GeographicLib: TransverseMercatorExact.hpp Source File

GeographicLib 2.7

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TransverseMercatorExact.hpp
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1/**
2 * \file TransverseMercatorExact.hpp
3 * \brief Header for GeographicLib::TransverseMercatorExact class
4 *
5 * Copyright (c) Charles Karney (2008-2023) <karney@alum.mit.edu> 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
16namespace 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 * <a href="https://archive.org/details/conformalproject0000leel/page/92">
31 * borrow from archive.org</a>.
32 * - C. F. F. Karney,
33 * <a href="https://doi.org/10.1007/s00190-011-0445-3">
34 * Transverse Mercator with an accuracy of a few nanometers,</a>
35 * J. Geodesy 85(8), 475--485 (Aug. 2011);
36 * preprint
37 * <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a>.
38 *
39 * Lee gives the correct results for forward and reverse transformations
40 * subject to the branch cut rules (see the description of the \e extendp
41 * argument to the constructor). The maximum error is about 8 nm (8
42 * nanometers), ground distance, for the forward and reverse transformations.
43 * The error in the convergence is 2 &times; 10<sup>&minus;15</sup>&quot;,
44 * the relative error in the scale is 7 &times; 10<sup>&minus;12</sup>%%.
45 * See Sec. 3 of
46 * <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for details.
47 * The method is "exact" in the sense that the errors are close to the
48 * round-off limit and that no changes are needed in the algorithms for them
49 * to be used with reals of a higher precision. Thus the errors using long
50 * double (with a 64-bit fraction) are about 2000 times smaller than using
51 * double (with a 53-bit fraction).
52 *
53 * This algorithm is about 4.5 times slower than the 6th-order Kr&uuml;ger
54 * method, TransverseMercator, taking about 11 us for a combined forward and
55 * reverse projection on a 2.66 GHz Intel machine (g++, version 4.3.0, -O3).
56 *
57 * The ellipsoid parameters and the central scale are set in the constructor.
58 * The central meridian (which is a trivial shift of the longitude) is
59 * specified as the \e lon0 argument of the TransverseMercatorExact::Forward
60 * and TransverseMercatorExact::Reverse functions. The latitude of origin is
61 * taken to be the equator. See the documentation on TransverseMercator for
62 * how to include a false easting, false northing, or a latitude of origin.
63 *
64 * See <a href="https://geographiclib.sourceforge.io/tm-grid.kmz"
65 * type="application/vnd.google-earth.kmz"> tm-grid.kmz</a>, for an
66 * illustration of the transverse Mercator grid in Google Earth.
67 *
68 * This class also returns the meridian convergence \e gamma and scale \e k.
69 * The meridian convergence is the bearing of grid north (the \e y axis)
70 * measured clockwise from true north.
71 *
72 * See TransverseMercatorExact.cpp for more information on the
73 * implementation.
74 *
75 * See \ref transversemercator for a discussion of this projection.
76 *
77 * Example of use:
78 * \include example-TransverseMercatorExact.cpp
79 *
80 * <a href="TransverseMercatorProj.1.html">TransverseMercatorProj</a> is a
81 * command-line utility providing access to the functionality of
82 * TransverseMercator and TransverseMercatorExact.
83 **********************************************************************/
84
85 class GEOGRAPHICLIB_EXPORT TransverseMercatorExact {
86 private:
87 typedef Math::real real;
88 friend class TransverseMercator; // Allow TM to call the default constructor
89 // Private default constructor to support TransverseMercator(a, f, exact)
90 TransverseMercatorExact() {}; // Do nothing; used with exact = false.
91 static const int numit_ = 10;
92 real tol_, tol2_, taytol_;
93 real _a, _f, _k0, _mu, _mv, _e;
94 bool _extendp;
95 EllipticFunction _eEu, _eEv;
96
97 void zeta(real u, real snu, real cnu, real dnu,
98 real v, real snv, real cnv, real dnv,
99 real& taup, real& lam) const;
100
101 void dwdzeta(real u, real snu, real cnu, real dnu,
102 real v, real snv, real cnv, real dnv,
103 real& du, real& dv) const;
104
105 bool zetainv0(real psi, real lam, real& u, real& v) const;
106 void zetainv(real taup, real lam, real& u, real& v) const;
107
108 void sigma(real u, real snu, real cnu, real dnu,
109 real v, real snv, real cnv, real dnv,
110 real& xi, real& eta) const;
111
112 void dwdsigma(real u, real snu, real cnu, real dnu,
113 real v, real snv, real cnv, real dnv,
114 real& du, real& dv) const;
115
116 bool sigmainv0(real xi, real eta, real& u, real& v) const;
117 void sigmainv(real xi, real eta, real& u, real& v) const;
118
119 void Scale(real tau, real lam,
120 real snu, real cnu, real dnu,
121 real snv, real cnv, real dnv,
122 real& gamma, real& k) const;
123 public:
124
125 /**
126 * Constructor for an ellipsoid with
127 *
128 * @param[in] a equatorial radius (meters).
