Rhumb.hpp

/**
 * \file Rhumb.hpp
 * \brief Header for GeographicLib::Rhumb and GeographicLib::RhumbLine classes
 *
 * Copyright (c) Charles Karney (2014-2019) <charles@karney.com> and licensed
 * under the MIT/X11 License.  For more information, see
 * https://geographiclib.sourceforge.io/
 **********************************************************************/

#if !defined(GEOGRAPHICLIB_RHUMB_HPP)
#define GEOGRAPHICLIB_RHUMB_HPP 1

#include <GeographicLib/Constants.hpp>
#include <GeographicLib/Ellipsoid.hpp>

#if !defined(GEOGRAPHICLIB_RHUMBAREA_ORDER)
/**
 * The order of the series approximation used in rhumb area calculations.
 * GEOGRAPHICLIB_RHUMBAREA_ORDER can be set to any integer in [4, 8].
 **********************************************************************/
#  define GEOGRAPHICLIB_RHUMBAREA_ORDER \
  (GEOGRAPHICLIB_PRECISION == 2 ? 6 : \
   (GEOGRAPHICLIB_PRECISION == 1 ? 4 : 8))
#endif

namespace GeographicLib {

  class RhumbLine;
  template <class T> class PolygonAreaT;

  /**
   * \brief Solve of the direct and inverse rhumb problems.
   *
   * The path of constant azimuth between two points on a ellipsoid at (\e
   * lat1, \e lon1) and (\e lat2, \e lon2) is called the rhumb line (also
   * called the loxodrome).  Its length is \e s12 and its azimuth is \e azi12.
   * (The azimuth is the heading measured clockwise from north.)
   *
   * Given \e lat1, \e lon1, \e azi12, and \e s12, we can determine \e lat2,
   * and \e lon2.  This is the \e direct rhumb problem and its solution is
   * given by the function Rhumb::Direct.
   *
   * Given \e lat1, \e lon1, \e lat2, and \e lon2, we can determine \e azi12
   * and \e s12.  This is the \e inverse rhumb problem, whose solution is given
   * by Rhumb::Inverse.  This finds the shortest such rhumb line, i.e., the one
   * that wraps no more than half way around the earth.  If the end points are
   * on opposite meridians, there are two shortest rhumb lines and the
   * east-going one is chosen.
   *
   * These routines also optionally calculate the area under the rhumb line, \e
   * S12.  This is the area, measured counter-clockwise, of the rhumb line
   * quadrilateral with corners (<i>lat1</i>,<i>lon1</i>), (0,<i>lon1</i>),
   * (0,<i>lon2</i>), and (<i>lat2</i>,<i>lon2</i>).
   *
   * Note that rhumb lines may be appreciably longer (up to 50%) than the
   * corresponding Geodesic.  For example the distance between London Heathrow
   * and Tokyo Narita via the rhumb line is 11400 km which is 18% longer than
   * the geodesic distance 9600 km.
   *
   * For more information on rhumb lines see \ref rhumb.
   *
   * Example of use:
   * \include example-Rhumb.cpp
   **********************************************************************/

  class  GEOGRAPHICLIB_EXPORT Rhumb {
  private:
    typedef Math::real real;
    friend class RhumbLine;
    template <class T> friend class PolygonAreaT;
    Ellipsoid _ell;
    bool _exact;
    real _c2;
    static const int tm_maxord = GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER;
    static const int maxpow_ = GEOGRAPHICLIB_RHUMBAREA_ORDER;
    // _R[0] unused
    real _R[maxpow_ + 1];
    static real gd(real x)
    { using std::atan; using std::sinh; return atan(sinh(x)); }

