text: /**
   * \file TransverseMercatorExact.hpp
   * \brief Header for GeographicLib::TransverseMercatorExact class
   *
   * Copyright (c) Charles Karney (2008-2011) <charles@karney.com> and licensed
   * under the MIT/X11 License.  For more information, see
   * http://geographiclib.sourceforge.net/
   **********************************************************************/
  
  #if !defined(GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP)
  #define GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP 1
  
  #include <GeographicLib/Constants.hpp>
  #include <GeographicLib/EllipticFunction.hpp>
  
  namespace GeographicLib {
  
    /**
     * \brief An exact implementation of the transverse Mercator projection
     *
     * Implementation of the Transverse Mercator Projection given in
     *  - L. P. Lee,
     *    <a href="http://dx.doi.org/10.3138/X687-1574-4325-WM62"> Conformal
     *    Projections Based On Jacobian Elliptic Functions</a>, Part V of
     *    Conformal Projections Based on Elliptic Functions,
     *    (B. V. Gutsell, Toronto, 1976), 128pp.,
     *    ISBN: 0919870163
     *    (also appeared as:
     *    Monograph 16, Suppl. No. 1 to Canadian Cartographer, Vol 13).
     *  - C. F. F. Karney,
     *    <a href="http://dx.doi.org/10.1007/s00190-011-0445-3">
     *    Transverse Mercator with an accuracy of a few nanometers,</a>
     *    J. Geodesy 85(8), 475--485 (Aug. 2011);
     *    preprint
     *    <a href="http://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>.
     *
     * Lee gives the correct results for forward and reverse transformations
     * subject to the branch cut rules (see the description of the \e extendp
     * argument to the constructor).  The maximum error is about 8 nm (8
     * nanometers), ground distance, for the forward and reverse transformations.
     * The error in the convergence is 2 &times; 10<sup>&minus;15</sup>&quot;,
     * the relative error in the scale is 7 &times; 10<sup>&minus;12</sup>%%.
     * See Sec. 3 of
     * <a href="http://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for details.
     * The method is "exact" in the sense that the errors are close to the
     * round-off limit and that no changes are needed in the algorithms for them
     * to be used with reals of a higher precision.  Thus the errors using long
     * double (with a 64-bit fraction) are about 2000 times smaller than using
     * double (with a 53-bit fraction).
     *
     * This algorithm is about 4.5 times slower than the 6th-order Kr&uuml;ger
     * method, TransverseMercator, taking about 11 us for a combined forward and
     * reverse projection on a 2.66 GHz Intel machine (g++, version 4.3.0, -O3).
     *
     * The ellipsoid parameters and the central scale are set in the constructor.
     * The central meridian (which is a trivial shift of the longitude) is
     * specified as the \e lon0 argument of the TransverseMercatorExact::Forward
     * and TransverseMercatorExact::Reverse functions.  The latitude of origin is
     * taken to be the equator.  See the documentation on TransverseMercator for
     * how to include a false easting, false northing, or a latitude of origin.
     *
     * See <a href="http://geographiclib.sourceforge.net/tm-grid.kmz"
     * type="application/vnd.google-earth.kmz"> tm-grid.kmz</a>, for an
     * illustration of the transverse Mercator grid in Google Earth.
     *
     * See TransverseMercatorExact.cpp for more information on the
     * implementation.
     *
     * See \ref transversemercator for a discussion of this projection.
     *
     * Example of use:
     * \include example-TransverseMercatorExact.cpp
     *
     * <a href="TransverseMercatorProj.1.html">TransverseMercatorProj</a> is a
     * command-line utility providing access to the functionality of
     * TransverseMercator and TransverseMercatorExact.
     **********************************************************************/
  
