1 /** 2 * \file TransverseMercatorExact.hpp 3 * \brief Header for GeographicLib::TransverseMercatorExact class 4 * 5 * Copyright (c) Charles Karney (2008-2020) <charles@karney.com> and licensed 6 * under the MIT/X11 License. For more information, see 7 * https://geographiclib.sourceforge.io/ 8 **********************************************************************/ 9 10 #if !defined(GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP) 11 #define GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP 1 12 13 #include <GeographicLib/Constants.hpp> 14 #include <GeographicLib/EllipticFunction.hpp> 15 16 namespace GeographicLib { 17 18 /** 19 * \brief An exact implementation of the transverse Mercator projection 20 * 21 * Implementation of the Transverse Mercator Projection given in 22 * - L. P. Lee, 23 * <a href="https://doi.org/10.3138/X687-1574-4325-WM62"> Conformal 24 * Projections Based On Jacobian Elliptic Functions</a>, Part V of 25 * Conformal Projections Based on Elliptic Functions, 26 * (B. V. Gutsell, Toronto, 1976), 128pp., 27 * ISBN: 0919870163 28 * (also appeared as: 29 * Monograph 16, Suppl. No. 1 to Canadian Cartographer, Vol 13). 30 * - C. F. F. Karney, 31 * <a href="https://doi.org/10.1007/s00190-011-0445-3"> 32 * Transverse Mercator with an accuracy of a few nanometers,</a> 33 * J. Geodesy 85(8), 475--485 (Aug. 2011); 34 * preprint 35 * <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>. 36 * 37 * Lee gives the correct results for forward and reverse transformations 38 * subject to the branch cut rules (see the description of the \e extendp 39 * argument to the constructor). The maximum error is about 8 nm (8 40 * nanometers), ground distance, for the forward and reverse transformations. 41 * The error in the convergence is 2 × 10<sup>−15</sup>", 42 * the relative error in the scale is 7 × 10<sup>−12</sup>%%. 43 * See Sec. 3 of 44 * <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for details. 45 * The method is "exact" in the sense that the errors are close to the 46 * round-off limit and that no changes are needed in the algorithms for them 47 * to be used with reals of a higher precision. Thus the errors using long 48 * double (with a 64-bit fraction) are about 2000 times smaller than using 49 * double (with a 53-bit fraction). 50 * 51 * This algorithm is about 4.5 times slower than the 6th-order Krüger 52 * method, TransverseMercator, taking about 11 us for a combined forward and 53 * reverse projection on a 2.66 GHz Intel machine (g++, version 4.3.0, -O3). 54 * 55 * The ellipsoid parameters and the central scale are set in the constructor. 56 * The central meridian (which is a trivial shift of the longitude) is 57 * specified as the \e lon0 argument of the TransverseMercatorExact::Forward 58 * and TransverseMercatorExact::Reverse functions. The latitude of origin is 59 * taken to be the equator. See the documentation on TransverseMercator for 60 * how to include a false easting, false northing, or a latitude of origin. 61 * 62 * See <a href="https://geographiclib.sourceforge.io/tm-grid.kmz" 63 * type="application/vnd.google-earth.kmz"> tm-grid.kmz</a>, for an 64 * illustration of the transverse Mercator grid in Google Earth. 65 * 66 * This class also returns the meridian convergence \e gamma and scale \e k. 67 * The meridian convergence is the bearing of grid north (the \e y axis) 68 * measured clockwise from true north. 69 * 70 * See TransverseMercatorExact.cpp for more information on the 71 * implementation. 72 * 73 * See \ref transversemercator for a discussion of this projection. 74 * 75 * Example of use: 76 * \include example-TransverseMercatorExact.cpp 77 * 78 * <a href="TransverseMercatorProj.1.html">TransverseMercatorProj</a> is a 79 * command-line utility providing access to the functionality of 80 * TransverseMercator and TransverseMercatorExact. 