    // Use divided differences to determine (mu2 - mu1) / (psi2 - psi1)
    // accurately
    //
    // Definition: Df(x,y,d) = (f(x) - f(y)) / (x - y)
    // See:
    //   W. M. Kahan and R. J. Fateman,
    //   Symbolic computation of divided differences,
    //   SIGSAM Bull. 33(3), 7-28 (1999)
    //   https://doi.org/10.1145/334714.334716
    //   http://www.cs.berkeley.edu/~fateman/papers/divdiff.pdf

    static real Dlog(real x, real y) {
      using std::sqrt;
      real t = x - y;
      // Change
      //
      //   atanh(t / (x + y))
      //
      // to
      //
      //   asinh(t / (2 * sqrt(x*y)))
      //
      // to avoid taking atanh(1) when x is large and y is 1.  N.B., this
      // routine is invoked with positive x and y, so no need to guard against
      // taking the sqrt of a negative quantity.  This fixes bogus results for
      // the area being returning when an endpoint is at a pole.
      return t != 0 ? 2 * Math::asinh(t / (2 * sqrt(x*y))) / t : 1 / x;
    }
    // N.B., x and y are in degrees
    static real Dtan(real x, real y) {
      real d = x - y, tx = Math::tand(x), ty = Math::tand(y), txy = tx * ty;
      return d != 0 ?
        (2 * txy > -1 ? (1 + txy) * Math::tand(d) : tx - ty) /
        (d * Math::degree()) :
        1 + txy;
    }
    static real Datan(real x, real y) {
      using std::atan;
      real d = x - y, xy = x * y;
      return d != 0 ?
        (2 * xy > -1 ? atan( d / (1 + xy) ) : atan(x) - atan(y)) / d :
        1 / (1 + xy);
    }
    static real Dsin(real x, real y) {
      using std::sin; using std::cos;
      real d = (x - y) / 2;
      return cos((x + y)/2) * (d != 0 ? sin(d) / d : 1);
    }
    static real Dsinh(real x, real y) {
      using std::sinh; using std::cosh;
      real d = (x - y) / 2;
      return cosh((x + y) / 2) * (d != 0 ? sinh(d) / d : 1);
    }
    static real Dcosh(real x, real y) {
      using std::sinh;
      real d = (x - y) / 2;
      return sinh((x + y) / 2) * (d != 0 ? sinh(d) / d : 1);
    }
    static real Dasinh(real x, real y) {
      real d = x - y,
        hx = Math::hypot(real(1), x), hy = Math::hypot(real(1), y);
      return d != 0 ?
        Math::asinh(x*y > 0 ? d * (x + y) / (x*hy + y*hx) : x*hy - y*hx) / d :
        1 / hx;
    }
    static real Dgd(real x, real y) {
      using std::sinh;
      return Datan(sinh(x), sinh(y)) * Dsinh(x, y);
    }
    // N.B., x and y are the tangents of the angles
    static real Dgdinv(real x, real y)
    { return Dasinh(x, y) / Datan(x, y); }
    // Copied from LambertConformalConic...
    // Deatanhe(x,y) = eatanhe((x-y)/(1-e^2*x*y))/(x-y)
    real Deatanhe(real x, real y) const {
      real t = x - y, d = 1 - _ell._e2 * x * y;
      return t != 0 ? Math::eatanhe(t / d, _ell._es) / t : _ell._e2 / d;
    }
    // (E(x) - E(y)) / (x - y) -- E = incomplete elliptic integral of 2nd kind
    real DE(real x, real y) const;
    // (mux - muy) / (phix - phiy) using elliptic integrals
    real DRectifying(real latx, real laty) const;
    // (psix - psiy) / (phix - phiy)
    real DIsometric(real latx, real laty) const;

    // (sum(c[j]*sin(2*j*x),j=1..n) - sum(c[j]*sin(2*j*x),j=1..n)) / (x - y)
    static real SinCosSeries(bool sinp,
                             real x, real y, const real c[], int n);
    // (mux - muy) / (chix - chiy) using Krueger's series
    real DConformalToRectifying(real chix, real chiy) const;
    // (chix - chiy) / (mux - muy) using Krueger's series
    real DRectifyingToConformal(real mux, real muy) const;

