/** * \file TransverseMercatorExact.hpp * \brief Header for GeographicLib::TransverseMercatorExact class * * Copyright (c) Charles Karney (2008-2015) 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 #include namespace GeographicLib { /** * \brief An exact implementation of the transverse Mercator projection * * Implementation of the Transverse Mercator Projection given in * - L. P. Lee, * Conformal * Projections Based On Jacobian Elliptic Functions, 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, * * Transverse Mercator with an accuracy of a few nanometers, * J. Geodesy 85(8), 475--485 (Aug. 2011); * preprint * arXiv:1002.1417. * * 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 × 10−15", * the relative error in the scale is 7 × 10−12%%. * See Sec. 3 of * arXiv:1002.1417 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 Krüger * 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 tm-grid.kmz, 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 * * TransverseMercatorProj 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 int numit_ = 10; real tol_, tol1_, tol2_, taytol_; real _a, _f, _k0, _mu, _mv, _e; bool _extendp; EllipticFunction _Eu, _Ev; void zeta(real u, real snu, real cnu, real dnu, real v, real snv, real cnv, real dnv, real& taup, real& lam) const; void dwdzeta(real u, real snu, real cnu, real dnu, real v, real snv, real cnv, real dnv, real& du, real& dv) const; bool zetainv0(real psi, real lam, real& u, real& v) const; void zetainv(real taup, real lam, real& u, real& v) const; void sigma(real u, real snu, real cnu, real dnu, real v, real snv, real cnv, real dnv, real& xi, real& eta) const; void dwdsigma(real u, real snu, real cnu, real dnu, real v, real snv, real cnv, real dnv, real& du, real& dv) const; bool sigmainv0(real xi, real eta, real& u, real& v) const; void sigmainv(real xi, real eta, real& u, real& v) const; void Scale(real tau, real lam, real snu, real cnu, real dnu, real snv, real cnv, real dnv, real& gamma, real& k) const; public: /** * Constructor for a ellipsoid with * * @param[in] a equatorial radius (meters). * @param[in] f flattening of ellipsoid. * @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 − \e lon0 = 90 (1 − \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 ± \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 = −7131237.022729 m and \e x = * 29735142.378357 m, \e y = 4235043.607933 m both map to \e lat = * −2°, \e lon = 88°. * * 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 − \e lon0 in [0, 90] * - \e lat in (-90, 0] and \e lon − \e lon0 in [90 (1 − \e e), 90] * - the union of * - x/(\e k0 \e a) in [0, ∞) and * y/(\e k0 \e a) in [0, E(e2)] * - x/(\e k0 \e a) in [K(1 − e2) − * E(1 − e2), ∞) and y/(\e k0 \e * a) in (−∞, 0] * . * See Sec. 5 of * arXiv:1002.1417 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::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 * [−90°, 90°]. **********************************************************************/ void Forward(real lon0, real lat, real lon, real& x, real& y, real& gamma, real& k) const; /** * 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. The value of \e lon returned is * in the range [−180°, 180°). **********************************************************************/ void Reverse(real lon0, real x, real y, real& lat, real& lon, real& gamma, real& k) const; /** * TransverseMercatorExact::Forward without returning the convergence and * scale. **********************************************************************/ void Forward(real lon0, real lat, real lon, real& x, real& y) const { 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 { 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 { 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 \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 { 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