GeographicLib 1.52
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TransverseMercator.cpp
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1/**
2 * \file TransverseMercator.cpp
3 * \brief Implementation for GeographicLib::TransverseMercator class
4 *
5 * Copyright (c) Charles Karney (2008-2020) <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
40namespace GeographicLib {
41
42 using namespace std;
43
45 : _a(a)
46 , _f(f)
47 , _k0(k0)
48 , _e2(_f * (2 - _f))
49 , _es((_f < 0 ? -1 : 1) * sqrt(abs(_e2)))
50 , _e2m(1 - _e2)
51 // _c = sqrt( pow(1 + _e, 1 + _e) * pow(1 - _e, 1 - _e) ) )
52 // See, for example, Lee (1976), p 100.
53 , _c( sqrt(_e2m) * exp(Math::eatanhe(real(1), _es)) )
54 , _n(_f / (2 - _f))
55 {
56 if (!(isfinite(_a) && _a > 0))
57 throw GeographicErr("Equatorial radius is not positive");
58 if (!(isfinite(_f) && _f < 1))
59 throw GeographicErr("Polar semi-axis is not positive");
60 if (!(isfinite(_k0) && _k0 > 0))
61 throw GeographicErr("Scale is not positive");
62
63 // Generated by Maxima on 2015-05-14 22:55:13-04:00
64#if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2
65 static const real b1coeff[] = {
66 // b1*(n+1), polynomial in n2 of order 2
67 1, 16, 64, 64,
68 }; // count = 4
69#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3
70 static const real b1coeff[] = {
71 // b1*(n+1), polynomial in n2 of order 3
72 1, 4, 64, 256, 256,
73 }; // count = 5
74#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4
75 static const real b1coeff[] = {
76 // b1*(n+1), polynomial in n2 of order 4
77 25, 64, 256, 4096, 16384, 16384,
78 }; // count = 6
79#else
80#error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER"
81#endif
82
83#if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
84 static const real alpcoeff[] = {
85 // alp[1]/n^1, polynomial in n of order 3
86 164, 225, -480, 360, 720,
87 // alp[2]/n^2, polynomial in n of order 2
88 557, -864, 390, 1440,
89 // alp[3]/n^3, polynomial in n of order 1
90 -1236, 427, 1680,
91 // alp[4]/n^4, polynomial in n of order 0
92 49561, 161280,
93 }; // count = 14
94#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
95 static const real alpcoeff[] = {
96 // alp[1]/n^1, polynomial in n of order 4
97 -635, 328, 450, -960, 720, 1440,
98 // alp[2]/n^2, polynomial in n of order 3
99 4496, 3899, -6048, 2730, 10080,
100 // alp[3]/n^3, polynomial in n of order 2
101 15061, -19776, 6832, 26880,
102 // alp[4]/n^4, polynomial in n of order 1
103 -171840, 49561, 161280,
104 // alp[5]/n^5, polynomial in n of order 0
105 34729, 80640,
106 }; // count = 20
107#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
108 static const real alpcoeff[] = {
109 // alp[1]/n^1, polynomial in n of order 5
110 31564, -66675, 34440, 47250, -100800, 75600, 151200,
111 // alp[2]/n^2, polynomial in n of order 4
112 -1983433, 863232, 748608, -1161216, 524160, 1935360,
113 // alp[3]/n^3, polynomial in n of order 3
114 670412, 406647, -533952, 184464, 725760,
115 // alp[4]/n^4, polynomial in n of order 2
116 6601661, -7732800, 2230245, 7257600,
117 // alp[5]/n^5, polynomial in n of order 1
118 -13675556, 3438171, 7983360,
119 // alp[6]/n^6, polynomial in n of order 0
120 212378941, 319334400,
121 }; // count = 27
122#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
123 static const real alpcoeff[] = {
124 // alp[1]/n^1, polynomial in n of order 6
125 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800,
126 // alp[2]/n^2, polynomial in n of order 5
127 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800,
128 // alp[3]/n^3, polynomial in n of order 4
129 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400,
130 // alp[4]/n^4, polynomial in n of order 3
131 155912000, 72618271, -85060800, 24532695, 79833600,
132 // alp[5]/n^5, polynomial in n of order 2
