GeographicLib  1.43
TransverseMercatorExact.cpp
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1 /**
2  * \file TransverseMercatorExact.cpp
3  * \brief Implementation for GeographicLib::TransverseMercatorExact class
4  *
5  * Copyright (c) Charles Karney (2008-2015) <charles@karney.com> and licensed
6  * under the MIT/X11 License. For more information, see
7  * http://geographiclib.sourceforge.net/
8  *
9  * The relevant section of Lee's paper is part V, pp 67--101,
10  * <a href="https://dx.doi.org/10.3138/X687-1574-4325-WM62">Conformal
11  * Projections Based On Jacobian Elliptic Functions</a>.
12  *
13  * The method entails using the Thompson Transverse Mercator as an
14  * intermediate projection. The projections from the intermediate
15  * coordinates to [\e phi, \e lam] and [\e x, \e y] are given by elliptic
16  * functions. The inverse of these projections are found by Newton's method
17  * with a suitable starting guess.
18  *
19  * This implementation and notation closely follows Lee, with the following
20  * exceptions:
21  * <center><table>
22  * <tr><th>Lee <th>here <th>Description
23  * <tr><td>x/a <td>xi <td>Northing (unit Earth)
24  * <tr><td>y/a <td>eta <td>Easting (unit Earth)
25  * <tr><td>s/a <td>sigma <td>xi + i * eta
26  * <tr><td>y <td>x <td>Easting
27  * <tr><td>x <td>y <td>Northing
28  * <tr><td>k <td>e <td>eccentricity
29  * <tr><td>k^2 <td>mu <td>elliptic function parameter
30  * <tr><td>k'^2 <td>mv <td>elliptic function complementary parameter
31  * <tr><td>m <td>k <td>scale
32  * <tr><td>zeta <td>zeta <td>complex longitude = Mercator = chi in paper
33  * <tr><td>s <td>sigma <td>complex GK = zeta in paper
34  * </table></center>
35  *
36  * Minor alterations have been made in some of Lee's expressions in an
37  * attempt to control round-off. For example atanh(sin(phi)) is replaced by
38  * asinh(tan(phi)) which maintains accuracy near phi = pi/2. Such changes
39  * are noted in the code.
40  **********************************************************************/
41 
43 
44 #if defined(_MSC_VER)
45 // Squelch warnings about constant conditional expressions
46 # pragma warning (disable: 4127)
47 #endif
48 
49 namespace GeographicLib {
50 
51  using namespace std;
52 
54  bool extendp)
55  : tol_(numeric_limits<real>::epsilon())
56  , tol1_(real(0.1) * sqrt(tol_))
57  , tol2_(real(0.1) * tol_)
58  , taytol_(pow(tol_, real(0.6)))
59  , _a(a)
60  , _f(f <= 1 ? f : 1/f)
61  , _k0(k0)
62  , _mu(_f * (2 - _f)) // e^2
63  , _mv(1 - _mu) // 1 - e^2
64  , _e(sqrt(_mu))
65  , _extendp(extendp)
66  , _Eu(_mu)
67  , _Ev(_mv)
68  {
69  if (!(Math::isfinite(_a) && _a > 0))
70  throw GeographicErr("Major radius is not positive");
71  if (!(_f > 0))
72  throw GeographicErr("Flattening is not positive");
73  if (!(_f < 1))
74  throw GeographicErr("Minor radius is not positive");
75  if (!(Math::isfinite(_k0) && _k0 > 0))
76  throw GeographicErr("Scale is not positive");
77  }
78 
83  return utm;
84  }
85 
86  void TransverseMercatorExact::zeta(real /*u*/, real snu, real cnu, real dnu,
87  real /*v*/, real snv, real cnv, real dnv,
88  real& taup, real& lam) const {
89  // Lee 54.17 but write
90  // atanh(snu * dnv) = asinh(snu * dnv / sqrt(cnu^2 + _mv * snu^2 * snv^2))
91  // atanh(_e * snu / dnv) =
92  // asinh(_e * snu / sqrt(_mu * cnu^2 + _mv * cnv^2))
93  real
94  d1 = sqrt(Math::sq(cnu) + _mv * Math::sq(snu * snv)),
95  d2 = sqrt(_mu * Math::sq(cnu) + _mv * Math::sq(cnv)),
96  t1 = (d1 ? snu * dnv / d1 : (snu < 0 ? -overflow() : overflow())),
97  t2 = (d2 ? sinh( _e * Math::asinh(_e * snu / d2) ) :
98  (snu < 0 ? -overflow() : overflow()));
99  // psi = asinh(t1) - asinh(t2)
100  // taup = sinh(psi)
101  taup = t1 * Math::hypot(real(1), t2) - t2 * Math::hypot(real(1), t1);
102  lam = (d1 != 0 && d2 != 0) ?
