCE | AZIMUTH |
                        REDUCEDLENGTH | GEODESICSCALE,
                        s12, azi1, azi2, m12, M12, M21, t);
    }
    ///@}
 
    /** \name General version of inverse geodesic solution.
     **********************************************************************/
    ///@{
    /**
     * The general inverse geodesic calculation.  GeodesicExact::Inverse is
     * defined in terms of this function.
     *
     * @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] outmask a bitor'ed combination of GeodesicExact::mask values
     *   specifying which of the following parameters should be set.
     * @param[out] s12 distance between point 1 and point 2 (meters).
     * @param[out] azi1 azimuth at point 1 (degrees).
     * @param[out] azi2 (forward) azimuth at point 2 (degrees).
     * @param[out] m12 reduced length of geodesic (meters).
     * @param[out] M12 geodesic scale of point 2 relative to point 1
     *   (dimensionless).
     * @param[out] M21 geodesic scale of point 1 relative to point 2
     *   (dimensionless).
     * @param[out] S12 area under the geodesic (meters<sup>2</sup>).
     * @return \e a12 arc length of between point 1 and point 2 (degrees).
     *
     * The GeodesicExact::mask values possible for \e outmask are
     * - \e outmask |= GeodesicExact::DISTANCE for the distance \e s12;
     * - \e outmask |= GeodesicExact::AZIMUTH for the latitude \e azi2;
     * - \e outmask |= GeodesicExact::REDUCEDLENGTH for the reduced length \e
     *   m12;
     * - \e outmask |= GeodesicExact::GEODESICSCALE for the geodesic scales \e
     *   M12 and \e M21;
     * - \e outmask |= GeodesicExact::AREA for the area \e S12;
     * - \e outmask |= GeodesicExact::ALL for all of the above.
     * .
     * The arc length is always computed and returned as the function value.
     **********************************************************************/
    Math::real GenInverse(real lat1, real lon1, real lat2, real lon2,
                          unsigned outmask,
                          real& s12, real& azi1, real& azi2,
                          real& m12, real& M12, real& M21, real& S12) const;
    ///@}
 
    /** \name Interface to GeodesicLineExact.
     **********************************************************************/
    ///@{
 
    /**
     * Typedef for the class for computing multiple points on a geodesic.
     **********************************************************************/
    typedef GeodesicLineExact LineClass;
 
    /**
     * Set up to compute several points on a single geodesic.
     *
     * @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 GeodesicExact::mask values
     *   specifying the capabilities the GeodesicLineExact object should
     *   possess, i.e., which quantities can be returned in calls to
     *   GeodesicLineExact::Position.
     * @return a GeodesicLineExact object.
     *
     * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
     *
     * The GeodesicExact::mask values are
     * - \e caps |= GeodesicExact::LATITUDE for the latitude \e lat2; this is
     *   added automatically;
     * - \e caps |= GeodesicExact::LONGITUDE for the latitude \e lon2;
     * - \e caps |= GeodesicExact::AZIMUTH for the azimuth \e azi2; this is
     *   added automatically;
     * - \e caps |= GeodesicExact::DISTANCE for the distance \e s12;
     * - \e caps |= GeodesicExact::REDUCEDLENGTH for the reduced length \e m12;
     * - \e caps |= GeodesicExact::GEODESICSCALE for the geodesic scales \e M12
     *   and \e M21;
     * - \e caps |= GeodesicExact::AREA for the area \e S12;
     * - \e caps |= GeodesicExact::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;
     * - \e caps |= GeodesicExact::ALL for all of the above.
     * .
     * The default value of \e caps is GeodesicExact::ALL which turns on all
     * the capabilities.
     *
     * If the point is at a pole, the azimuth is defined by keeping \e lon1
     * fixed, writing \e lat1 = &plusmn;(90 &minus; &epsilon;), and taking the
     * limit &epsilon; &rarr; 0+.
     **********************************************************************/
    GeodesicLineExact Line(real lat1, real lon1, real azi1,
                           unsigned caps = ALL) const;
 
