GeographicLib  2.1
TransverseMercator.cpp
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1 /**
2  * \file TransverseMercator.cpp
3  * \brief Implementation for GeographicLib::TransverseMercator class
4  *
5  * Copyright (c) Charles Karney (2008-2022) <charles@karney.com> and licensed
6  * under the MIT/X11 License. For more information, see
7  * https://geographiclib.sourceforge.io/
8  *
9  * This implementation follows closely JHS 154, ETRS89 -
10  * j&auml;rjestelm&auml;&auml;n liittyv&auml;t karttaprojektiot,
11  * tasokoordinaatistot ja karttalehtijako</a> (Map projections, plane
12  * coordinates, and map sheet index for ETRS89), published by JUHTA, Finnish
13  * Geodetic Institute, and the National Land Survey of Finland (2006).
14  *
15  * The relevant section is available as the 2008 PDF file
16  * http://docs.jhs-suositukset.fi/jhs-suositukset/JHS154/JHS154_liite1.pdf
17  *
18  * This is a straight transcription of the formulas in this paper with the
19  * following exceptions:
20  * - use of 6th order series instead of 4th order series. This reduces the
21  * error to about 5nm for the UTM range of coordinates (instead of 200nm),
22  * with a speed penalty of only 1%;
23  * - use Newton's method instead of plain iteration to solve for latitude in
24  * terms of isometric latitude in the Reverse method;
25  * - use of Horner's representation for evaluating polynomials and Clenshaw's
26  * method for summing trigonometric series;
27  * - several modifications of the formulas to improve the numerical accuracy;
28  * - evaluating the convergence and scale using the expression for the
29  * projection or its inverse.
30  *
31  * If the preprocessor variable GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER is set
32  * to an integer between 4 and 8, then this specifies the order of the series
33  * used for the forward and reverse transformations. The default value is 6.
34  * (The series accurate to 12th order is given in \ref tmseries.)
35  **********************************************************************/
36 
37 #include <complex>
39 
40 #if defined(_MSC_VER)
41 // Squelch warnings about enum-float expressions
42 # pragma warning (disable: 5055)
43 #endif
44 
45 namespace GeographicLib {
46 
47  using namespace std;
48 
49  TransverseMercator::TransverseMercator(real a, real f, real k0)
50  : _a(a)
51  , _f(f)
52  , _k0(k0)
53  , _e2(_f * (2 - _f))
54  , _es((_f < 0 ? -1 : 1) * sqrt(fabs(_e2)))
55  , _e2m(1 - _e2)
56  // _c = sqrt( pow(1 + _e, 1 + _e) * pow(1 - _e, 1 - _e) ) )
57  // See, for example, Lee (1976), p 100.
58  , _c( sqrt(_e2m) * exp(Math::eatanhe(real(1), _es)) )
59  , _n(_f / (2 - _f))
60  {
61  if (!(isfinite(_a) && _a > 0))
62  throw GeographicErr("Equatorial radius is not positive");
63  if (!(isfinite(_f) && _f < 1))
64  throw GeographicErr("Polar semi-axis is not positive");
65  if (!(isfinite(_k0) && _k0 > 0))
66  throw GeographicErr("Scale is not positive");
67 
68  // Generated by Maxima on 2015-05-14 22:55:13-04:00
69 #if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2
70  static const real b1coeff[] = {
71  // b1*(n+1), polynomial in n2 of order 2
72  1, 16, 64, 64,
73  }; // count = 4
74 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3
75  static const real b1coeff[] = {
76  // b1*(n+1), polynomial in n2 of order 3
77  1, 4, 64, 256, 256,
78  }; // count = 5
79 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4
80  static const real b1coeff[] = {
81  // b1*(n+1), polynomial in n2 of order 4
82  25, 64, 256, 4096, 16384, 16384,
83  }; // count = 6
84 #else
85 #error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER"
86 #endif
87 
88 #if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
89  static const real alpcoeff[] = {
90  // alp[1]/n^1, polynomial in n of order 3