129 * @param[in] f flattening of ellipsoid.
130 * @param[in] k0 central scale factor.
131 * @param[in] extendp if true, use extended domain (default false).
132 * @exception GeographicErr if \e a, \e f, or \e k0 is not positive.
133 *
134 * The transverse Mercator projection has a branch point singularity at \e
135 * lat = 0 and \e lon &minus; \e lon0 = 90 (1 &minus; \e e) or (for
136 * TransverseMercatorExact::UTM) x = 18381 km, y = 0m. The \e extendp
137 * argument governs where the branch cut is placed. With \e extendp =
138 * false, the "standard" convention is followed, namely the cut is placed
139 * along \e x > 18381 km, \e y = 0m. Forward can be called with any \e lat
140 * and \e lon then produces the transformation shown in Lee, Fig 46.
141 * Reverse analytically continues this in the &plusmn; \e x direction. As
142 * a consequence, Reverse may map multiple points to the same geographic
143 * location; for example, for TransverseMercatorExact::UTM, \e x =
144 * 22051449.037349 m, \e y = &minus;7131237.022729 m and \e x =
145 * 29735142.378357 m, \e y = 4235043.607933 m both map to \e lat =
146 * &minus;2&deg;, \e lon = 88&deg;.
147 *
148 * With \e extendp = true, the branch cut is moved to the lower left
149 * quadrant. The various symmetries of the transverse Mercator projection
150 * can be used to explore the projection on any sheet. In this mode the
151 * domains of \e lat, \e lon, \e x, and \e y are restricted to
152 * - the union of
153 * - \e lat in [0, 90] and \e lon &minus; \e lon0 in [0, 90]
154 * - \e lat in (-90, 0] and \e lon &minus; \e lon0 in [90 (1 &minus; \e
155 e), 90]
156 * - the union of
157 * - <i>x</i>/(\e k0 \e a) in [0, &infin;) and
158 * <i>y</i>/(\e k0 \e a) in [0, E(<i>e</i><sup>2</sup>)]
159 * - <i>x</i>/(\e k0 \e a) in [K(1 &minus; <i>e</i><sup>2</sup>) &minus;
160 * E(1 &minus; <i>e</i><sup>2</sup>), &infin;) and <i>y</i>/(\e k0 \e
161 * a) in (&minus;&infin;, 0]
162 * .
163 * See Sec. 5 of
164 * <a href="https://arxiv.org/abs/1002.1417">arXiv:1002.1417</a> for a full
165 * discussion of the treatment of the branch cut.
166 *
167 * The method will work for all ellipsoids used in terrestrial geodesy.
168 * The method cannot be applied directly to the case of a sphere (\e f = 0)
169 * because some the constants characterizing this method diverge in that
170 * limit, and in practice, \e f should be larger than about
171 * numeric_limits<real>::epsilon(). However, TransverseMercator treats the
172 * sphere exactly.
173 **********************************************************************/
174 TransverseMercatorExact(real a, real f, real k0, bool extendp = false);
175
176 /**
177 * Forward projection, from geographic to transverse Mercator.
178 *
179 * @param[in] lon0 central meridian of the projection (degrees).
180 * @param[in] lat latitude of point (degrees).
181 * @param[in] lon longitude of point (degrees).
182 * @param[out] x easting of point (meters).
183 * @param[out] y northing of point (meters).
184 * @param[out] gamma meridian convergence at point (degrees).
185 * @param[out] k scale of projection at point.
186 *
187 * No false easting or northing is added. \e lat should be in the range
188 * [&minus;90&deg;, 90&deg;].
