CassiniSoldner.hpp

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/**
 * \file CassiniSoldner.hpp
 * \brief Header for GeographicLib::CassiniSoldner class
 *
 * Copyright (c) Charles Karney (2009-2022) <charles@karney.com> and licensed
 * under the MIT/X11 License.  For more information, see
 * https://geographiclib.sourceforge.io/
 **********************************************************************/
 
#if !defined(GEOGRAPHICLIB_CASSINISOLDNER_HPP)
#define GEOGRAPHICLIB_CASSINISOLDNER_HPP 1
 
#include <GeographicLib/Geodesic.hpp>
#include <GeographicLib/GeodesicLine.hpp>
#include <GeographicLib/Constants.hpp>
 
namespace GeographicLib {
 
  /**
   * \brief Cassini-Soldner projection
   *
   * Cassini-Soldner projection centered at an arbitrary position, \e lat0, \e
   * lon0, on the ellipsoid.  This projection is a transverse cylindrical
   * equidistant projection.  The projection from (\e lat, \e lon) to easting
   * and northing (\e x, \e y) is defined by geodesics as follows.  Go north
   * along a geodesic a distance \e y from the central point; then turn
   * clockwise 90&deg; and go a distance \e x along a geodesic.
   * (Although the initial heading is north, this changes to south if the pole
   * is crossed.)  This procedure uniquely defines the reverse projection.  The
   * forward projection is constructed as follows.  Find the point (\e lat1, \e
   * lon1) on the meridian closest to (\e lat, \e lon).  Here we consider the
   * full meridian so that \e lon1 may be either \e lon0 or \e lon0 +
   * 180&deg;.  \e x is the geodesic distance from (\e lat1, \e lon1) to
   * (\e lat, \e lon), appropriately signed according to which side of the
   * central meridian (\e lat, \e lon) lies.  \e y is the shortest distance
   * along the meridian from (\e lat0, \e lon0) to (\e lat1, \e lon1), again,
   * appropriately signed according to the initial heading.  [Note that, in the
   * case of prolate ellipsoids, the shortest meridional path from (\e lat0, \e
   * lon0) to (\e lat1, \e lon1) may not be the shortest path.]  This procedure
   * uniquely defines the forward projection except for a small class of points
   * for which there may be two equally short routes for either leg of the
   * path.
   *
   * Because of the properties of geodesics, the (\e x, \e y) grid is
   * orthogonal.  The scale in the easting direction is unity.  The scale, \e
   * k, in the northing direction is unity on the central meridian and
   * increases away from the central meridian.  The projection routines return
   * \e azi, the true bearing of the easting direction, and \e rk = 1/\e k, the
   * reciprocal of the scale in the northing direction.
   *
   * The conversions all take place using a Geodesic object (by default
   * Geodesic::WGS84()).  For more information on geodesics see \ref geodesic.
   * The determination of (\e lat1, \e lon1) in the forward projection is by
   * solving the inverse geodesic problem for (\e lat, \e lon) and its twin
   * obtained by reflection in the meridional plane.  The scale is found by
   * determining where two neighboring geodesics intersecting the central
   * meridian at \e lat1 and \e lat1 + \e dlat1 intersect and taking the ratio
   * of the reduced lengths for the two geodesics between that point and,
   * respectively, (\e lat1, \e lon1) and (\e lat, \e lon).
   *
   * Example of use:
   * \include example-CassiniSoldner.cpp
   *
   * <a href="GeodesicProj.1.html">GeodesicProj</a> is a command-line utility
   * providing access to the functionality of AzimuthalEquidistant, Gnomonic,
   * and CassiniSoldner.
   **********************************************************************/
 
  class GEOGRAPHICLIB_EXPORT CassiniSoldner {
  private:
    typedef Math::real real;
    Geodesic _earth;
    GeodesicLine _meridian;
    real _sbet0, _cbet0;
    static const unsigned maxit_ = 10;
 
  public:
 
    /**
     * Constructor for CassiniSoldner.
     *
     * @param[in] earth the Geodesic object to use for geodesic calculations.
     *   By default this uses the WGS84 ellipsoid.
     *
     * This constructor makes an "uninitialized" object.  Call Reset to set the
     * central latitude and longitude, prior to calling Forward and Reverse.
     **********************************************************************/
    explicit CassiniSoldner(const Geodesic& earth = Geodesic::WGS84());
 