91  164, 225, -480, 360, 720,
92  // alp[2]/n^2, polynomial in n of order 2
93  557, -864, 390, 1440,
94  // alp[3]/n^3, polynomial in n of order 1
95  -1236, 427, 1680,
96  // alp[4]/n^4, polynomial in n of order 0
97  49561, 161280,
98  }; // count = 14
99 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
100  static const real alpcoeff[] = {
101  // alp[1]/n^1, polynomial in n of order 4
102  -635, 328, 450, -960, 720, 1440,
103  // alp[2]/n^2, polynomial in n of order 3
104  4496, 3899, -6048, 2730, 10080,
105  // alp[3]/n^3, polynomial in n of order 2
106  15061, -19776, 6832, 26880,
107  // alp[4]/n^4, polynomial in n of order 1
108  -171840, 49561, 161280,
109  // alp[5]/n^5, polynomial in n of order 0
110  34729, 80640,
111  }; // count = 20
112 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
113  static const real alpcoeff[] = {
114  // alp[1]/n^1, polynomial in n of order 5
115  31564, -66675, 34440, 47250, -100800, 75600, 151200,
116  // alp[2]/n^2, polynomial in n of order 4
117  -1983433, 863232, 748608, -1161216, 524160, 1935360,
118  // alp[3]/n^3, polynomial in n of order 3
119  670412, 406647, -533952, 184464, 725760,
120  // alp[4]/n^4, polynomial in n of order 2
121  6601661, -7732800, 2230245, 7257600,
122  // alp[5]/n^5, polynomial in n of order 1
123  -13675556, 3438171, 7983360,
124  // alp[6]/n^6, polynomial in n of order 0
125  212378941, 319334400,
126  }; // count = 27
127 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
128  static const real alpcoeff[] = {
129  // alp[1]/n^1, polynomial in n of order 6
130  1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800,
131  // alp[2]/n^2, polynomial in n of order 5
132  4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800,
133  // alp[3]/n^3, polynomial in n of order 4
134  -67102379, 26816480, 16265880, -21358080, 7378560, 29030400,
135  // alp[4]/n^4, polynomial in n of order 3
136  155912000, 72618271, -85060800, 24532695, 79833600,
137  // alp[5]/n^5, polynomial in n of order 2
138  102508609, -109404448, 27505368, 63866880,
139  // alp[6]/n^6, polynomial in n of order 1
140  -12282192400LL, 2760926233LL, 4151347200LL,
141  // alp[7]/n^7, polynomial in n of order 0
142  1522256789, 1383782400,
143  }; // count = 35
144 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
145  static const real alpcoeff[] = {
146  // alp[1]/n^1, polynomial in n of order 7
147  -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200,
148  101606400, 203212800,
149  // alp[2]/n^2, polynomial in n of order 6
150  148003883, 83274912, -178508970, 77690880, 67374720, -104509440,
151  47174400, 174182400,
152  // alp[3]/n^3, polynomial in n of order 5
153  318729724, -738126169, 294981280, 178924680, -234938880, 81164160,
154  319334400,
155  // alp[4]/n^4, polynomial in n of order 4
156  -40176129013LL, 14967552000LL, 6971354016LL, -8165836800LL, 2355138720LL,
157  7664025600LL,
158  // alp[5]/n^5, polynomial in n of order 3
159  10421654396LL, 3997835751LL, -4266773472LL, 1072709352, 2490808320LL,
160  // alp[6]/n^6, polynomial in n of order 2
161  175214326799LL, -171950693600LL, 38652967262LL, 58118860800LL,
162  // alp[7]/n^7, polynomial in n of order 1
163  -67039739596LL, 13700311101LL, 12454041600LL,
164  // alp[8]/n^8, polynomial in n of order 0
165  1424729850961LL, 743921418240LL,
166  }; // count = 44
167 #else
168 #error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER"
169 #endif
170 
171 #if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
172  static const real betcoeff[] = {
173  // bet[1]/n^1, polynomial in n of order 3
174  -4, 555, -960, 720, 1440,
175  // bet[2]/n^2, polynomial in n of order 2
176  -437, 96, 30, 1440,
177  // bet[3]/n^3, polynomial in n of order 1
178  -148, 119, 3360,