    // (mux - muy) / (psix - psiy)
    // N.B., psix and psiy are in degrees
    real DIsometricToRectifying(real psix, real psiy) const;
    // (psix - psiy) / (mux - muy)
    real DRectifyingToIsometric(real mux, real muy) const;

    real MeanSinXi(real psi1, real psi2) const;

    // The following two functions (with lots of ignored arguments) mimic the
    // interface to the corresponding Geodesic function.  These are needed by
    // PolygonAreaT.
    void GenDirect(real lat1, real lon1, real azi12,
                   bool, real s12, unsigned outmask,
                   real& lat2, real& lon2, real&, real&, real&, real&, real&,
                   real& S12) const {
      GenDirect(lat1, lon1, azi12, s12, outmask, lat2, lon2, S12);
    }
    void GenInverse(real lat1, real lon1, real lat2, real lon2,
                    unsigned outmask, real& s12, real& azi12,
                    real&, real& , real& , real& , real& S12) const {
      GenInverse(lat1, lon1, lat2, lon2, outmask, s12, azi12, S12);
    }
  public:

    /**
     * Bit masks for what calculations to do.  They specify which results to
     * return in the general routines Rhumb::GenDirect and Rhumb::GenInverse
     * routines.  RhumbLine::mask is a duplication of this enum.
     **********************************************************************/
    enum mask {
      /**
       * No output.
       * @hideinitializer
       **********************************************************************/
      NONE          = 0U,
      /**
       * Calculate latitude \e lat2.
       * @hideinitializer
       **********************************************************************/
      LATITUDE      = 1U<<7,
      /**
       * Calculate longitude \e lon2.
       * @hideinitializer
       **********************************************************************/
      LONGITUDE     = 1U<<8,
      /**
       * Calculate azimuth \e azi12.
       * @hideinitializer
       **********************************************************************/
      AZIMUTH       = 1U<<9,
      /**
       * Calculate distance \e s12.
       * @hideinitializer
       **********************************************************************/
      DISTANCE      = 1U<<10,
      /**
       * Calculate area \e S12.
       * @hideinitializer
       **********************************************************************/
      AREA          = 1U<<14,
      /**
       * Unroll \e lon2 in the direct calculation.
       * @hideinitializer
       **********************************************************************/
      LONG_UNROLL   = 1U<<15,
      /**
       * Calculate everything.  (LONG_UNROLL is not included in this mask.)
       * @hideinitializer
       **********************************************************************/
      ALL           = 0x7F80U,
    };

    /**
     * Constructor for a 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.
     * @param[in] exact if true (the default) use an addition theorem for
     *   elliptic integrals to compute divided differences; otherwise use
     *   series expansion (accurate for |<i>f</i>| < 0.01).
     * @exception GeographicErr if \e a or (1 &minus; \e f) \e a is not
     *   positive.
     *
     * See \ref rhumb, for a detailed description of the \e exact parameter.
     **********************************************************************/
    Rhumb(real a, real f, bool exact = true);

    /**
     * Solve the direct rhumb problem returning also the area.
     *
     * @param[in] lat1 latitude of point 1 (degrees).
     * @param[in] lon1 longitude of point 1 (degrees).
     * @param[in] azi12 azimuth of the rhumb line (degrees).
     * @param[in] s12 distance between point 1 and point 2 (meters); it can be
     *   negative.
     * @param[out] lat2 latitude of point 2 (degrees).
     * @param[out] lon2 longitude of point 2 (degrees).
     * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
     *
     * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].  The value of
     * \e lon2 returned is in the range [&minus;180&deg;, 180&deg;].
     *
     * If point 1 is a pole, the cosine of its latitude is taken to be
     * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>).  This
     * position, which is extremely close to the actual pole, allows the
     * calculation to be carried out in finite terms.  If \e s12 is large
     * enough that the rhumb line crosses a pole, the longitude of point 2
     * is indeterminate (a NaN is returned for \e lon2 and \e S12).
     **********************************************************************/
    void Direct(real lat1, real lon1, real azi12, real s12,
                real& lat2, real& lon2, real& S12) const {
      GenDirect(lat1, lon1, azi12, s12,
                LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
    }