103  atan2(dnu * snv, cnu * cnv) - _e * atan2(_e * cnu * snv, dnu * cnv) :
104  0;
105  }
106 
107  void TransverseMercatorExact::dwdzeta(real /*u*/,
108  real snu, real cnu, real dnu,
109  real /*v*/,
110  real snv, real cnv, real dnv,
111  real& du, real& dv) const {
112  // Lee 54.21 but write (1 - dnu^2 * snv^2) = (cnv^2 + _mu * snu^2 * snv^2)
113  // (see A+S 16.21.4)
114  real d = _mv * Math::sq(Math::sq(cnv) + _mu * Math::sq(snu * snv));
115  du = cnu * dnu * dnv * (Math::sq(cnv) - _mu * Math::sq(snu * snv)) / d;
116  dv = -snu * snv * cnv * (Math::sq(dnu * dnv) + _mu * Math::sq(cnu)) / d;
117  }
118 
119  // Starting point for zetainv
120  bool TransverseMercatorExact::zetainv0(real psi, real lam, real& u, real& v)
121  const {
122  bool retval = false;
123  if (psi < -_e * Math::pi()/4 &&
124  lam > (1 - 2 * _e) * Math::pi()/2 &&
125  psi < lam - (1 - _e) * Math::pi()/2) {
126  // N.B. this branch is normally not taken because psi < 0 is converted
127  // psi > 0 by Forward.
128  //
129  // There's a log singularity at w = w0 = Eu.K() + i * Ev.K(),
130  // corresponding to the south pole, where we have, approximately
131  //
132  // psi = _e + i * pi/2 - _e * atanh(cos(i * (w - w0)/(1 + _mu/2)))
133  //
134  // Inverting this gives:
135  real
136  psix = 1 - psi / _e,
137  lamx = (Math::pi()/2 - lam) / _e;
138  u = Math::asinh(sin(lamx) / Math::hypot(cos(lamx), sinh(psix))) *
139  (1 + _mu/2);
140  v = atan2(cos(lamx), sinh(psix)) * (1 + _mu/2);
141  u = _Eu.K() - u;
142  v = _Ev.K() - v;
143  } else if (psi < _e * Math::pi()/2 &&
144  lam > (1 - 2 * _e) * Math::pi()/2) {
145  // At w = w0 = i * Ev.K(), we have
146  //
147  // zeta = zeta0 = i * (1 - _e) * pi/2
148  // zeta' = zeta'' = 0
149  //
150  // including the next term in the Taylor series gives:
151  //
152  // zeta = zeta0 - (_mv * _e) / 3 * (w - w0)^3
153  //
154  // When inverting this, we map arg(w - w0) = [-90, 0] to
155  // arg(zeta - zeta0) = [-90, 180]
156  real
157  dlam = lam - (1 - _e) * Math::pi()/2,
158  rad = Math::hypot(psi, dlam),
159  // atan2(dlam-psi, psi+dlam) + 45d gives arg(zeta - zeta0) in range
160  // [-135, 225). Subtracting 180 (since multiplier is negative) makes
161  // range [-315, 45). Multiplying by 1/3 (for cube root) gives range
162  // [-105, 15). In particular the range [-90, 180] in zeta space maps
163  // to [-90, 0] in w space as required.
