1

2# -*- coding: utf-8 -*-

3

4 u'''A pure Python version of I{Karney}'s C++ class U{TransverseMercator

5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1TransverseMercator.html>}

6 based on I{Krüger} series. See also I{Karney}'s utility U{TransverseMercatorProj

7<https://GeographicLib.SourceForge.io/C++/doc/TransverseMercatorProj.1.html>}.

8

9 Following and further below is a copy of I{Karney}'s U{TransverseMercator.hpp

10<https://GeographicLib.SourceForge.io/C++/doc/TransverseMercator_8hpp_source.html>}

11 file C{Header}.

12

13 This implementation follows closely JHS 154, ETRS89 - I{järjestelmään liittyvät

14 karttaprojektiot, tasokoordinaatistot ja karttalehtijako} (Map projections, plane

15 coordinates, and map sheet index for ETRS89), published by JUHTA, Finnish Geodetic

16 Institute, and the National Land Survey of Finland (2006). The relevant section

17 is available as the U{2008 PDF file

18<http://Docs.JHS-suositukset.FI/jhs-suositukset/JHS154/JHS154_liite1.pdf>}.

19

20 This is a straight transcription of the formulas in this paper with the

21 following exceptions:

22

23 - Use of 6th order series instead of 4th order series. This reduces the

24 error to about 5 nm for the UTM range of coordinates (instead of 200 nm),

25 with a speed penalty of only 1%,

26

27 - Use Newton's method instead of plain iteration to solve for latitude

28 in terms of isometric latitude in the Reverse method,

29

30 - Use of Horner's representation for evaluating polynomials and Clenshaw's

31 method for summing trigonometric series,

32

33 - Several modifications of the formulas to improve the numerical accuracy,

34

35 - Evaluating the convergence and scale using the expression for the

36 projection or its inverse.

37

38 Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2023)

39 and licensed under the MIT/X11 License. For more information, see the

40 U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.

41'''

42# make sure int/int division yields float quotient

43 from __future__ import division as _; del _ # PYCHOK semicolon

44

45 from pygeodesy.basics import copysign0, isodd, neg, neg_, \

46 _reverange, _xinstanceof

47 from pygeodesy.constants import INF, _K0_UTM, PI, PI_2, _0_0s, _0_0, \

48 _1_0, _90_0, _copysignINF

49 from pygeodesy.datums import Datum, _spherical_datum, _WGS84, _EWGS84

50# from pygeodesy.ellipsoids import _EWGS84 # from .datums

51 from pygeodesy.errors import _ValueError, _xkwds_pop2, _Xorder

52 from pygeodesy.fmath import fdot_, hypot, hypot1

53 from pygeodesy.interns import _COMMASPACE_, _singular_

54 from pygeodesy.karney import _atan2d, _diff182, _fix90, _norm180, \

55 _polynomial, _unsigned2

56# from pygeodesy.lazily import _ALL_LAZY # from .named

57 from pygeodesy.named import _NamedBase, pairs, _ALL_LAZY

58 from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple

59 from pygeodesy.props import property_doc_, Property, Property_RO, \

60 _update_all

61# from pygeodesy.streprs import pairs # from .named

62 from pygeodesy.units import Degrees, Scalar_, _1mm as _TOL_10 # PYCHOK used!

63 from pygeodesy.utily import atan1d, _loneg, sincos2, sincos2d_

64

65 from cmath import polar

66 from math import atan2, asinh, cos, cosh, degrees, fabs, sin, sinh, sqrt, tanh

67

68__all__ = _ALL_LAZY.ktm

69__version__ = '24.11.11'

70

71

72 class KTMError(_ValueError):

73 '''Error raised for L{KTransverseMercator} and L{KTransverseMercator.forward} issues.

74 '''

75 pass

76

77

78 class KTransverseMercator(_NamedBase):

79 '''I{Karney}'s C++ class U{TransverseMercator<https://GeographicLib.SourceForge.io/

80 C++/doc/classGeographicLib_1_1TransverseMercator.html>} transcoded to pure

81 Python, following is a partial copy of I{Karney}'s documentation.

82

83 Transverse Mercator projection based on Krüger's method which evaluates the

84 projection and its inverse in terms of a series.