179  // bet[4]/n^4, polynomial in n of order 0
180  4397, 161280,
181  }; // count = 14
182 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
183  static const real betcoeff[] = {
184  // bet[1]/n^1, polynomial in n of order 4
185  -3645, -64, 8880, -15360, 11520, 23040,
186  // bet[2]/n^2, polynomial in n of order 3
187  4416, -3059, 672, 210, 10080,
188  // bet[3]/n^3, polynomial in n of order 2
189  -627, -592, 476, 13440,
190  // bet[4]/n^4, polynomial in n of order 1
191  -3520, 4397, 161280,
192  // bet[5]/n^5, polynomial in n of order 0
193  4583, 161280,
194  }; // count = 20
195 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
196  static const real betcoeff[] = {
197  // bet[1]/n^1, polynomial in n of order 5
198  384796, -382725, -6720, 932400, -1612800, 1209600, 2419200,
199  // bet[2]/n^2, polynomial in n of order 4
200  -1118711, 1695744, -1174656, 258048, 80640, 3870720,
201  // bet[3]/n^3, polynomial in n of order 3
202  22276, -16929, -15984, 12852, 362880,
203  // bet[4]/n^4, polynomial in n of order 2
204  -830251, -158400, 197865, 7257600,
205  // bet[5]/n^5, polynomial in n of order 1
206  -435388, 453717, 15966720,
207  // bet[6]/n^6, polynomial in n of order 0
208  20648693, 638668800,
209  }; // count = 27
210 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
211  static const real betcoeff[] = {
212  // bet[1]/n^1, polynomial in n of order 6
213  -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600,
214  38707200,
215  // bet[2]/n^2, polynomial in n of order 5
216  829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600,
217  // bet[3]/n^3, polynomial in n of order 4
218  9261899, 3564160, -2708640, -2557440, 2056320, 58060800,
219  // bet[4]/n^4, polynomial in n of order 3
220  14928352, -9132761, -1742400, 2176515, 79833600,
221  // bet[5]/n^5, polynomial in n of order 2
222  -8005831, -1741552, 1814868, 63866880,
223  // bet[6]/n^6, polynomial in n of order 1
224  -261810608, 268433009, 8302694400LL,
225  // bet[7]/n^7, polynomial in n of order 0
226  219941297, 5535129600LL,
227  }; // count = 35
228 #elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
229  static const real betcoeff[] = {
230  // bet[1]/n^1, polynomial in n of order 7
231  31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600,
232  135475200, 270950400,
233  // bet[2]/n^2, polynomial in n of order 6
234  24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600,
235  348364800,
236  // bet[3]/n^3, polynomial in n of order 5
237  -232468668, 101880889, 39205760, -29795040, -28131840, 22619520,
238  638668800,
239  // bet[4]/n^4, polynomial in n of order 4
240  324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600LL,
241  // bet[5]/n^5, polynomial in n of order 3
242  457888660, -312227409, -67920528, 70779852, 2490808320LL,
243  // bet[6]/n^6, polynomial in n of order 2
244  -19841813847LL, -3665348512LL, 3758062126LL, 116237721600LL,
245  // bet[7]/n^7, polynomial in n of order 1
246  -1989295244, 1979471673, 49816166400LL,
247  // bet[8]/n^8, polynomial in n of order 0
248  191773887257LL, 3719607091200LL,
249  }; // count = 44
250 #else
251 #error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER"
252 #endif
253 
254  static_assert(sizeof(b1coeff) / sizeof(real) == maxpow_/2 + 2,
255  "Coefficient array size mismatch for b1");
256  static_assert(sizeof(alpcoeff) / sizeof(real) ==
257  (maxpow_ * (maxpow_ + 3))/2,
258  "Coefficient array size mismatch for alp");
259  static_assert(sizeof(betcoeff) / sizeof(real) ==
260  (maxpow_ * (maxpow_ + 3))/2,
261  "Coefficient array size mismatch for bet");
262  int m = maxpow_/2;
263  _b1 = Math::polyval(m, b1coeff, Math::sq(_n)) / (b1coeff[m + 1] * (1+_n));
264  // _a1 is the equivalent radius for computing the circumference of
265  // ellipse.