164  ang = atan2(dlam-psi, psi+dlam) - real(0.75) * Math::pi();
165  // Error using this guess is about 0.21 * (rad/e)^(5/3)
166  retval = rad < _e * taytol_;
167  rad = Math::cbrt(3 / (_mv * _e) * rad);
168  ang /= 3;
169  u = rad * cos(ang);
170  v = rad * sin(ang) + _Ev.K();
171  } else {
172  // Use spherical TM, Lee 12.6 -- writing atanh(sin(lam) / cosh(psi)) =
173  // asinh(sin(lam) / hypot(cos(lam), sinh(psi))). This takes care of the
174  // log singularity at zeta = Eu.K() (corresponding to the north pole)
175  v = Math::asinh(sin(lam) / Math::hypot(cos(lam), sinh(psi)));
176  u = atan2(sinh(psi), cos(lam));
177  // But scale to put 90,0 on the right place
178  u *= _Eu.K() / (Math::pi()/2);
179  v *= _Eu.K() / (Math::pi()/2);
180  }
181  return retval;
182  }
183 
184  // Invert zeta using Newton's method
185  void TransverseMercatorExact::zetainv(real taup, real lam, real& u, real& v)
186  const {
187  real
188  psi = Math::asinh(taup),
189  scal = 1/Math::hypot(real(1), taup);
190  if (zetainv0(psi, lam, u, v))
191  return;
192  real stol2 = tol2_ / Math::sq(max(psi, real(1)));
193  // min iterations = 2, max iterations = 6; mean = 4.0
194  for (int i = 0, trip = 0; i < numit_ || GEOGRAPHICLIB_PANIC; ++i) {
195  real snu, cnu, dnu, snv, cnv, dnv;
196  _Eu.sncndn(u, snu, cnu, dnu);
197  _Ev.sncndn(v, snv, cnv, dnv);
198  real tau1, lam1, du1, dv1;
199  zeta(u, snu, cnu, dnu, v, snv, cnv, dnv, tau1, lam1);
200  dwdzeta(u, snu, cnu, dnu, v, snv, cnv, dnv, du1, dv1);
201  tau1 -= taup;
202  lam1 -= lam;
203  tau1 *= scal;
204  real
205  delu = tau1 * du1 - lam1 * dv1,
206  delv = tau1 * dv1 + lam1 * du1;
207  u -= delu;
208  v -= delv;
209  if (trip)
210  break;
211  real delw2 = Math::sq(delu) + Math::sq(delv);
212  if (!(delw2 >= stol2))
213  ++trip;
214  }
215  }
216 
217  void TransverseMercatorExact::sigma(real /*u*/, real snu, real cnu, real dnu,
218  real v, real snv, real cnv, real dnv,
219  real& xi, real& eta) const {
220  // Lee 55.4 writing
221  // dnu^2 + dnv^2 - 1 = _mu * cnu^2 + _mv * cnv^2
222  real d = _mu * Math::sq(cnu) + _mv * Math::sq(cnv);
223  xi = _Eu.E(snu, cnu, dnu) - _mu * snu * cnu * dnu / d;
224  eta = v - _Ev.E(snv, cnv, dnv) + _mv * snv * cnv * dnv / d;
225  }
226 
227  void TransverseMercatorExact::dwdsigma(real /*u*/,
228  real snu, real cnu, real dnu,
229  real /*v*/,
230  real snv, real cnv, real dnv,
231  real& du, real& dv) const {
232  // Reciprocal of 55.9: dw/ds = dn(w)^2/_mv, expanding complex dn(w) using
233  // A+S 16.21.4
234  real d = _mv * Math::sq(Math::sq(cnv) + _mu * Math::sq(snu * snv));
235  real
236  dnr = dnu * cnv * dnv,
237  dni = - _mu * snu * cnu * snv;
238  du = (Math::sq(dnr) - Math::sq(dni)) / d;
239  dv = 2 * dnr * dni / d;
240  }
241 
242  // Starting point for sigmainv
243  bool TransverseMercatorExact::sigmainv0(real xi, real eta, real& u, real& v)
244  const {
245  bool retval = false;
246  if (eta > real(1.25) * _Ev.KE() ||
247  (xi < -real(0.25) * _Eu.E() && xi < eta - _Ev.KE())) {
248  // sigma as a simple pole at w = w0 = Eu.K() + i * Ev.K() and sigma is
249  // approximated by
250  //
251  // sigma = (Eu.E() + i * Ev.KE()) + 1/(w - w0)
252  real
253  x = xi - _Eu.E(),
254  y = eta - _Ev.KE(),
255  r2 = Math::sq(x) + Math::sq(y);
256  u = _Eu.K() + x/r2;
257  v = _Ev.K() - y/r2;
258  } else if ((eta > real(0.75) * _Ev.KE() && xi < real(0.25) * _Eu.E())
259  || eta > _Ev.KE()) {
260  // At w = w0 = i * Ev.K(), we have
261  //
262  // sigma = sigma0 = i * Ev.KE()
263  // sigma' = sigma'' = 0
264  //
265  // including the next term in the Taylor series gives:
266  //
267  // sigma = sigma0 - _mv / 3 * (w - w0)^3
268  //
269  // When inverting this, we map arg(w - w0) = [-pi/2, -pi/6] to
270  // arg(sigma - sigma0) = [-pi/2, pi/2]
271  // mapping arg = [-pi/2, -pi/6] to [-pi/2, pi/2]
272  real
273  deta = eta - _Ev.KE(),
274  rad = Math::hypot(xi, deta),
275  // Map the range [-90, 180] in sigma space to [-90, 0] in w space. See
276  // discussion in zetainv0 on the cut for ang.