    /**
     * Solve the direct rhumb problem without the area.
     **********************************************************************/
    void Direct(real lat1, real lon1, real azi12, real s12,
                real& lat2, real& lon2) const {
      real t;
      GenDirect(lat1, lon1, azi12, s12, LATITUDE | LONGITUDE, lat2, lon2, t);
    }

    /**
     * The general direct rhumb problem.  Rhumb::Direct 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] azi12 azimuth of the rhumb line (degrees).
     * @param[in] s12 distance between point 1 and point 2 (meters); it can be
     *   negative.
     * @param[in] outmask a bitor'ed combination of Rhumb::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] S12 area under the rhumb line (meters<sup>2</sup>).
     *
     * The Rhumb::mask values possible for \e outmask are
     * - \e outmask |= Rhumb::LATITUDE for the latitude \e lat2;
     * - \e outmask |= Rhumb::LONGITUDE for the latitude \e lon2;
     * - \e outmask |= Rhumb::AREA for the area \e S12;
     * - \e outmask |= Rhumb::ALL for all of the above;
     * - \e outmask |= Rhumb::LONG_UNROLL to unroll \e lon2 instead of wrapping
     *   it into the range [&minus;180&deg;, 180&deg;].
     * .
     * With the Rhumb::LONG_UNROLL bit set, the quantity \e lon2 &minus;
     * \e lon1 indicates how many times and in what sense the rhumb line
     * encircles the ellipsoid.
     **********************************************************************/
    void GenDirect(real lat1, real lon1, real azi12, real s12,
                   unsigned outmask, real& lat2, real& lon2, real& S12) const;

    /**
     * Solve the inverse rhumb problem returning also the area.
     *
     * @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 rhumb distance between point 1 and point 2 (meters).
     * @param[out] azi12 azimuth of the rhumb line (degrees).
     * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
     *
     * The shortest rhumb line is found.  If the end points are on opposite
     * meridians, there are two shortest rhumb lines and the east-going one is
     * chosen.  \e lat1 and \e lat2 should be in the range [&minus;90&deg;,
     * 90&deg;].  The value of \e azi12 returned is in the range
     * [&minus;180&deg;, 180&deg;].
     *
     * If either point is a pole, the cosine of its latitude is taken to be
     * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>).  This
     * position, which is extremely close to the actual pole, allows the
     * calculation to be carried out in finite terms.
     **********************************************************************/
    void Inverse(real lat1, real lon1, real lat2, real lon2,
                 real& s12, real& azi12, real& S12) const {
      GenInverse(lat1, lon1, lat2, lon2,
                 DISTANCE | AZIMUTH | AREA, s12, azi12, S12);
    }

    /**
     * Solve the inverse rhumb problem without the area.
     **********************************************************************/
    void Inverse(real lat1, real lon1, real lat2, real lon2,
                 real& s12, real& azi12) const {
      real t;
      GenInverse(lat1, lon1, lat2, lon2, DISTANCE | AZIMUTH, s12, azi12, t);
    }

    /**
     * The general inverse rhumb problem.  Rhumb::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 Rhumb::mask values
     *   specifying which of the following parameters should be set.
     * @param[out] s12 rhumb distance between point 1 and point 2 (meters).
     * @param[out] azi12 azimuth of the rhumb line (degrees).
     * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
     *
     * The Rhumb::mask values possible for \e outmask are
     * - \e outmask |= Rhumb::DISTANCE for the latitude \e s12;
     * - \e outmask |= Rhumb::AZIMUTH for the latitude \e azi12;
     * - \e outmask |= Rhumb::AREA for the area \e S12;
     * - \e outmask |= Rhumb::ALL for all of the above;
     **********************************************************************/
    void GenInverse(real lat1, real lon1, real lat2, real lon2,
                    unsigned outmask,
                    real& s12, real& azi12, real& S12) const;