85

86 There's a singularity in the projection at I{phi = 0, lam - lam0 = +/- (1 - e)

87 90}, about +/- 82.6 degrees for WGS84, where I{e} is the eccentricity. Beyond

88 this point, the series ceases to converge and the results from this method

89 will be garbage. I{To be on the safe side, don't use this method if the

90 angular distance from the central meridian exceeds (1 - 2e) x 90}, about 75

91 degrees for the WGS84 ellipsoid.

92

93 Class L{ExactTransverseMercator} is an alternative implementation of the

94 projection using I{exact} formulas which yield accurate (to 8 nm) results

95 over the entire ellipsoid.

96

97 The ellipsoid parameters and the central scale are set in the constructor.

98 The central meridian (which is a trivial shift of the longitude) is specified

99 as the C{lon0} keyword argument of the L{KTransverseMercator.forward} and

100 L{KTransverseMercator.reverse} methods. The latitude of origin is taken to

101 be the equator. There is no provision in this class for specifying a false

102 easting or false northing or a different latitude of origin. However these

103 are can be simply included by the calling function.

104

105 The L{KTransverseMercator.forward} and L{KTransverseMercator.reverse} methods

106 also return the meridian convergence C{gamma} and scale C{k}. The meridian

107 convergence is the bearing of grid North, the C{y axis}, measured clockwise

108 from true North.

109 '''

110 _datum = _WGS84

111 _k0 = _K0_UTM # central scale factor

112 _lat0 = _0_0 # central parallel

113 _lon0 = _0_0 # central meridian

114 _mTM = 6

115 _raiser = False # throw Error

116

117 def __init__(self, a_earth=_EWGS84, f=None, lon0=0, k0=_K0_UTM,

118 raiser=False, **TMorder_name):

119 '''New L{KTransverseMercator}.

120

121 @kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple},

122 L{Datum}, 2-tuple (C{a, f})) or the equatorial radius (C{meter}).

123 @kwarg f: The ellipsoid's flattening (C{scalar}), required if B{C{a_earth}} is

124 is C{meter}, ignored otherwise.

125 @kwarg lon0: The central meridian (C{degrees180}).

126 @kwarg k0: Central scale factor (C{scalar}).

127 @kwarg raiser: If C{True}, throw a L{KTMError} for C{forward} singularities (C{bool}).

128 @kwarg TMorder_name: Optional C{B{name}=NN} (C{str}) and optional keyword argument

129 C{B{TMorder}=6} for the order of this L{KTransverseMercator}, see

130 property C{TMorder}.

131

132 @raise KTMError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}.

133 '''

134 if TMorder_name:

135 M = self._mTM

136 m, name = _xkwds_pop2(TMorder_name, TMorder=M)

137 if m != M:

138 self.TMorder = m

139 if name:

140 self.name = name

141

142 if f is not None:

143 self.ellipsoid = a_earth, f

144 elif a_earth in (_EWGS84, _WGS84, None):

145 pass

146 elif isinstance(a_earth, Datum):

147 self.datum = a_earth

148 else:

149 self.ellipsoid = a_earth

150

151 self.lon0 = lon0

152 self.k0 = k0

153 if raiser:

154 self.raiser = True

155

156 @Property_RO

157 def _Alp(self):

158 return _Xs(_AlpCoeffs, self.TMorder, self.ellipsoid)

159

160 @Property_RO

161 def _b1(self):

162 n = self.ellipsoid.n

163 if n: # isEllipsoidal

164 m = self.TMorder // 2

165 B1 = _B1Coeffs[m]

166 m += 1

167 b1 = _polynomial(n**2, B1, 0, m) / (B1[m] * (n + _1_0))

168 else: # isSpherical

169 b1 = _1_0 # B1[m - 1] / B1[m1] == 1, always

170 return b1

171

172 @Property_RO

173 def _Bet(self):

174 C = _Xs(_BetCoeffs, self.TMorder, self.ellipsoid)

175 return tuple(map(neg, C)) if self.f else C # negated if isEllipsoidal

176

177 @property

178 def datum(self):

179 '''Get this rhumb's datum (L{Datum}).