    class GEOGRAPHICLIB_EXPORT TransverseMercatorExact {
    private:
      typedef Math::real real;
      static const real tol_;
      static const real tol1_;
      static const real tol2_;
      static const real taytol_;
      static const real overflow_;
      static const int numit_ = 10;
      real _a, _f, _k0, _mu, _mv, _e;
      bool _extendp;
      EllipticFunction _Eu, _Ev;
      // tan(x) for x in [-pi/2, pi/2] ensuring that the sign is right
      static inline real tanx(real x) throw() {
        real t = std::tan(x);
        // Write the tests this way to ensure that tanx(NaN()) is NaN()
        return x >= 0 ? (!(t < 0) ? t : overflow_) : (!(t >= 0) ? t : -overflow_);
      }
  
      real taup(real tau) const throw();
      real taupinv(real taup) const throw();
  
      void zeta(real u, real snu, real cnu, real dnu,
                real v, real snv, real cnv, real dnv,
                real& taup, real& lam) const throw();
  
      void dwdzeta(real u, real snu, real cnu, real dnu,
                   real v, real snv, real cnv, real dnv,
                   real& du, real& dv) const throw();
  
      bool zetainv0(real psi, real lam, real& u, real& v) const throw();
      void zetainv(real taup, real lam, real& u, real& v) const throw();
  
      void sigma(real u, real snu, real cnu, real dnu,
                 real v, real snv, real cnv, real dnv,
                 real& xi, real& eta) const throw();
  
      void dwdsigma(real u, real snu, real cnu, real dnu,
                    real v, real snv, real cnv, real dnv,
                    real& du, real& dv) const throw();
  
      bool sigmainv0(real xi, real eta, real& u, real& v) const throw();
      void sigmainv(real xi, real eta, real& u, real& v) const throw();
  
      void Scale(real tau, real lam,
                 real snu, real cnu, real dnu,
                 real snv, real cnv, real dnv,
                 real& gamma, real& k) const throw();
  
    public:
  
      /**
       * Constructor for a ellipsoid with
       *
       * @param[in] a equatorial radius (meters).
       * @param[in] f flattening of ellipsoid.  If \e f > 1, set flattening
       *   to 1/\e f.
       * @param[in] k0 central scale factor.
       * @param[in] extendp use extended domain.
       * @exception GeographicErr if \e a, \e f, or \e k0 is not positive.
       *
       * The transverse Mercator projection has a branch point singularity at \e
       * lat = 0 and \e lon &minus; \e lon0 = 90 (1 &minus; \e e) or (for
       * TransverseMercatorExact::UTM) x = 18381 km, y = 0m.  The \e extendp
       * argument governs where the branch cut is placed.  With \e extendp =
       * false, the "standard" convention is followed, namely the cut is placed
       * along \e x > 18381 km, \e y = 0m.  Forward can be called with any \e lat
       * and \e lon then produces the transformation shown in Lee, Fig 46.
       * Reverse analytically continues this in the &plusmn; \e x direction.  As
       * a consequence, Reverse may map multiple points to the same geographic
       * location; for example, for TransverseMercatorExact::UTM, \e x =
       * 22051449.037349 m, \e y = &minus;7131237.022729 m and \e x =
       * 29735142.378357 m, \e y = 4235043.607933 m both map to \e lat =
       * &minus;2&deg;, \e lon = 88&deg;.
       *
       * With \e extendp = true, the branch cut is moved to the lower left
       * quadrant.  The various symmetries of the transverse Mercator projection
       * can be used to explore the projection on any sheet.  In this mode the
       * domains of \e lat, \e lon, \e x, and \e y are restricted to
       * - the union of
       *   - \e lat in [0, 90] and \e lon &minus; \e lon0 in [0, 90]
       *   - \e lat in (-90, 0] and \e lon &minus; \e lon0 in [90 (1 &minus; \e
             e), 90]
       * - the union of
       *   - <i>x</i>/(\e k0 \e a) in [0, &infin;) and
       *     <i>y</i>/(\e k0 \e a) in [0, E(<i>e</i><sup>2</sup>)]
       *   - <i>x</i>/(\e k0 \e a) in [K(1 &minus; <i>e</i><sup>2</sup>) &minus;
       *     E(1 &minus; <i>e</i><sup>2</sup>), &infin;) and <i>y</i>/(\e k0 \e
       *     a) in (&minus;&infin;, 0]
       * .
       * See Sec. 5 of
       * <a href="http://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for a full
       * discussion of the treatment of the branch cut.
       *
       * The method will work for all ellipsoids used in terrestrial geodesy.
       * The method cannot be applied directly to the case of a sphere (\e f = 0)
       * because some the constants characterizing this method diverge in that
       * limit, and in practice, \e f should be larger than about
       * numeric_limits<real>::epsilon().  However, TransverseMercator treats the
       * sphere exactly.
       **********************************************************************/
      TransverseMercatorExact(real a, real f, real k0, bool extendp = false);
  