277  ang = atan2(deta-xi, xi+deta) - real(0.75) * Math::pi();
278  // Error using this guess is about 0.068 * rad^(5/3)
279  retval = rad < 2 * taytol_;
280  rad = Math::cbrt(3 / _mv * rad);
281  ang /= 3;
282  u = rad * cos(ang);
283  v = rad * sin(ang) + _Ev.K();
284  } else {
285  // Else use w = sigma * Eu.K/Eu.E (which is correct in the limit _e -> 0)
286  u = xi * _Eu.K()/_Eu.E();
287  v = eta * _Eu.K()/_Eu.E();
288  }
289  return retval;
290  }
291 
292  // Invert sigma using Newton's method
293  void TransverseMercatorExact::sigmainv(real xi, real eta, real& u, real& v)
294  const {
295  if (sigmainv0(xi, eta, u, v))
296  return;
297  // min iterations = 2, max iterations = 7; mean = 3.9
298  for (int i = 0, trip = 0; i < numit_ || GEOGRAPHICLIB_PANIC; ++i) {
299  real snu, cnu, dnu, snv, cnv, dnv;
300  _Eu.sncndn(u, snu, cnu, dnu);
301  _Ev.sncndn(v, snv, cnv, dnv);
302  real xi1, eta1, du1, dv1;
303  sigma(u, snu, cnu, dnu, v, snv, cnv, dnv, xi1, eta1);
304  dwdsigma(u, snu, cnu, dnu, v, snv, cnv, dnv, du1, dv1);
305  xi1 -= xi;
306  eta1 -= eta;
307  real
308  delu = xi1 * du1 - eta1 * dv1,
309  delv = xi1 * dv1 + eta1 * du1;
310  u -= delu;
311  v -= delv;
312  if (trip)
313  break;
314  real delw2 = Math::sq(delu) + Math::sq(delv);
315  if (!(delw2 >= tol2_))
316  ++trip;
317  }
318  }
319 
320  void TransverseMercatorExact::Scale(real tau, real /*lam*/,
321  real snu, real cnu, real dnu,
322  real snv, real cnv, real dnv,
323  real& gamma, real& k) const {
324  real sec2 = 1 + Math::sq(tau); // sec(phi)^2
325  // Lee 55.12 -- negated for our sign convention. gamma gives the bearing
326  // (clockwise from true north) of grid north
327  gamma = atan2(_mv * snu * snv * cnv, cnu * dnu * dnv);
328  // Lee 55.13 with nu given by Lee 9.1 -- in sqrt change the numerator
329  // from
330  //
331  // (1 - snu^2 * dnv^2) to (_mv * snv^2 + cnu^2 * dnv^2)
332  //
333  // to maintain accuracy near phi = 90 and change the denomintor from
334  //
335  // (dnu^2 + dnv^2 - 1) to (_mu * cnu^2 + _mv * cnv^2)
336  //
337  // to maintain accuracy near phi = 0, lam = 90 * (1 - e). Similarly
338  // rewrite sqrt term in 9.1 as
339  //
340  // _mv + _mu * c^2 instead of 1 - _mu * sin(phi)^2
341  k = sqrt(_mv + _mu / sec2) * sqrt(sec2) *
342  sqrt( (_mv * Math::sq(snv) + Math::sq(cnu * dnv)) /
343  (_mu * Math::sq(cnu) + _mv * Math::sq(cnv)) );
344  }
345 
346  void TransverseMercatorExact::Forward(real lon0, real lat, real lon,
347  real& x, real& y, real& gamma, real& k)
348  const {
350  // Explicitly enforce the parity
351  int
352  latsign = (!_extendp && lat < 0) ? -1 : 1,
353  lonsign = (!_extendp && lon < 0) ? -1 : 1;
354  lon *= lonsign;
355  lat *= latsign;
356  bool backside = !_extendp && lon > 90;
357  if (backside) {
358  if (lat == 0)
359  latsign = -1;
360  lon = 180 - lon;
361  }
362  real
363  lam = lon * Math::degree(),
364  tau = Math::tand(lat);
365 
366  // u,v = coordinates for the Thompson TM, Lee 54
367  real u, v;
368  if (lat == 90) {
369  u = _Eu.K();
370  v = 0;
371  } else if (lat == 0 && lon == 90 * (1 - _e)) {
372  u = 0;
373  v = _Ev.K();
374  } else
375  // tau = tan(phi), taup = sinh(psi)
376  zetainv(Math::taupf(tau, _e), lam, u, v);
377 
378  real snu, cnu, dnu, snv, cnv, dnv;
379  _Eu.sncndn(u, snu, cnu, dnu);
380  _Ev.sncndn(v, snv, cnv, dnv);
381 
382  real xi, eta;