133 102508609, -109404448, 27505368, 63866880,
134 // alp[6]/n^6, polynomial in n of order 1
135 -12282192400LL, 2760926233LL, 4151347200LL,
136 // alp[7]/n^7, polynomial in n of order 0
137 1522256789, 1383782400,
138 }; // count = 35
139#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
140 static const real alpcoeff[] = {
141 // alp[1]/n^1, polynomial in n of order 7
142 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200,
143 101606400, 203212800,
144 // alp[2]/n^2, polynomial in n of order 6
145 148003883, 83274912, -178508970, 77690880, 67374720, -104509440,
146 47174400, 174182400,
147 // alp[3]/n^3, polynomial in n of order 5
148 318729724, -738126169, 294981280, 178924680, -234938880, 81164160,
149 319334400,
150 // alp[4]/n^4, polynomial in n of order 4
151 -40176129013LL, 14967552000LL, 6971354016LL, -8165836800LL, 2355138720LL,
152 7664025600LL,
153 // alp[5]/n^5, polynomial in n of order 3
154 10421654396LL, 3997835751LL, -4266773472LL, 1072709352, 2490808320LL,
155 // alp[6]/n^6, polynomial in n of order 2
156 175214326799LL, -171950693600LL, 38652967262LL, 58118860800LL,
157 // alp[7]/n^7, polynomial in n of order 1
158 -67039739596LL, 13700311101LL, 12454041600LL,
159 // alp[8]/n^8, polynomial in n of order 0
160 1424729850961LL, 743921418240LL,
161 }; // count = 44
162#else
163#error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER"
164#endif
165
166#if GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4
167 static const real betcoeff[] = {
168 // bet[1]/n^1, polynomial in n of order 3
169 -4, 555, -960, 720, 1440,
170 // bet[2]/n^2, polynomial in n of order 2
171 -437, 96, 30, 1440,
172 // bet[3]/n^3, polynomial in n of order 1
173 -148, 119, 3360,
174 // bet[4]/n^4, polynomial in n of order 0
175 4397, 161280,
176 }; // count = 14
177#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5
178 static const real betcoeff[] = {
179 // bet[1]/n^1, polynomial in n of order 4
180 -3645, -64, 8880, -15360, 11520, 23040,
181 // bet[2]/n^2, polynomial in n of order 3
182 4416, -3059, 672, 210, 10080,
183 // bet[3]/n^3, polynomial in n of order 2
184 -627, -592, 476, 13440,
185 // bet[4]/n^4, polynomial in n of order 1
186 -3520, 4397, 161280,
187 // bet[5]/n^5, polynomial in n of order 0
188 4583, 161280,
189 }; // count = 20
190#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6
191 static const real betcoeff[] = {
192 // bet[1]/n^1, polynomial in n of order 5
193 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200,
194 // bet[2]/n^2, polynomial in n of order 4
195 -1118711, 1695744, -1174656, 258048, 80640, 3870720,
196 // bet[3]/n^3, polynomial in n of order 3
197 22276, -16929, -15984, 12852, 362880,
198 // bet[4]/n^4, polynomial in n of order 2
199 -830251, -158400, 197865, 7257600,
200 // bet[5]/n^5, polynomial in n of order 1
201 -435388, 453717, 15966720,
202 // bet[6]/n^6, polynomial in n of order 0
203 20648693, 638668800,
204 }; // count = 27
205#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7
206 static const real betcoeff[] = {
207 // bet[1]/n^1, polynomial in n of order 6
208 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600,
209 38707200,
210 // bet[2]/n^2, polynomial in n of order 5
211 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600,
212 // bet[3]/n^3, polynomial in n of order 4
213 9261899, 3564160, -2708640, -2557440, 2056320, 58060800,
214 // bet[4]/n^4, polynomial in n of order 3
215 14928352, -9132761, -1742400, 2176515, 79833600,
216 // bet[5]/n^5, polynomial in n of order 2
217 -8005831, -1741552, 1814868, 63866880,
218 // bet[6]/n^6, polynomial in n of order 1
219 -261810608, 268433009, 8302694400LL,
220 // bet[7]/n^7, polynomial in n of order 0
221 219941297, 5535129600LL,
222 }; // count = 35
223#elif GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8