266  _a1 = _b1 * _a;
267  int o = 0;
268  real d = _n;
269  for (int l = 1; l <= maxpow_; ++l) {
270  m = maxpow_ - l;
271  _alp[l] = d * Math::polyval(m, alpcoeff + o, _n) / alpcoeff[o + m + 1];
272  _bet[l] = d * Math::polyval(m, betcoeff + o, _n) / betcoeff[o + m + 1];
273  o += m + 2;
274  d *= _n;
275  }
276  // Post condition: o == sizeof(alpcoeff) / sizeof(real) &&
277  // o == sizeof(betcoeff) / sizeof(real)
278  }
279 
281  static const TransverseMercator utm(Constants::WGS84_a(),
284  return utm;
285  }
286 
287  // Engsager and Poder (2007) use trigonometric series to convert between phi
288  // and phip. Here are the series...
289  //
290  // Conversion from phi to phip:
291  //
292  // phip = phi + sum(c[j] * sin(2*j*phi), j, 1, 6)
293  //
294  // c[1] = - 2 * n
295  // + 2/3 * n^2
296  // + 4/3 * n^3
297  // - 82/45 * n^4
298  // + 32/45 * n^5
299  // + 4642/4725 * n^6;
300  // c[2] = 5/3 * n^2
301  // - 16/15 * n^3
302  // - 13/9 * n^4
303  // + 904/315 * n^5
304  // - 1522/945 * n^6;
305  // c[3] = - 26/15 * n^3
306  // + 34/21 * n^4
307  // + 8/5 * n^5
308  // - 12686/2835 * n^6;
309  // c[4] = 1237/630 * n^4
310  // - 12/5 * n^5
311  // - 24832/14175 * n^6;
312  // c[5] = - 734/315 * n^5
313  // + 109598/31185 * n^6;
314  // c[6] = 444337/155925 * n^6;
315  //
316  // Conversion from phip to phi:
317  //
318  // phi = phip + sum(d[j] * sin(2*j*phip), j, 1, 6)
319  //
320  // d[1] = 2 * n
321  // - 2/3 * n^2
322  // - 2 * n^3
323  // + 116/45 * n^4
324  // + 26/45 * n^5
325  // - 2854/675 * n^6;
326  // d[2] = 7/3 * n^2
327  // - 8/5 * n^3
328  // - 227/45 * n^4
329  // + 2704/315 * n^5
330  // + 2323/945 * n^6;
331  // d[3] = 56/15 * n^3
332  // - 136/35 * n^4
333  // - 1262/105 * n^5
334  // + 73814/2835 * n^6;
335  // d[4] = 4279/630 * n^4
336  // - 332/35 * n^5
337  // - 399572/14175 * n^6;
338  // d[5] = 4174/315 * n^5
339  // - 144838/6237 * n^6;
340  // d[6] = 601676/22275 * n^6;
341  //
342  // In order to maintain sufficient relative accuracy close to the pole use
343  //
344  // S = sum(c[i]*sin(2*i*phi),i,1,6)
345  // taup = (tau + tan(S)) / (1 - tau * tan(S))
346 
347  // In Math::taupf and Math::tauf we evaluate the forward transform explicitly
348  // and solve the reverse one by Newton's method.