81 **********************************************************************/ 82 83 class GEOGRAPHICLIB_EXPORT TransverseMercatorExact { 84 private: 85 typedef Math::real real; 86 static const int numit_ = 10; 87 real tol_, tol2_, taytol_; 88 real _a, _f, _k0, _mu, _mv, _e; 89 bool _extendp; 90 EllipticFunction _Eu, _Ev; 91 92 void zeta(real u, real snu, real cnu, real dnu, 93 real v, real snv, real cnv, real dnv, 94 real& taup, real& lam) const; 95 96 void dwdzeta(real u, real snu, real cnu, real dnu, 97 real v, real snv, real cnv, real dnv, 98 real& du, real& dv) const; 99 100 bool zetainv0(real psi, real lam, real& u, real& v) const; 101 void zetainv(real taup, real lam, real& u, real& v) const; 102 103 void sigma(real u, real snu, real cnu, real dnu, 104 real v, real snv, real cnv, real dnv, 105 real& xi, real& eta) const; 106 107 void dwdsigma(real u, real snu, real cnu, real dnu, 108 real v, real snv, real cnv, real dnv, 109 real& du, real& dv) const; 110 111 bool sigmainv0(real xi, real eta, real& u, real& v) const; 112 void sigmainv(real xi, real eta, real& u, real& v) const; 113 114 void Scale(real tau, real lam, 115 real snu, real cnu, real dnu, 116 real snv, real cnv, real dnv, 117 real& gamma, real& k) const; 118 119 public: 120 121 /** 122 * Constructor for a ellipsoid with 123 * 124 * @param[in] a equatorial radius (meters). 125 * @param[in] f flattening of ellipsoid. 126 * @param[in] k0 central scale factor. 127 * @param[in] extendp use extended domain. 128 * @exception GeographicErr if \e a, \e f, or \e k0 is not positive. 129 * 130 * The transverse Mercator projection has a branch point singularity at \e 131 * lat = 0 and \e lon − \e lon0 = 90 (1 − \e e) or (for 132 * TransverseMercatorExact::UTM) x = 18381 km, y = 0m. The \e extendp 133 * argument governs where the branch cut is placed. With \e extendp = 134 * false, the "standard" convention is followed, namely the cut is placed 135 * along \e x > 18381 km, \e y = 0m. Forward can be called with any \e lat 136 * and \e lon then produces the transformation shown in Lee, Fig 46. 137 * Reverse analytically continues this in the ± \e x direction. As 138 * a consequence, Reverse may map multiple points to the same geographic 139 * location; for example, for TransverseMercatorExact::UTM, \e x = 140 * 22051449.037349 m, \e y = −7131237.022729 m and \e x = 141 * 29735142.378357 m, \e y = 4235043.607933 m both map to \e lat = 142 * −2°, \e lon = 88°. 143 * 144 * With \e extendp = true, the branch cut is moved to the lower left 145 * quadrant. The various symmetries of the transverse Mercator projection 146 * can be used to explore the projection on any sheet. In this mode the 147 * domains of \e lat, \e lon, \e x, and \e y are restricted to 148 * - the union of 149 * - \e lat in [0, 90] and \e lon − \e lon0 in [0, 90] 150 * - \e lat in (-90, 0] and \e lon − \e lon0 in [90 (1 − \e 151 e), 90] 152 * - the union of 153 * - <i>x</i>/(\e k0 \e a) in [0, ∞) and 154 * <i>y</i>/(\e k0 \e a) in [0, E(<i>e</i><sup>2</sup>)] 155 * - <i>x</i>/(\e k0 \e a) in [K(1 − <i>e</i><sup>2</sup>) − 156 * E(1 − <i>e</i><sup>2</sup>), ∞) and <i>y</i>/(\e k0 \e 157 * a) in (−∞, 0] 158 * . 159 * See Sec. 5 of 160 * <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for a full 161 * discussion of the treatment of the branch cut. 162 * 163 * The method will work for all ellipsoids used in terrestrial geodesy. 164 * The method cannot be applied directly to the case of a sphere (\e f = 0) 165 * because some the constants characterizing this method diverge in that 166 * limit, and in practice, \e f should be larger than about 167 * numeric_limits<real>::epsilon(). However, TransverseMercator treats the 168 * sphere exactly. 169 **********************************************************************/ 170 TransverseMercatorExact(real a, real f, real k0, bool extendp = false); 171 172 /** 173 * Forward projection, from geographic to transverse Mercator. 