224 static const real betcoeff[] = {
225 // bet[1]/n^1, polynomial in n of order 7
226 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600,
227 135475200, 270950400,
228 // bet[2]/n^2, polynomial in n of order 6
229 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600,
230 348364800,
231 // bet[3]/n^3, polynomial in n of order 5
232 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520,
233 638668800,
234 // bet[4]/n^4, polynomial in n of order 4
235 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600LL,
236 // bet[5]/n^5, polynomial in n of order 3
237 457888660, -312227409, -67920528, 70779852, 2490808320LL,
238 // bet[6]/n^6, polynomial in n of order 2
239 -19841813847LL, -3665348512LL, 3758062126LL, 116237721600LL,
240 // bet[7]/n^7, polynomial in n of order 1
241 -1989295244, 1979471673, 49816166400LL,
242 // bet[8]/n^8, polynomial in n of order 0
243 191773887257LL, 3719607091200LL,
244 }; // count = 44
245#else
246#error "Bad value for GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER"
247#endif
248
249 static_assert(sizeof(b1coeff) / sizeof(real) == maxpow_/2 + 2,
250 "Coefficient array size mismatch for b1");
251 static_assert(sizeof(alpcoeff) / sizeof(real) ==
252 (maxpow_ * (maxpow_ + 3))/2,
253 "Coefficient array size mismatch for alp");
254 static_assert(sizeof(betcoeff) / sizeof(real) ==
255 (maxpow_ * (maxpow_ + 3))/2,
256 "Coefficient array size mismatch for bet");
257 int m = maxpow_/2;
258 _b1 = Math::polyval(m, b1coeff, Math::sq(_n)) / (b1coeff[m + 1] * (1+_n));
259 // _a1 is the equivalent radius for computing the circumference of
260 // ellipse.
261 _a1 = _b1 * _a;
262 int o = 0;
263 real d = _n;
264 for (int l = 1; l <= maxpow_; ++l) {
265 m = maxpow_ - l;
266 _alp[l] = d * Math::polyval(m, alpcoeff + o, _n) / alpcoeff[o + m + 1];
267 _bet[l] = d * Math::polyval(m, betcoeff + o, _n) / betcoeff[o + m + 1];
268 o += m + 2;
269 d *= _n;
270 }
271 // Post condition: o == sizeof(alpcoeff) / sizeof(real) &&
272 // o == sizeof(betcoeff) / sizeof(real)
273 }
274
276 static const TransverseMercator utm(Constants::WGS84_a(),
279 return utm;
280 }
281
282 // Engsager and Poder (2007) use trigonometric series to convert between phi
283 // and phip. Here are the series...
284 //
285 // Conversion from phi to phip:
286 //
287 // phip = phi + sum(c[j] * sin(2*j*phi), j, 1, 6)
288 //
289 // c[1] = - 2 * n
290 // + 2/3 * n^2
291 // + 4/3 * n^3
292 // - 82/45 * n^4
293 // + 32/45 * n^5
294 // + 4642/4725 * n^6;
295 // c[2] = 5/3 * n^2
296 // - 16/15 * n^3
297 // - 13/9 * n^4
298 // + 904/315 * n^5
299 // - 1522/945 * n^6;
300 // c[3] = - 26/15 * n^3
301 // + 34/21 * n^4
302 // + 8/5 * n^5
303 // - 12686/2835 * n^6;
304 // c[4] = 1237/630 * n^4
305 // - 12/5 * n^5
306 // - 24832/14175 * n^6;
307 // c[5] = - 734/315 * n^5
308 // + 109598/31185 * n^6;
309 // c[6] = 444337/155925 * n^6;
310 //
311 // Conversion from phip to phi:
312 //
313 // phi = phip + sum(d[j] * sin(2*j*phip), j, 1, 6)
314 //
315 // d[1] = 2 * n
316 // - 2/3 * n^2
317 // - 2 * n^3
318 // + 116/45 * n^4
319 // + 26/45 * n^5
320 // - 2854/675 * n^6;
321 // d[2] = 7/3 * n^2
322 // - 8/5 * n^3
323 // - 227/45 * n^4
324 // + 2704/315 * n^5
325 // + 2323/945 * n^6;
326 // d[3] = 56/15 * n^3
327 // - 136/35 * n^4
328 // - 1262/105 * n^5
329 // + 73814/2835 * n^6;
330 // d[4] = 4279/630 * n^4
331 // - 332/35 * n^5
332 // - 399572/14175 * n^6;
333 // d[5] = 4174/315 * n^5
334 // - 144838/6237 * n^6;
335 // d[6] = 601676/22275 * n^6;
336 //
337 // In order to maintain sufficient relative accuracy close to the pole use
338 //
339 // S = sum(c[i]*sin(2*i*phi),i,1,6)
340 // taup = (tau + tan(S)) / (1 - tau * tan(S))
341
342 // In Math::taupf and Math::tauf we evaluate the forward transform explicitly
343 // and solve the reverse one by Newton's method.