349  //
350  // There are adapted from TransverseMercatorExact (taup and taupinv). tau =
351  // tan(phi), taup = sinh(psi)
352 
353  void TransverseMercator::Forward(real lon0, real lat, real lon,
354  real& x, real& y,
355  real& gamma, real& k) const {
356  lat = Math::LatFix(lat);
357  lon = Math::AngDiff(lon0, lon);
358  // Explicitly enforce the parity
359  int
360  latsign = signbit(lat) ? -1 : 1,
361  lonsign = signbit(lon) ? -1 : 1;
362  lon *= lonsign;
363  lat *= latsign;
364  bool backside = lon > Math::qd;
365  if (backside) {
366  if (lat == 0)
367  latsign = -1;
368  lon = Math::hd - lon;
369  }
370  real sphi, cphi, slam, clam;
371  Math::sincosd(lat, sphi, cphi);
372  Math::sincosd(lon, slam, clam);
373  // phi = latitude
374  // phi' = conformal latitude
375  // psi = isometric latitude
376  // tau = tan(phi)
377  // tau' = tan(phi')
378  // [xi', eta'] = Gauss-Schreiber TM coordinates
379  // [xi, eta] = Gauss-Krueger TM coordinates
380  //
381  // We use
382  // tan(phi') = sinh(psi)
383  // sin(phi') = tanh(psi)
384  // cos(phi') = sech(psi)
385  // denom^2 = 1-cos(phi')^2*sin(lam)^2 = 1-sech(psi)^2*sin(lam)^2
386  // sin(xip) = sin(phi')/denom = tanh(psi)/denom
387  // cos(xip) = cos(phi')*cos(lam)/denom = sech(psi)*cos(lam)/denom
388  // cosh(etap) = 1/denom = 1/denom
389  // sinh(etap) = cos(phi')*sin(lam)/denom = sech(psi)*sin(lam)/denom
390  real etap, xip;
391  if (lat != Math::qd) {
392  real
393  tau = sphi / cphi,
394  taup = Math::taupf(tau, _es);
395  xip = atan2(taup, clam);
396  // Used to be
397  // etap = Math::atanh(sin(lam) / cosh(psi));
398  etap = asinh(slam / hypot(taup, clam));
399  // convergence and scale for Gauss-Schreiber TM (xip, etap) -- gamma0 =
400  // atan(tan(xip) * tanh(etap)) = atan(tan(lam) * sin(phi'));
401  // sin(phi') = tau'/sqrt(1 + tau'^2)
402  // Krueger p 22 (44)
403  gamma = Math::atan2d(slam * taup, clam * hypot(real(1), taup));
404  // k0 = sqrt(1 - _e2 * sin(phi)^2) * (cos(phi') / cos(phi)) * cosh(etap)
405  // Note 1/cos(phi) = cosh(psip);
406  // and cos(phi') * cosh(etap) = 1/hypot(sinh(psi), cos(lam))
407  //
408  // This form has cancelling errors. This property is lost if cosh(psip)
409  // is replaced by 1/cos(phi), even though it's using "primary" data (phi
410  // instead of psip).
411  k = sqrt(_e2m + _e2 * Math::sq(cphi)) * hypot(real(1), tau)
412  / hypot(taup, clam);
413  } else {
414  xip = Math::pi()/2;
415  etap = 0;
416  gamma = lon;
417  k = _c;
418  }
419  // {xi',eta'} is {northing,easting} for Gauss-Schreiber transverse Mercator
420  // (for eta' = 0, xi' = bet). {xi,eta} is {northing,easting} for transverse
421  // Mercator with constant scale on the central meridian (for eta = 0, xip =
422  // rectifying latitude). Define
423  //
424  // zeta = xi + i*eta
425  // zeta' = xi' + i*eta'
426  //
427  // The conversion from conformal to rectifying latitude can be expressed as
428  // a series in _n:
429  //
430  // zeta = zeta' + sum(h[j-1]' * sin(2 * j * zeta'), j = 1..maxpow_)
431  //
432  // where h[j]' = O(_n^j). The reversion of this series gives
433  //
434  // zeta' = zeta - sum(h[j-1] * sin(2 * j * zeta), j = 1..maxpow_)
435  //
436  // which is used in Reverse.