180 '''

181 return self._datum

182

183 @datum.setter # PYCHOK setter!

184 def datum(self, datum):

185 '''Set this rhumb's datum (L{Datum}).

186 '''

187 _xinstanceof(Datum, datum=datum)

188 if self._datum != datum:

189 _update_all(self)

190 self._datum = datum

191

192 @Property

193 def ellipsoid(self):

194 '''Get the ellipsoid (L{Ellipsoid}).

195 '''

196 return self.datum.ellipsoid

197

198 @ellipsoid.setter # PYCHOK setter!

199 def ellipsoid(self, a_earth_f):

200 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum},

201 L{a_f2Tuple} or 2-tuple C{(a, f)}).

202 '''

203 self.datum = _spherical_datum(a_earth_f, Error=KTMError)

204

205 @Property_RO

206 def equatoradius(self):

207 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}).

208 '''

209 return self.ellipsoid.a

210

211 a = equatoradius

212

213 @Property_RO

214 def flattening(self):

215 '''Get the C{ellipsoid}'s flattening (C{scalar}).

216 '''

217 return self.ellipsoid.f

218

219 f = flattening

220

221 def forward(self, lat, lon, lon0=None, **name):

222 '''Forward projection, from geographic to transverse Mercator.

223

224 @arg lat: Latitude of point (C{degrees90}).

225 @arg lon: Longitude of point (C{degrees180}).

226 @arg lon0: Central meridian of the projection (C{degrees180}).

227 @kwarg name: Optional C{B{name}=NN} (C{str}).

228

229 @return: L{Forward4Tuple}C{(easting, northing, gamma, scale)} with

230 C{easting} and C{northing} in C{meter}, unfalsed, the

231 meridian convergence C{gamma} at point in C{degrees180}

232 and the C{scale} of projection at point C{scalar}. Any

233 value may be C{NAN}, C{NINF} or C{INF} for singularities.

234

235 @raise KTMError: For singularities, iff property C{raiser} is C{True}.

236 '''

237 lat, _lat = _unsigned2(_fix90(lat - self._lat0))

238 lon, _ = _diff182((self.lon0 if lon0 is None else lon0), lon)

239 lon, _lon = _unsigned2(lon)

240 backside = lon > 90

241 if backside: # PYCHOK no cover

242 lon = _loneg(lon)

243 if lat == 0:

244 _lat = True

245

246 sphi, cphi, slam, clam = sincos2d_(lat, lon)

247 E = self.ellipsoid

248 if cphi and lat != 90:

249 t = sphi / cphi

250 tp = E.es_taupf(t)

251 h = hypot(tp, clam)

252 if h:

253 xip = atan2(tp, clam)

254 etap = asinh(slam / h) # atanh(sin(lam) / cosh(psi))

255 g = _atan2d(slam * tp, clam * hypot1(tp)) # Krueger p 22 (44)

256 k = sqrt(cphi**2 * E.e2 + E.e21) * hypot1(t) / h

257 elif self.raiser:

258 raise KTMError(lat=lat, lon=lon, lon0=lon0, txt=_singular_)

259 else: # PYCHOK no cover

260 xip, etap = _0_0, _copysignINF(slam)

261 g, k = copysign0(_90_0, slam), INF

262 else: # PYCHOK no cover

263 xip, etap = PI_2, _0_0

264 g, k = lon, E.es_c

265 y, x, d, t = _Cyxgk4(E, xip, etap, self._Alp)

266 g -= d

267 k *= t * self._k0_b1

268

269 if backside: # PYCHOK no cover

270 y, g = (PI - y), _loneg(g)

271 y *= self._k0_a1

272 x *= self._k0_a1

273 if _lat:

274 y, g = neg_(y, g)

275 if _lon:

276 x, g = neg_(x, g)

277 return Forward4Tuple(x, y, _norm180(g), k, name=self._name__(name))

278

279 @property_doc_(''' the central scale factor (C{float}).''')