344 //
345 // There are adapted from TransverseMercatorExact (taup and taupinv). tau =
346 // tan(phi), taup = sinh(psi)
347
348 void TransverseMercator::Forward(real lon0, real lat, real lon,
349 real& x, real& y,
350 real& gamma, real& k) const {
351 lat = Math::LatFix(lat);
352 lon = Math::AngDiff(lon0, lon);
353 // Explicitly enforce the parity
354 int
355 latsign = (lat < 0) ? -1 : 1,
356 lonsign = (lon < 0) ? -1 : 1;
357 lon *= lonsign;
358 lat *= latsign;
359 bool backside = lon > 90;
360 if (backside) {
361 if (lat == 0)
362 latsign = -1;
363 lon = 180 - lon;
364 }
365 real sphi, cphi, slam, clam;
366 Math::sincosd(lat, sphi, cphi);
367 Math::sincosd(lon, slam, clam);
368 // phi = latitude
369 // phi' = conformal latitude
370 // psi = isometric latitude
371 // tau = tan(phi)
372 // tau' = tan(phi')
373 // [xi', eta'] = Gauss-Schreiber TM coordinates
374 // [xi, eta] = Gauss-Krueger TM coordinates
375 //
376 // We use
377 // tan(phi') = sinh(psi)
378 // sin(phi') = tanh(psi)
379 // cos(phi') = sech(psi)
380 // denom^2 = 1-cos(phi')^2*sin(lam)^2 = 1-sech(psi)^2*sin(lam)^2
381 // sin(xip) = sin(phi')/denom = tanh(psi)/denom
382 // cos(xip) = cos(phi')*cos(lam)/denom = sech(psi)*cos(lam)/denom
383 // cosh(etap) = 1/denom = 1/denom
384 // sinh(etap) = cos(phi')*sin(lam)/denom = sech(psi)*sin(lam)/denom
385 real etap, xip;
386 if (lat != 90) {
387 real
388 tau = sphi / cphi,
389 taup = Math::taupf(tau, _es);
390 xip = atan2(taup, clam);
391 // Used to be
392 // etap = Math::atanh(sin(lam) / cosh(psi));
393 etap = asinh(slam / hypot(taup, clam));
394 // convergence and scale for Gauss-Schreiber TM (xip, etap) -- gamma0 =
395 // atan(tan(xip) * tanh(etap)) = atan(tan(lam) * sin(phi'));
396 // sin(phi') = tau'/sqrt(1 + tau'^2)
397 // Krueger p 22 (44)
398 gamma = Math::atan2d(slam * taup, clam * hypot(real(1), taup));
399 // k0 = sqrt(1 - _e2 * sin(phi)^2) * (cos(phi') / cos(phi)) * cosh(etap)
400 // Note 1/cos(phi) = cosh(psip);
401 // and cos(phi') * cosh(etap) = 1/hypot(sinh(psi), cos(lam))
402 //
403 // This form has cancelling errors. This property is lost if cosh(psip)
404 // is replaced by 1/cos(phi), even though it's using "primary" data (phi
405 // instead of psip).
406 k = sqrt(_e2m + _e2 * Math::sq(cphi)) * hypot(real(1), tau)
407 / hypot(taup, clam);
408 } else {
409 xip = Math::pi()/2;
410 etap = 0;
411 gamma = lon;
412 k = _c;
413 }
414 // {xi',eta'} is {northing,easting} for Gauss-Schreiber transverse Mercator
415 // (for eta' = 0, xi' = bet). {xi,eta} is {northing,easting} for transverse
416 // Mercator with constant scale on the central meridian (for eta = 0, xip =
417 // rectifying latitude). Define
418 //
419 // zeta = xi + i*eta
420 // zeta' = xi' + i*eta'
421 //
422 // The conversion from conformal to rectifying latitude can be expressed as
423 // a series in _n:
424 //
425 // zeta = zeta' + sum(h[j-1]' * sin(2 * j * zeta'), j = 1..maxpow_)
426 //
427 // where h[j]' = O(_n^j). The reversion of this series gives
428 //
429 // zeta' = zeta - sum(h[j-1] * sin(2 * j * zeta), j = 1..maxpow_)
430 //
431 // which is used in Reverse.