174 * 175 * @param[in] lon0 central meridian of the projection (degrees). 176 * @param[in] lat latitude of point (degrees). 177 * @param[in] lon longitude of point (degrees). 178 * @param[out] x easting of point (meters). 179 * @param[out] y northing of point (meters). 180 * @param[out] gamma meridian convergence at point (degrees). 181 * @param[out] k scale of projection at point. 182 * 183 * No false easting or northing is added. \e lat should be in the range 184 * [−90°, 90°]. 185 **********************************************************************/ 186 void Forward(real lon0, real lat, real lon, 187 real& x, real& y, real& gamma, real& k) const; 188 189 /** 190 * Reverse projection, from transverse Mercator to geographic. 191 * 192 * @param[in] lon0 central meridian of the projection (degrees). 193 * @param[in] x easting of point (meters). 194 * @param[in] y northing of point (meters). 195 * @param[out] lat latitude of point (degrees). 196 * @param[out] lon longitude of point (degrees). 197 * @param[out] gamma meridian convergence at point (degrees). 198 * @param[out] k scale of projection at point. 199 * 200 * No false easting or northing is added. The value of \e lon returned is 201 * in the range [−180°, 180°]. 202 **********************************************************************/ 203 void Reverse(real lon0, real x, real y, 204 real& lat, real& lon, real& gamma, real& k) const; 205 206 /** 207 * TransverseMercatorExact::Forward without returning the convergence and 208 * scale. 209 **********************************************************************/ Forward(real lon0,real lat,real lon,real & x,real & y) const210 void Forward(real lon0, real lat, real lon, 211 real& x, real& y) const { 212 real gamma, k; 213 Forward(lon0, lat, lon, x, y, gamma, k); 214 } 215 216 /** 217 * TransverseMercatorExact::Reverse without returning the convergence and 218 * scale. 219 **********************************************************************/ Reverse(real lon0,real x,real y,real & lat,real & lon) const220 void Reverse(real lon0, real x, real y, 221 real& lat, real& lon) const { 222 real gamma, k; 223 Reverse(lon0, x, y, lat, lon, gamma, k); 224 } 225 226 /** \name Inspector functions 227 **********************************************************************/ 228 ///@{ 229 /** 230 * @return \e a the equatorial radius of the ellipsoid (meters). This is 231 * the value used in the constructor. 232 **********************************************************************/ EquatorialRadius() const233 Math::real EquatorialRadius() const { return _a; } 234 235 /** 236 * @return \e f the flattening of the ellipsoid. This is the value used in 237 * the constructor. 238 **********************************************************************/ Flattening() const239 Math::real Flattening() const { return _f; } 240 241 /** 242 * @return \e k0 central scale for the projection. This is the value of \e 243 * k0 used in the constructor and is the scale on the central meridian. 244 **********************************************************************/ CentralScale() const245 Math::real CentralScale() const { return _k0; } 246 247 /** 248 * \deprecated An old name for EquatorialRadius(). 249 **********************************************************************/ 250 GEOGRAPHICLIB_DEPRECATED("Use EquatorialRadius()") MajorRadius() const251 Math::real MajorRadius() const { return EquatorialRadius(); } 252 ///@} 253 254 /** 255 * A global instantiation of TransverseMercatorExact with the WGS84 256 * ellipsoid and the UTM scale factor. However, unlike UTM, no false 257 * easting or northing is added. 258 **********************************************************************/ 259 static const TransverseMercatorExact& UTM(); 260 }; 261 262 } // namespace GeographicLib 263 264 #endif // GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP 265