280 def k0(self):

281 '''Get the central scale factor (C{float}), aka I{C{scale0}}.

282 '''

283 return self._k0 # aka scale0

284

285 @k0.setter # PYCHOK setter!

286 def k0(self, k0):

287 '''Set the central scale factor (C{float}), aka I{C{scale0}}.

288

289 @raise KTMError: Invalid B{C{k0}}.

290 '''

291 k0 = Scalar_(k0=k0, Error=KTMError, low=_TOL_10, high=_1_0)

292 if self._k0 != k0: # PYCHOK no cover

293 KTransverseMercator._k0_a1._update(self) # redo ._k0_a1

294 KTransverseMercator._k0_b1._update(self) # redo ._k0_b1

295 self._k0 = k0

296

297 @Property_RO

298 def _k0_a1(self):

299 '''(INTERNAL) Cache C{k0 * _b1 * equatoradius}.

300 '''

301 return self._k0_b1 * self.equatoradius

302

303 @Property_RO

304 def _k0_b1(self):

305 '''(INTERNAL) Cache C{k0 * _b1}.

306 '''

307 return self.k0 * self._b1

308

309 @property_doc_(''' the central meridian (C{degrees180}).''')

310 def lon0(self):

311 '''Get the central meridian (C{degrees180}).

312 '''

313 return self._lon0

314

315 @lon0.setter # PYCHOK setter!

316 def lon0(self, lon0):

317 '''Set the central meridian (C{degrees180}).

318

319 @raise KTMError: Invalid B{C{lon0}}.

320 '''

321 self._lon0 = _norm180(Degrees(lon0=lon0, Error=KTMError))

322

323 @property_doc_(''' raise a L{KTMError} for C{forward} singularities (C{bool}).''')

324 def raiser(self):

325 '''Get the error setting (C{bool}).

326 '''

327 return self._raiser

328

329 @raiser.setter # PYCHOK setter!

330 def raiser(self, raiser):

331 '''Set the error setting (C{bool}), to C{True} to throw a L{KTMError}

332 for C{forward} singularities.

333 '''

334 self._raiser = bool(raiser)

335

336 def reset(self, lat0, lon0):

337 '''Set the central parallel and meridian.

338

339 @arg lat0: Latitude of the central parallel (C{degrees90}).

340 @arg lon0: Longitude of the central parallel (C{degrees180}).

341

342 @return: 2-Tuple C{(lat0, lon0)} with the previous central

343 parallel and meridian.

344

345 @raise KTMError: Invalid B{C{lat0}} or B{C{lon0}}.

346 '''

347 t = self._lat0, self.lon0

348 self._lat0 = _fix90(Degrees(lat0=lat0, Error=KTMError))

349 self. lon0 = lon0

350 return t

351

352 def reverse(self, x, y, lon0=None, **name):

353 '''Reverse projection, from transverse Mercator to geographic.

354

355 @arg x: Easting of point (C{meter}).

356 @arg y: Northing of point (C{meter}).

357 @arg lon0: Central meridian of the projection (C{degrees180}).

358 @kwarg name: Optional C{B{name}=NN} (C{str}).

359

360 @return: L{Reverse4Tuple}C{(lat, lon, gamma, scale)} with

361 C{lat}- and C{lon}gitude in C{degrees}, I{unfalsed}.

362 '''

363 eta, _lon = _unsigned2(x / self._k0_a1)

364 xi, _lat = _unsigned2(y / self._k0_a1)

365 backside = xi > PI_2

366 if backside: # PYCHOK no cover

367 xi = PI - xi

368

369 E = self.ellipsoid

370 xip, etap, g, k = _Cyxgk4(E, xi, eta, self._Bet)