437  //
438  // Evaluate sums via Clenshaw method. See
439  // https://en.wikipedia.org/wiki/Clenshaw_algorithm
440  //
441  // Let
442  //
443  // S = sum(a[k] * phi[k](x), k = 0..n)
444  // phi[k+1](x) = alpha[k](x) * phi[k](x) + beta[k](x) * phi[k-1](x)
445  //
446  // Evaluate S with
447  //
448  // b[n+2] = b[n+1] = 0
449  // b[k] = alpha[k](x) * b[k+1] + beta[k+1](x) * b[k+2] + a[k]
450  // S = (a[0] + beta[1](x) * b[2]) * phi[0](x) + b[1] * phi[1](x)
451  //
452  // Here we have
453  //
454  // x = 2 * zeta'
455  // phi[k](x) = sin(k * x)
456  // alpha[k](x) = 2 * cos(x)
457  // beta[k](x) = -1
458  // [ sin(A+B) - 2*cos(B)*sin(A) + sin(A-B) = 0, A = k*x, B = x ]
459  // n = maxpow_
460  // a[k] = _alp[k]
461  // S = b[1] * sin(x)
462  //
463  // For the derivative we have
464  //
465  // x = 2 * zeta'
466  // phi[k](x) = cos(k * x)
467  // alpha[k](x) = 2 * cos(x)
468  // beta[k](x) = -1
469  // [ cos(A+B) - 2*cos(B)*cos(A) + cos(A-B) = 0, A = k*x, B = x ]
470  // a[0] = 1; a[k] = 2*k*_alp[k]
471  // S = (a[0] - b[2]) + b[1] * cos(x)
472  //
473  // Matrix formulation (not used here):
474  // phi[k](x) = [sin(k * x); k * cos(k * x)]
475  // alpha[k](x) = 2 * [cos(x), 0; -sin(x), cos(x)]
476  // beta[k](x) = -1 * [1, 0; 0, 1]
477  // a[k] = _alp[k] * [1, 0; 0, 1]
478  // b[n+2] = b[n+1] = [0, 0; 0, 0]
479  // b[k] = alpha[k](x) * b[k+1] + beta[k+1](x) * b[k+2] + a[k]
480  // N.B., for all k: b[k](1,2) = 0; b[k](1,1) = b[k](2,2)
481  // S = (a[0] + beta[1](x) * b[2]) * phi[0](x) + b[1] * phi[1](x)
482  // phi[0](x) = [0; 0]
483  // phi[1](x) = [sin(x); cos(x)]
484  real
485  c0 = cos(2 * xip), ch0 = cosh(2 * etap),
486  s0 = sin(2 * xip), sh0 = sinh(2 * etap);
487  complex<real> a(2 * c0 * ch0, -2 * s0 * sh0); // 2 * cos(2*zeta')
488  int n = maxpow_;
489  complex<real>
490  y0(n & 1 ? _alp[n] : 0), y1, // default initializer is 0+i0
491  z0(n & 1 ? 2*n * _alp[n] : 0), z1;
492  if (n & 1) --n;
493  while (n) {
494  y1 = a * y0 - y1 + _alp[n];
495  z1 = a * z0 - z1 + 2*n * _alp[n];
496  --n;
497  y0 = a * y1 - y0 + _alp[n];
498  z0 = a * z1 - z0 + 2*n * _alp[n];
499  --n;
500  }
501  a /= real(2); // cos(2*zeta')
502  z1 = real(1) - z1 + a * z0;
503  a = complex<real>(s0 * ch0, c0 * sh0); // sin(2*zeta')
504  y1 = complex<real>(xip, etap) + a * y0;
505  // Fold in change in convergence and scale for Gauss-Schreiber TM to
506  // Gauss-Krueger TM.
507  gamma -= Math::atan2d(z1.imag(), z1.real());
508  k *= _b1 * abs(z1);
509  real xi = y1.real(), eta = y1.imag();
510  y = _a1 * _k0 * (backside ? Math::pi() - xi : xi) * latsign;
511  x = _a1 * _k0 * eta * lonsign;
512  if (backside)
513  gamma = Math::hd - gamma;
514  gamma *= latsign * lonsign;
515  gamma = Math::AngNormalize(gamma);
516  k *= _k0;
517  }
518 
519  void TransverseMercator::Reverse(real lon0, real x, real y,
520  real& lat, real& lon,
521  real& gamma, real& k) const {
522  // This undoes the steps in Forward. The wrinkles are: (1) Use of the
523  // reverted series to express zeta' in terms of zeta. (2) Newton's method
524  // to solve for phi in terms of tan(phi).