189 **********************************************************************/
190 void Forward(real lon0, real lat, real lon,
191 real& x, real& y, real& gamma, real& k) const;
192
193 /**
194 * Reverse projection, from transverse Mercator to geographic.
195 *
196 * @param[in] lon0 central meridian of the projection (degrees).
197 * @param[in] x easting of point (meters).
198 * @param[in] y northing of point (meters).
199 * @param[out] lat latitude of point (degrees).
200 * @param[out] lon longitude of point (degrees).
201 * @param[out] gamma meridian convergence at point (degrees).
202 * @param[out] k scale of projection at point.
203 *
204 * No false easting or northing is added. The value of \e lon returned is
205 * in the range [&minus;180&deg;, 180&deg;].
206 **********************************************************************/
207 void Reverse(real lon0, real x, real y,
208 real& lat, real& lon, real& gamma, real& k) const;
209
210 /**
211 * TransverseMercatorExact::Forward without returning the convergence and
212 * scale.
213 **********************************************************************/
214 void Forward(real lon0, real lat, real lon,
215 real& x, real& y) const {
216 real gamma, k;
217 Forward(lon0, lat, lon, x, y, gamma, k);
218 }
219
220 /**
221 * TransverseMercatorExact::Reverse without returning the convergence and
222 * scale.
223 **********************************************************************/
224 void Reverse(real lon0, real x, real y,
225 real& lat, real& lon) const {
226 real gamma, k;
227 Reverse(lon0, x, y, lat, lon, gamma, k);
228 }
229
230 /** \name Inspector functions
231 **********************************************************************/
232 ///@{
233 /**
234 * @return \e a the equatorial radius of the ellipsoid (meters). This is
235 * the value used in the constructor.
236 **********************************************************************/
237 Math::real EquatorialRadius() const { return _a; }
238
239 /**
240 * @return \e f the flattening of the ellipsoid. This is the value used in
241 * the constructor.
242 **********************************************************************/
243 Math::real Flattening() const { return _f; }
244
245 /**
246 * @return \e k0 central scale for the projection. This is the value of \e
247 * k0 used in the constructor and is the scale on the central meridian.
248 **********************************************************************/
249 Math::real CentralScale() const { return _k0; }
250 ///@}
251
252 /**
253 * A global instantiation of TransverseMercatorExact with the WGS84
254 * ellipsoid and the UTM scale factor. However, unlike UTM, no false
255 * easting or northing is added.
256 **********************************************************************/
257 static const TransverseMercatorExact& UTM();
258 };
259
260} // namespace GeographicLib
261
262#endif // GEOGRAPHICLIB_TRANSVERSEMERCATOREXACT_HPP
Constants.hpp
Header for GeographicLib::Constants class.
GEOGRAPHICLIB_EXPORT
#define GEOGRAPHICLIB_EXPORT
Definition Constants.hpp:59
EllipticFunction.hpp
Header for GeographicLib::EllipticFunction class.
real
GeographicLib::Math::real real
Definition Geod3Solve.cpp:25
GeographicLib::EllipticFunction
Elliptic integrals and functions.
Definition EllipticFunction.hpp:63
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib::TransverseMercatorExact
An exact implementation of the transverse Mercator projection.
Definition TransverseMercatorExact.hpp:85
GeographicLib::TransverseMercatorExact::TransverseMercator
friend class TransverseMercator
Definition TransverseMercatorExact.hpp:88
GeographicLib::TransverseMercatorExact::CentralScale
Math::real CentralScale() const
Definition TransverseMercatorExact.hpp:249
GeographicLib::TransverseMercatorExact::Forward
void Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const
Definition TransverseMercatorExact.cpp:353
GeographicLib::TransverseMercatorExact::Reverse
void Reverse(real lon0, real x, real y, real &lat, real &lon) const
Definition TransverseMercatorExact.hpp:224
GeographicLib::TransverseMercatorExact::EquatorialRadius
Math::real EquatorialRadius() const
Definition TransverseMercatorExact.hpp:237
GeographicLib::TransverseMercatorExact::Forward
void Forward(real lon0, real lat, real lon, real &x, real &y) const
Definition TransverseMercatorExact.hpp:214
GeographicLib::TransverseMercatorExact::Flattening
Math::real Flattening() const
Definition TransverseMercatorExact.hpp:243
GeographicLib::TransverseMercatorExact::Reverse
void Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const
Definition TransverseMercatorExact.cpp:413
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12