      /**
       * Forward projection, from geographic to transverse Mercator.
       *
       * @param[in] lon0 central meridian of the projection (degrees).
       * @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] gamma meridian convergence at point (degrees).
       * @param[out] k scale of projection at point.
       *
       * No false easting or northing is added. \e lat should be in the range
       * [&minus;90&deg;, 90&deg;]; \e lon and \e lon0 should be in the
       * range [&minus;540&deg;, 540&deg;).
       **********************************************************************/
      void Forward(real lon0, real lat, real lon,
                   real& x, real& y, real& gamma, real& k) const throw();
  
      /**
       * Reverse projection, from transverse Mercator to geographic.
       *
       * @param[in] lon0 central meridian of the projection (degrees).
       * @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] gamma meridian convergence at point (degrees).
       * @param[out] k scale of projection at point.
       *
       * No false easting or northing is added.  \e lon0 should be in the range
       * [&minus;540&deg;, 540&deg;).  The value of \e lon returned is in
       * the range [&minus;180&deg;, 180&deg;).
       **********************************************************************/
      void Reverse(real lon0, real x, real y,
                   real& lat, real& lon, real& gamma, real& k) const throw();
  
      /**
       * TransverseMercatorExact::Forward without returning the convergence and
       * scale.
       **********************************************************************/
      void Forward(real lon0, real lat, real lon,
                   real& x, real& y) const throw() {
        real gamma, k;
        Forward(lon0, lat, lon, x, y, gamma, k);
      }
  
      /**
       * TransverseMercatorExact::Reverse without returning the convergence and
       * scale.
       **********************************************************************/
      void Reverse(real lon0, real x, real y,
                   real& lat, real& lon) const throw() {
        real gamma, k;
        Reverse(lon0, x, y, lat, lon, gamma, k);
      }
  
      /** \name Inspector functions
       **********************************************************************/
      ///@{
      /**
       * @return \e a the equatorial radius of the ellipsoid (meters).  This is
       *   the value used in the constructor.
       **********************************************************************/
      Math::real MajorRadius() const throw() { return _a; }
  
      /**
       * @return \e f the flattening of the ellipsoid.  This is the value used in
       *   the constructor.
       **********************************************************************/
      Math::real Flattening() const throw() { return _f; }
  
      /// \cond SKIP
      /**
       * <b>DEPRECATED</b>
       * @return \e r the inverse flattening of the ellipsoid.
       **********************************************************************/
      Math::real InverseFlattening() const throw() { return 1/_f; }
      /// \endcond
  
      /**
       * @return \e k0 central scale for the projection.  This is the value of \e
       *   k0 used in the constructor and is the scale on the central meridian.
       **********************************************************************/
      Math::real CentralScale() const throw() { return _k0; }
      ///@}
  
      /**
       * A global instantiation of TransverseMercatorExact with the WGS84
       * ellipsoid and the UTM scale factor.  However, unlike UTM, no false
       * easting or northing is added.
       **********************************************************************/
      static const TransverseMercatorExact UTM;
    };
  
  } // namespace GeographicLib
  
  #endif  // GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP
  