525  real
526  xi = y / (_a1 * _k0),
527  eta = x / (_a1 * _k0);
528  // Explicitly enforce the parity
529  int
530  xisign = signbit(xi) ? -1 : 1,
531  etasign = signbit(eta) ? -1 : 1;
532  xi *= xisign;
533  eta *= etasign;
534  bool backside = xi > Math::pi()/2;
535  if (backside)
536  xi = Math::pi() - xi;
537  real
538  c0 = cos(2 * xi), ch0 = cosh(2 * eta),
539  s0 = sin(2 * xi), sh0 = sinh(2 * eta);
540  complex<real> a(2 * c0 * ch0, -2 * s0 * sh0); // 2 * cos(2*zeta)
541  int n = maxpow_;
542  complex<real>
543  y0(n & 1 ? -_bet[n] : 0), y1, // default initializer is 0+i0
544  z0(n & 1 ? -2*n * _bet[n] : 0), z1;
545  if (n & 1) --n;
546  while (n) {
547  y1 = a * y0 - y1 - _bet[n];
548  z1 = a * z0 - z1 - 2*n * _bet[n];
549  --n;
550  y0 = a * y1 - y0 - _bet[n];
551  z0 = a * z1 - z0 - 2*n * _bet[n];
552  --n;
553  }
554  a /= real(2); // cos(2*zeta)
555  z1 = real(1) - z1 + a * z0;
556  a = complex<real>(s0 * ch0, c0 * sh0); // sin(2*zeta)
557  y1 = complex<real>(xi, eta) + a * y0;
558  // Convergence and scale for Gauss-Schreiber TM to Gauss-Krueger TM.
559  gamma = Math::atan2d(z1.imag(), z1.real());
560  k = _b1 / abs(z1);
561  // JHS 154 has
562  //
563  // phi' = asin(sin(xi') / cosh(eta')) (Krueger p 17 (25))
564  // lam = asin(tanh(eta') / cos(phi')
565  // psi = asinh(tan(phi'))
566  real
567  xip = y1.real(), etap = y1.imag(),
568  s = sinh(etap),
569  c = fmax(real(0), cos(xip)), // cos(pi/2) might be negative
570  r = hypot(s, c);
571  if (r != 0) {
572  lon = Math::atan2d(s, c); // Krueger p 17 (25)
573  // Use Newton's method to solve for tau
574  real
575  sxip = sin(xip),
576  tau = Math::tauf(sxip/r, _es);
577  gamma += Math::atan2d(sxip * tanh(etap), c); // Krueger p 19 (31)
578  lat = Math::atand(tau);
579  // Note cos(phi') * cosh(eta') = r
580  k *= sqrt(_e2m + _e2 / (1 + Math::sq(tau))) *
581  hypot(real(1), tau) * r;
582  } else {
583  lat = Math::qd;
584  lon = 0;
585  k *= _c;
586  }
587  lat *= xisign;
588  if (backside)
589  lon = Math::hd - lon;
590  lon *= etasign;
591  lon = Math::AngNormalize(lon + lon0);
592  if (backside)
593  gamma = Math::hd - gamma;
594  gamma *= xisign * etasign;
595  gamma = Math::AngNormalize(gamma);
596  k *= _k0;
597  }
598 
599 } // namespace GeographicLib
Header for GeographicLib::TransverseMercator class.
Exception handling for GeographicLib.
Definition: Constants.hpp:316
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
static T LatFix(T x)
Definition: Math.hpp:300
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:106
static T atan2d(T y, T x)
Definition: Math.cpp:183
static T sq(T x)
Definition: Math.hpp:212
static T tauf(T taup, T es)
Definition: Math.cpp:219
static T AngNormalize(T x)
Definition: Math.cpp:71
static T atand(T x)
Definition: Math.cpp:202
static T taupf(T tau, T es)
Definition: Math.cpp:209
static T pi()
Definition: Math.hpp:190
static T polyval(int N, const T p[], T x)
Definition: Math.hpp:271
static T AngDiff(T x, T y, T &e)
Definition: Math.cpp:82
@ hd
degrees per half turn
Definition: Math.hpp:144
@ qd
degrees per quarter turn
Definition: Math.hpp:141
Transverse Mercator projection.
void Reverse(real lon0, real x, real y, real &lat, real &lon, real &gamma, real &k) const
static const TransverseMercator & UTM()
TransverseMercator(real a, real f, real k0)
void Forward(real lon0, real lat, real lon, real &x, real &y, real &gamma, real &k) const
Namespace for GeographicLib.
Definition: Accumulator.cpp:12