432 //
433 // Evaluate sums via Clenshaw method. See
434 // https://en.wikipedia.org/wiki/Clenshaw_algorithm
435 //
436 // Let
437 //
438 // S = sum(a[k] * phi[k](x), k = 0..n)
439 // phi[k+1](x) = alpha[k](x) * phi[k](x) + beta[k](x) * phi[k-1](x)
440 //
441 // Evaluate S with
442 //
443 // b[n+2] = b[n+1] = 0
444 // b[k] = alpha[k](x) * b[k+1] + beta[k+1](x) * b[k+2] + a[k]
445 // S = (a[0] + beta[1](x) * b[2]) * phi[0](x) + b[1] * phi[1](x)
446 //
447 // Here we have
448 //
449 // x = 2 * zeta'
450 // phi[k](x) = sin(k * x)
451 // alpha[k](x) = 2 * cos(x)
452 // beta[k](x) = -1
453 // [ sin(A+B) - 2*cos(B)*sin(A) + sin(A-B) = 0, A = k*x, B = x ]
454 // n = maxpow_
455 // a[k] = _alp[k]
456 // S = b[1] * sin(x)
457 //
458 // For the derivative we have
459 //
460 // x = 2 * zeta'
461 // phi[k](x) = cos(k * x)
462 // alpha[k](x) = 2 * cos(x)
463 // beta[k](x) = -1
464 // [ cos(A+B) - 2*cos(B)*cos(A) + cos(A-B) = 0, A = k*x, B = x ]
465 // a[0] = 1; a[k] = 2*k*_alp[k]
466 // S = (a[0] - b[2]) + b[1] * cos(x)
467 //
468 // Matrix formulation (not used here):
469 // phi[k](x) = [sin(k * x); k * cos(k * x)]
470 // alpha[k](x) = 2 * [cos(x), 0; -sin(x), cos(x)]
471 // beta[k](x) = -1 * [1, 0; 0, 1]
472 // a[k] = _alp[k] * [1, 0; 0, 1]
473 // b[n+2] = b[n+1] = [0, 0; 0, 0]
474 // b[k] = alpha[k](x) * b[k+1] + beta[k+1](x) * b[k+2] + a[k]
475 // N.B., for all k: b[k](1,2) = 0; b[k](1,1) = b[k](2,2)
476 // S = (a[0] + beta[1](x) * b[2]) * phi[0](x) + b[1] * phi[1](x)
477 // phi[0](x) = [0; 0]
478 // phi[1](x) = [sin(x); cos(x)]
479 real
480 c0 = cos(2 * xip), ch0 = cosh(2 * etap),
481 s0 = sin(2 * xip), sh0 = sinh(2 * etap);
482 complex<real> a(2 * c0 * ch0, -2 * s0 * sh0); // 2 * cos(2*zeta')
483 int n = maxpow_;
484 complex<real>
485 y0(n & 1 ? _alp[n] : 0), y1, // default initializer is 0+i0
486 z0(n & 1 ? 2*n * _alp[n] : 0), z1;
487 if (n & 1) --n;
488 while (n) {
489 y1 = a * y0 - y1 + _alp[n];
490 z1 = a * z0 - z1 + 2*n * _alp[n];
491 --n;
492 y0 = a * y1 - y0 + _alp[n];
493 z0 = a * z1 - z0 + 2*n * _alp[n];
494 --n;
495 }
496 a /= real(2); // cos(2*zeta')
497 z1 = real(1) - z1 + a * z0;
498 a = complex<real>(s0 * ch0, c0 * sh0); // sin(2*zeta')
499 y1 = complex<real>(xip, etap) + a * y0;
500 // Fold in change in convergence and scale for Gauss-Schreiber TM to
501 // Gauss-Krueger TM.
502 gamma -= Math::atan2d(z1.imag(), z1.real());
503 k *= _b1 * abs(z1);
504 real xi = y1.real(), eta = y1.imag();
505 y = _a1 * _k0 * (backside ? Math::pi() - xi : xi) * latsign;
506 x = _a1 * _k0 * eta * lonsign;
507 if (backside)
508 gamma = 180 - gamma;
509 gamma *= latsign * lonsign;
510 gamma = Math::AngNormalize(gamma);
511 k *= _k0;
512 }
513
514 void TransverseMercator::Reverse(real lon0, real x, real y,
515 real& lat, real& lon,
516 real& gamma, real& k) const {
517 // This undoes the steps in Forward. The wrinkles are: (1) Use of the
518 // reverted series to express zeta' in terms of zeta. (2) Newton's method
519 // to solve for phi in terms of tan(phi).