    /**
     * Set up to compute several points on a single rhumb line.
     *
     * @param[in] lat1 latitude of point 1 (degrees).
     * @param[in] lon1 longitude of point 1 (degrees).
     * @param[in] azi12 azimuth of the rhumb line (degrees).
     * @return a RhumbLine object.
     *
     * \e lat1 should be in the range [&minus;90&deg;, 90&deg;].
     *
     * If point 1 is a pole, the cosine of its latitude is taken to be
     * 1/&epsilon;<sup>2</sup> (where &epsilon; is 2<sup>-52</sup>).  This
     * position, which is extremely close to the actual pole, allows the
     * calculation to be carried out in finite terms.
     **********************************************************************/
    RhumbLine Line(real lat1, real lon1, real azi12) 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 _ell.EquatorialRadius(); }

    /**
     * @return \e f the  flattening of the ellipsoid.  This is the
     *   value used in the constructor.
     **********************************************************************/
    Math::real Flattening() const { return _ell.Flattening(); }

    /**
     * @return total area of ellipsoid in meters<sup>2</sup>.  The area of a
     *   polygon encircling a pole can be found by adding
     *   Geodesic::EllipsoidArea()/2 to the sum of \e S12 for each side of the
     *   polygon.
     **********************************************************************/
    Math::real EllipsoidArea() const { return _ell.Area(); }

    /**
      * \deprecated An old name for EquatorialRadius().
      **********************************************************************/
    // GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()")
    Math::real MajorRadius() const { return EquatorialRadius(); }
    ///@}

    /**
     * A global instantiation of Rhumb with the parameters for the WGS84
     * ellipsoid.
     **********************************************************************/
    static const Rhumb& WGS84();
  };

  /**
   * \brief Find a sequence of points on a single rhumb line.
   *
   * RhumbLine facilitates the determination of a series of points on a single
   * rhumb line.  The starting point (\e lat1, \e lon1) and the azimuth \e
   * azi12 are specified in the call to Rhumb::Line which returns a RhumbLine
   * object.  RhumbLine.Position returns the location of point 2 (and,
   * optionally, the corresponding area, \e S12) a distance \e s12 along the
   * rhumb line.
   *
   * There is no public constructor for this class.  (Use Rhumb::Line to create
   * an instance.)  The Rhumb object used to create a RhumbLine must stay in
   * scope as long as the RhumbLine.
   *
   * Example of use:
   * \include example-RhumbLine.cpp
   **********************************************************************/

  class  GEOGRAPHICLIB_EXPORT RhumbLine {
  private:
    typedef Math::real real;
    friend class Rhumb;
    const Rhumb& _rh;
    bool _exact;
    real _lat1, _lon1, _azi12, _salp, _calp, _mu1, _psi1, _r1;
    RhumbLine& operator=(const RhumbLine&); // copy assignment not allowed
    RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12,
              bool exact);
  public:

    /**
     * This is a duplication of Rhumb::mask.
     **********************************************************************/
    enum mask {
      /**
       * No output.
       * @hideinitializer
       **********************************************************************/
      NONE          = Rhumb::NONE,
      /**
       * Calculate latitude \e lat2.
       * @hideinitializer
       **********************************************************************/
      LATITUDE      = Rhumb::LATITUDE,
      /**
       * Calculate longitude \e lon2.
       * @hideinitializer
       **********************************************************************/
      LONGITUDE     = Rhumb::LONGITUDE,
      /**
       * Calculate azimuth \e azi12.
       * @hideinitializer
       **********************************************************************/
      AZIMUTH       = Rhumb::AZIMUTH,
      /**
       * Calculate distance \e s12.
       * @hideinitializer
       **********************************************************************/
      DISTANCE      = Rhumb::DISTANCE,
      /**
       * Calculate area \e S12.
       * @hideinitializer
       **********************************************************************/
      AREA          = Rhumb::AREA,
      /**
       * Unroll \e lon2 in the direct calculation.
       * @hideinitializer
       **********************************************************************/
      LONG_UNROLL   = Rhumb::LONG_UNROLL,
      /**
       * Calculate everything.  (LONG_UNROLL is not included in this mask.)
       * @hideinitializer
       **********************************************************************/
      ALL           = Rhumb::ALL,
    };