    /**
     * Constructor for CassiniSoldner specifying a center point.
     *
     * @param[in] lat0 latitude of center point of projection (degrees).
     * @param[in] lon0 longitude of center point of projection (degrees).
     * @param[in] earth the Geodesic object to use for geodesic calculations.
     *   By default this uses the WGS84 ellipsoid.
     *
     * \e lat0 should be in the range [&minus;90&deg;, 90&deg;].
     **********************************************************************/
    CassiniSoldner(real lat0, real lon0,
                   const Geodesic& earth = Geodesic::WGS84());
 
    /**
     * Set the central point of the projection
     *
     * @param[in] lat0 latitude of center point of projection (degrees).
     * @param[in] lon0 longitude of center point of projection (degrees).
     *
     * \e lat0 should be in the range [&minus;90&deg;, 90&deg;].
     **********************************************************************/
    void Reset(real lat0, real lon0);
 
    /**
     * Forward projection, from geographic to Cassini-Soldner.
     *
     * @param[in] lat latitude of point (degrees).
     * @param[in] lon longitude of point (degrees).
     * @param[out] x easting of point (meters).
     * @param[out] y northing of point (meters).
     * @param[out] azi azimuth of easting direction at point (degrees).
     * @param[out] rk reciprocal of azimuthal northing scale at point.
     *
     * \e lat should be in the range [&minus;90&deg;, 90&deg;].  A call to
     * Forward followed by a call to Reverse will return the original (\e lat,
     * \e lon) (to within roundoff).  The routine does nothing if the origin
     * has not been set.
     **********************************************************************/
    void Forward(real lat, real lon,
                 real& x, real& y, real& azi, real& rk) const;
 
    /**
     * Reverse projection, from Cassini-Soldner to geographic.
     *
     * @param[in] x easting of point (meters).
     * @param[in] y northing of point (meters).
     * @param[out] lat latitude of point (degrees).
     * @param[out] lon longitude of point (degrees).
     * @param[out] azi azimuth of easting direction at point (degrees).
     * @param[out] rk reciprocal of azimuthal northing scale at point.
     *
     * A call to Reverse followed by a call to Forward will return the original
     * (\e x, \e y) (to within roundoff), provided that \e x and \e y are
     * sufficiently small not to "wrap around" the earth.  The routine does
     * nothing if the origin has not been set.
     **********************************************************************/
    void Reverse(real x, real y,
                 real& lat, real& lon, real& azi, real& rk) const;
 
    /**
     * CassiniSoldner::Forward without returning the azimuth and scale.
     **********************************************************************/
    void Forward(real lat, real lon,
                 real& x, real& y) const {
      real azi, rk;
      Forward(lat, lon, x, y, azi, rk);
    }
 
    /**
     * CassiniSoldner::Reverse without returning the azimuth and scale.
     **********************************************************************/
    void Reverse(real x, real y,
                 real& lat, real& lon) const {
      real azi, rk;
      Reverse(x, y, lat, lon, azi, rk);
    }
 
    /** \name Inspector functions
     **********************************************************************/
    ///@{
    /**
     * @return true if the object has been initialized.
     **********************************************************************/
    bool Init() const { return _meridian.Init(); }
 
    /**
     * @return \e lat0 the latitude of origin (degrees).
     **********************************************************************/
    Math::real LatitudeOrigin() const
    { return _meridian.Latitude(); }
 
    /**
     * @return \e lon0 the longitude of origin (degrees).
     **********************************************************************/
    Math::real LongitudeOrigin() const
    { return _meridian.Longitude(); }
 
    /**
     * @return \e a the equatorial radius of the ellipsoid (meters).  This is
     *   the value inherited from the Geodesic object used in the constructor.
     **********************************************************************/
    Math::real EquatorialRadius() const { return _earth.EquatorialRadius(); }
 
    /**
     * @return \e f the flattening of the ellipsoid.  This is the value
     *   inherited from the Geodesic object used in the constructor.
     **********************************************************************/
    Math::real Flattening() const { return _earth.Flattening(); }
    ///@}
 
  };
 
} // namespace GeographicLib
 
#endif  // GEOGRAPHICLIB_CASSINISOLDNER_HPP