383  sigma(u, snu, cnu, dnu, v, snv, cnv, dnv, xi, eta);
384  if (backside)
385  xi = 2 * _Eu.E() - xi;
386  y = xi * _a * _k0 * latsign;
387  x = eta * _a * _k0 * lonsign;
388 
389  if (lat == 90) {
390  gamma = lon;
391  k = 1;
392  } else {
393  // Recompute (tau, lam) from (u, v) to improve accuracy of Scale
394  zeta(u, snu, cnu, dnu, v, snv, cnv, dnv, tau, lam);
395  tau = Math::tauf(tau, _e);
396  Scale(tau, lam, snu, cnu, dnu, snv, cnv, dnv, gamma, k);
397  gamma /= Math::degree();
398  }
399  if (backside)
400  gamma = 180 - gamma;
401  gamma *= latsign * lonsign;
402  k *= _k0;
403  }
404 
405  void TransverseMercatorExact::Reverse(real lon0, real x, real y,
406  real& lat, real& lon,
407  real& gamma, real& k)
408  const {
409  // This undoes the steps in Forward.
410  real
411  xi = y / (_a * _k0),
412  eta = x / (_a * _k0);
413  // Explicitly enforce the parity
414  int
415  latsign = !_extendp && y < 0 ? -1 : 1,
416  lonsign = !_extendp && x < 0 ? -1 : 1;
417  xi *= latsign;
418  eta *= lonsign;
419  bool backside = !_extendp && xi > _Eu.E();
420  if (backside)
421  xi = 2 * _Eu.E()- xi;
422 
423  // u,v = coordinates for the Thompson TM, Lee 54
424  real u, v;
425  if (xi == 0 && eta == _Ev.KE()) {
426  u = 0;
427  v = _Ev.K();
428  } else
429  sigmainv(xi, eta, u, v);
430 
431  real snu, cnu, dnu, snv, cnv, dnv;
432  _Eu.sncndn(u, snu, cnu, dnu);
433  _Ev.sncndn(v, snv, cnv, dnv);
434  real phi, lam, tau;
435  if (v != 0 || u != _Eu.K()) {
436  zeta(u, snu, cnu, dnu, v, snv, cnv, dnv, tau, lam);
437  tau = Math::tauf(tau, _e);
438  phi = atan(tau);
439  lat = phi / Math::degree();
440  lon = lam / Math::degree();
441  Scale(tau, lam, snu, cnu, dnu, snv, cnv, dnv, gamma, k);
442  gamma /= Math::degree();
443  } else {
444  lat = 90;
445  lon = lam = gamma = 0;
446  k = 1;
447  }
448 
449  if (backside)
450  lon = 180 - lon;
451  lon *= lonsign;
452  lon = Math::AngNormalize(lon + Math::AngNormalize(lon0));
453  lat *= latsign;
454  if (backside)
455  gamma = 180 - gamma;
456  gamma *= latsign * lonsign;
457  k *= _k0;
458  }
459 
460 } // namespace GeographicLib
static T AngNormalize(T x)
Definition: Math.hpp:445
static T pi()
Definition: Math.hpp:214
static const TransverseMercatorExact & UTM()
An exact implementation of the transverse Mercator projection.
GeographicLib::Math::real real
Definition: GeodSolve.cpp:32
static T cbrt(T x)
Definition: Math.hpp:357
static bool isfinite(T x)
Definition: Math.hpp:614
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:102
Header for GeographicLib::TransverseMercatorExact class.
static T asinh(T x)
Definition: Math.hpp:323
static T hypot(T x, T y)
Definition: Math.hpp:255
TransverseMercatorExact(real a, real f, real k0, bool extendp=false)
static T sq(T x)
Definition: Math.hpp:244
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
void sncndn(real x, real &sn, real &cn, real &dn) const
void Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const
void Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const
static T degree()
Definition: Math.hpp:228
static T AngDiff(T x, T y)
Definition: Math.hpp:475
static T tand(T x)
Definition: Math.hpp:517
static T tauf(T taup, T es)
Exception handling for GeographicLib.
Definition: Constants.hpp:382
static T taupf(T tau, T es)
#define GEOGRAPHICLIB_PANIC
Definition: Math.hpp:87