371 t = self._k0_b1

372 k = (t / k) if k else _copysignINF(t) # _over(t, k)

373 h, c = sinh(etap), cos(xip)

374 if c > 0:

375 r = hypot(h, c)

376 else: # PYCHOK no cover

377 r = fabs(h)

378 c = _0_0

379 if r:

380 lon = _atan2d(h, c) # Krueger p 17 (25)

381 s = sin(xip) # Newton for tau

382 t = E.es_tauf(s / r)

383 lat = atan1d(t)

384 g += _atan2d(s * tanh(etap), c) # Krueger p 19 (31)

385 k *= sqrt(E.e2 / (t**2 + _1_0) + E.e21) * hypot1(t) * r

386 else: # PYCHOK no cover

387 lat = _90_0

388 lon = _0_0

389 k *= E.es_c

390

391 if backside: # PYCHOK no cover

392 lon, g = _loneg(lon), _loneg(g)

393 if _lat:

394 lat, g = neg_(lat, g)

395 if _lon:

396 lon, g = neg_(lon, g)

397 lat += self._lat0

398 lon += self._lon0 if lon0 is None else _norm180(lon0)

399 return Reverse4Tuple(lat, _norm180(lon), _norm180(g), k,

400 name=self._name__(name))

401

402 @Property

403 def TMorder(self):

404 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).

405 '''

406 return self._mTM

407

408 @TMorder.setter # PYCHOK setter!

409 def TMorder(self, order):

410 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8).

411 '''

412 m = _Xorder(_AlpCoeffs, KTMError, TMorder=order)

413 if self._mTM != m:

414 _update_all(self)

415 self._mTM = m

416

417 def toStr(self, **kwds):

418 '''Return a C{str} representation.

419

420 @arg kwds: Optional, overriding keyword arguments.

421 '''

422 d = dict(ellipsoid=self.ellipsoid, k0=self.k0, TMorder=self.TMorder)

423 if self.name: # PYCHOK no cover

424 d.update(name=self.name)

425 return _COMMASPACE_.join(pairs(d, **kwds))

426

427

428 def _cma(a, b0, b1, Cn):

429 '''(INTERNAL) Compute complex M{a * b0 - b1 + Cn} with complex

430 C{a}, C{b0} and C{b1} and scalar C{Cn}.

431

432 @see: CPython function U{_Py_c_prod<https://GitHub.com/python/

433 cpython/blob/main/Objects/complexobject.c>}.

434

435 @note: Python function C{cmath.fsum} no longer exists.

436 '''

437 r = fdot_(a.real, b0.real, -a.imag, b0.imag, -b1.real, _1_0, start=Cn)

438 j = fdot_(a.real, b0.imag, a.imag, b0.real, -b1.imag, _1_0)

439 return complex(r, j)

440

441

442 def _Cyxgk4(E, xi_, eta_, C):

443 '''(INTERNAL) Complex Clenshaw summation with C{B{C}=._Alp}

444 or C{B{C}=-._Bet}.

445 '''

446 x = complex(xi_, eta_)

447 if E.f: # isEllipsoidal

448 s, c = sincos2( xi_ * 2)

449 sh, ch = _sinhcosh2(eta_ * 2)

450 n = -s

451 s = complex(s * ch, c * sh) # sin(zeta * 2)

452 c = complex(c * ch, n * sh) # cos(zeta * 2)

453 a = c * 2 # cos(zeta * 2) * 2

454

455 y0 = y1 = \

456 z0 = z1 = complex(0) # 0+0j

457 n = len(C) - 1 # == .TMorder

458 if isodd(n):

459 Cn = C[n]

460 y0 = complex(Cn) # +0j

461 z0 = complex(Cn * (n * 2))

462 n -= 1

463 while n > 0:

464 Cn = C[n]

465 y1 = _cma(a, y0, y1, Cn)

466 z1 = _cma(a, z0, z1, Cn * (n * 2))

467 n -= 1

468 Cn = C[n]

469 y0 = _cma(a, y1, y0, Cn)

470 z0 = _cma(a, z1, z0, Cn * (n * 2))

471 n -= 1

472 # assert n == 0

473 x = _cma(s, y0, -x, _0_0)

474 c = _cma(c, z0, z1, _1_0)

475

476 # Gauss-Schreiber to Gauss-Krueger TM

477 # C{cmath.polar} handles INF, NAN, etc.