    /**
     * Compute the position of point 2 which is a distance \e s12 (meters) from
     * point 1.  The area is also computed.
     *
     * @param[in] s12 distance between point 1 and point 2 (meters); it can be
     *   negative.
     * @param[out] lat2 latitude of point 2 (degrees).
     * @param[out] lon2 longitude of point 2 (degrees).
     * @param[out] S12 area under the rhumb line (meters<sup>2</sup>).
     *
     * The value of \e lon2 returned is in the range [&minus;180&deg;,
     * 180&deg;].
     *
     * If \e s12 is large enough that the rhumb line crosses a pole, the
     * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
     * \e S12).
     **********************************************************************/
    void Position(real s12, real& lat2, real& lon2, real& S12) const {
      GenPosition(s12, LATITUDE | LONGITUDE | AREA, lat2, lon2, S12);
    }

    /**
     * Compute the position of point 2 which is a distance \e s12 (meters) from
     * point 1.  The area is not computed.
     **********************************************************************/
    void Position(real s12, real& lat2, real& lon2) const {
      real t;
      GenPosition(s12, LATITUDE | LONGITUDE, lat2, lon2, t);
    }

    /**
     * The general position routine.  RhumbLine::Position is defined in term so
     * this function.
     *
     * @param[in] s12 distance between point 1 and point 2 (meters); it can be
     *   negative.
     * @param[in] outmask a bitor'ed combination of RhumbLine::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] S12 area under the rhumb line (meters<sup>2</sup>).
     *
     * The RhumbLine::mask values possible for \e outmask are
     * - \e outmask |= RhumbLine::LATITUDE for the latitude \e lat2;
     * - \e outmask |= RhumbLine::LONGITUDE for the latitude \e lon2;
     * - \e outmask |= RhumbLine::AREA for the area \e S12;
     * - \e outmask |= RhumbLine::ALL for all of the above;
     * - \e outmask |= RhumbLine::LONG_UNROLL to unroll \e lon2 instead of
     *   wrapping it into the range [&minus;180&deg;, 180&deg;].
     * .
     * With the RhumbLine::LONG_UNROLL bit set, the quantity \e lon2 &minus; \e
     * lon1 indicates how many times and in what sense the rhumb line encircles
     * the ellipsoid.
     *
     * If \e s12 is large enough that the rhumb line crosses a pole, the
     * longitude of point 2 is indeterminate (a NaN is returned for \e lon2 and
     * \e S12).
     **********************************************************************/
    void GenPosition(real s12, unsigned outmask,
                     real& lat2, real& lon2, real& S12) const;

    /** \name Inspector functions
     **********************************************************************/
    ///@{

    /**
     * @return \e lat1 the latitude of point 1 (degrees).
     **********************************************************************/
    Math::real Latitude() const { return _lat1; }

    /**
     * @return \e lon1 the longitude of point 1 (degrees).
     **********************************************************************/
    Math::real Longitude() const { return _lon1; }

    /**
     * @return \e azi12 the azimuth of the rhumb line (degrees).
     **********************************************************************/
    Math::real Azimuth() const { return  _azi12; }

    /**
     * @return \e a the equatorial radius of the ellipsoid (meters).  This is
     *   the value inherited from the Rhumb object used in the constructor.
     **********************************************************************/
    Math::real EquatorialRadius() const { return _rh.EquatorialRadius(); }

    /**
     * @return \e f the flattening of the ellipsoid.  This is the value
     *   inherited from the Rhumb object used in the constructor.
     **********************************************************************/
    Math::real Flattening() const { return _rh.Flattening(); }
  };

} // namespace GeographicLib

#endif  // GEOGRAPHICLIB_RHUMB_HPP