    /**
     * Define a GeodesicLineExact in terms of the inverse geodesic problem.
     *
     * @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 GeodesicExact::mask values
     *   specifying the capabilities the GeodesicLineExact object should
     *   possess, i.e., which quantities can be returned in calls to
     *   GeodesicLineExact::Position.
     * @return a GeodesicLineExact object.
     *
     * This function sets point 3 of the GeodesicLineExact to correspond to
     * point 2 of the inverse geodesic problem.
     *
     * \e lat1 and \e lat2 should be in the range [&minus;90&deg;, 90&deg;].
     **********************************************************************/
    GeodesicLineExact InverseLine(real lat1, real lon1, real lat2, real lon2,
                                  unsigned caps = ALL) const;
 
    /**
     * Define a GeodesicLineExact in terms of the direct geodesic problem
     * specified in terms of distance.
     *
     * @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[in] caps bitor'ed combination of GeodesicExact::mask values
     *   specifying the capabilities the GeodesicLineExact object should
     *   possess, i.e., which quantities can be returned in calls to
     *   GeodesicLineExact::Position.
     * @return a GeodesicLineExact object.
     *
     * This function sets point 3 of the GeodesicLineExact to correspond to
     * point 2 of the direct geodesic problem.
     *
     * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
     **********************************************************************/
    GeodesicLineExact DirectLine(real lat1, real lon1, real azi1, real s12,
                                 unsigned caps = ALL) const;
 
    /**
     * Define a GeodesicLineExact in terms of the direct geodesic problem
     * specified in terms of arc length.
     *
     * @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] a12 arc length between point 1 and point 2 (degrees); it can
     *   be negative.
     * @param[in] caps bitor'ed combination of GeodesicExact::mask values
     *   specifying the capabilities the GeodesicLineExact object should
     *   possess, i.e., which quantities can be returned in calls to
     *   GeodesicLineExact::Position.
     * @return a GeodesicLineExact object.
     *
     * This function sets point 3 of the GeodesicLineExact to correspond to
     * point 2 of the direct geodesic problem.
     *
     * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
     **********************************************************************/
    GeodesicLineExact ArcDirectLine(real lat1, real lon1, real azi1, real a12,
                                    unsigned caps = ALL) const;
 
    /**
     * Define a GeodesicLineExact in terms of the direct geodesic problem
     * specified in terms of either distance or arc length.
     *
     * @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 boolean flag determining the meaning of the \e
     *   s12_a12.
     * @param[in] s12_a12 if \e arcmode is false, 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[in] caps bitor'ed combination of GeodesicExact::mask values
     *   specifying the capabilities the GeodesicLineExact object should
     *   possess, i.e., which quantities can be returned in calls to
     *   GeodesicLineExact::Position.
     * @return a GeodesicLineExact object.
     *
     * This function sets point 3 of the GeodesicLineExact to correspond to
     * point 2 of the direct geodesic problem.
     *
     * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
     **********************************************************************/
    GeodesicLineExact GenDirectLine(real lat1, real lon1, real azi1,
                                    bool arcmode, real s12_a12,
                                    unsigned caps = ALL) const;
    ///@}
 
    /** \name Inspector functions.
     **********************************************************************/
    ///@{
 
    /**
     * @return \e a the equatorial radius of the ellipsoid (meters).  This is
     *   the value used in the constructor.
     **********************************************************************/
    Math::real EquatorialRadius() const { return _a; }
 
    /**
     * @return \e f the  flattening of the ellipsoid.  This is the
     *   value used in the constructor.
     **********************************************************************/
    Math::real Flattening() const { return _f; }
 
    /**
     * @return total area of ellipsoid in meters<sup>2</sup>.  The area of a
     *   polygon encircling a pole can be found by adding
     *   GeodesicExact::EllipsoidArea()/2 to the sum of \e S12 for each side of
     *   the polygon.
     **********************************************************************/
    Math::real EllipsoidArea() const
    { return 4 * Math::pi() * _c2; }
    ///@}
 
    /**
     * A global instantiation of GeodesicExact with the parameters for the
     * WGS84 ellipsoid.
     **********************************************************************/
    static const GeodesicExact& WGS84();
 
  };
 
} // namespace GeographicLib
 
#endif  // GEOGRAPHICLIB_GEODESICEXACT_HPP