478 k, g = polar(c)

479 g = degrees(g)

480 else: # E.isSpherical

481 g, k = _0_0, _1_0

482

483 return x.real, x.imag, g, k

484

485

486 def _sinhcosh2(x):

487 '''(INTERNAL) Like C{sincos2}.

488 '''

489 return sinh(x), cosh(x)

490

491

492 def _Xs(_Coeffs, m, E, RA=False): # in .rhumb.ekx

493 '''(INTERNAL) Compute the C{A}, C{B} or C{RA} terms of order

494 B{C{m}} for I{Krüger} series and I{rhumb.ekx._sincosSeries},

495 return a tuple with C{B{m} + 1} terms C{X}, C{X[0]==0}.

496 '''

497 Cs = _Coeffs[m]

498 assert len(Cs) == (((m + 1) * (m + 4)) if RA else

499 ((m + 3) * m)) // 2

500 n = n_ = E.n

501 if n: # isEllipsoidal

502 X = [0] # X[0] never used, it's just an integration

503 # constant, it cancels when evaluating a definite

504 # integral. Don't bother computing it, it is unused

505 # in C{_Cyxgk4} above and C{rhumb.ekx._sincosSeries}.

506 i = (m + 2) if RA else 0

507 _p = _polynomial

508 for r in _reverange(m): # [m-1 ... 0]

509 j = i + r + 1

510 X.append(_p(n, Cs, i, j) * n_ / Cs[j])

511 i = j + 1

512 n_ *= n

513 X = tuple(X)

514 else: # isSpherical

515 X = _0_0s(m + 1)

516 return X

517

518

519# _Alp- and _BetCoeffs in .rhumb.ekx, .rhumb.bases

520_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00

521 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4

522 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3

523 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2

524 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1

525 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14

526 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5

527 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4

528 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3

529 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2

530 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1

531 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20

532 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6

533 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5

534 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4

535 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3

536 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2

537 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1

538 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27

539 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7

540 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6

541 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5

542 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4

543 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3

544 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2

545 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1

546 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35

547 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8

548 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7

549 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6

550 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5

551 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4

552 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3

553 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2

554 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1

555 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44

556}

557_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00

558 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2

559 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2

560 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3

561 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3

562 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4

563 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4

564}

565_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00

566 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4

567 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3

568 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2

569 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1

570 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14

571 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5

572 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4

573 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3

574 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2

575 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1

576 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20

577 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6

578 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5

579 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4

580 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3

581 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2

582 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1

583 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27

584 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7

585 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6

586 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5

587 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4

588 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3

589 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2

590 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1

591 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35

592 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8

593 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7

594 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6

595 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5

596 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4

597 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3

598 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2

599 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1

600 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44

601}

602

603 assert set(_AlpCoeffs.keys()) == set(_BetCoeffs.keys())

604

605 if __name__ == '__main__':

606

607 from pygeodesy.internals import _usage

608 from sys import argv, exit as _exit

609

610 _exit(_usage(*argv).replace('.ktm', '.etm -series'))

611

612# **) MIT License

613#

614# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.

615#

616# Permission is hereby granted, free of charge, to any person obtaining a

617# copy of this software and associated documentation files (the "Software"),

618# to deal in the Software without restriction, including without limitation

619# the rights to use, copy, modify, merge, publish, distribute, sublicense,

620# and/or sell copies of the Software, and to permit persons to whom the

621# Software is furnished to do so, subject to the following conditions:

622#

623# The above copyright notice and this permission notice shall be included

624# in all copies or substantial portions of the Software.

625#

626# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

627# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

628# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

629# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

630# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

631# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

632# OTHER DEALINGS IN THE SOFTWARE.