520 real
521 xi = y / (_a1 * _k0),
522 eta = x / (_a1 * _k0);
523 // Explicitly enforce the parity
524 int
525 xisign = (xi < 0) ? -1 : 1,
526 etasign = (eta < 0) ? -1 : 1;
527 xi *= xisign;
528 eta *= etasign;
529 bool backside = xi > Math::pi()/2;
530 if (backside)
531 xi = Math::pi() - xi;
532 real
533 c0 = cos(2 * xi), ch0 = cosh(2 * eta),
534 s0 = sin(2 * xi), sh0 = sinh(2 * eta);
535 complex<real> a(2 * c0 * ch0, -2 * s0 * sh0); // 2 * cos(2*zeta)
536 int n = maxpow_;
537 complex<real>
538 y0(n & 1 ? -_bet[n] : 0), y1, // default initializer is 0+i0
539 z0(n & 1 ? -2*n * _bet[n] : 0), z1;
540 if (n & 1) --n;
541 while (n) {
542 y1 = a * y0 - y1 - _bet[n];
543 z1 = a * z0 - z1 - 2*n * _bet[n];
544 --n;
545 y0 = a * y1 - y0 - _bet[n];
546 z0 = a * z1 - z0 - 2*n * _bet[n];
547 --n;
548 }
549 a /= real(2); // cos(2*zeta)
550 z1 = real(1) - z1 + a * z0;
551 a = complex<real>(s0 * ch0, c0 * sh0); // sin(2*zeta)
552 y1 = complex<real>(xi, eta) + a * y0;
553 // Convergence and scale for Gauss-Schreiber TM to Gauss-Krueger TM.
554 gamma = Math::atan2d(z1.imag(), z1.real());
555 k = _b1 / abs(z1);
556 // JHS 154 has
557 //
558 // phi' = asin(sin(xi') / cosh(eta')) (Krueger p 17 (25))
559 // lam = asin(tanh(eta') / cos(phi')
560 // psi = asinh(tan(phi'))
561 real
562 xip = y1.real(), etap = y1.imag(),
563 s = sinh(etap),
564 c = max(real(0), cos(xip)), // cos(pi/2) might be negative
565 r = hypot(s, c);
566 if (r != 0) {
567 lon = Math::atan2d(s, c); // Krueger p 17 (25)
568 // Use Newton's method to solve for tau
569 real
570 sxip = sin(xip),
571 tau = Math::tauf(sxip/r, _es);
572 gamma += Math::atan2d(sxip * tanh(etap), c); // Krueger p 19 (31)
573 lat = Math::atand(tau);
574 // Note cos(phi') * cosh(eta') = r
575 k *= sqrt(_e2m + _e2 / (1 + Math::sq(tau))) *
576 hypot(real(1), tau) * r;
577 } else {
578 lat = 90;
579 lon = 0;
580 k *= _c;
581 }
582 lat *= xisign;
583 if (backside)
584 lon = 180 - lon;
585 lon *= etasign;
586 lon = Math::AngNormalize(lon + lon0);
587 if (backside)
588 gamma = 180 - gamma;
589 gamma *= xisign * etasign;
590 gamma = Math::AngNormalize(gamma);
591 k *= _k0;
592 }
593
594} // namespace GeographicLib
Header for GeographicLib::TransverseMercator class.
Exception handling for GeographicLib.
Definition: Constants.hpp:315
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
static T AngNormalize(T x)
Definition: Math.hpp:420
static T LatFix(T x)
Definition: Math.hpp:433
static void sincosd(T x, T &sinx, T &cosx)
Definition: Math.cpp:126
static T atan2d(T y, T x)
Definition: Math.cpp:180
static T sq(T x)
Definition: Math.hpp:171
static T tauf(T taup, T es)
Definition: Math.cpp:224
static T atand(T x)
Definition: Math.cpp:205
static T taupf(T tau, T es)
Definition: Math.cpp:213
static T pi()
Definition: Math.hpp:149
static T polyval(int N, const T p[], T x)
Definition: Math.hpp:402
static T AngDiff(T x, T y, T &e)
Definition: Math.hpp:452
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