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_get, _Xorder, _pairs

52 from pygeodesy.fmath import hypot, hypot1

53 from pygeodesy.fsums import fsum1f_, _ALL_LAZY

54 from pygeodesy.interns import NN, _COMMASPACE_, _singular_

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

56 _polynomial, _unsigned2, _NamedBase

57# from pygeodesy.lazily import _ALL_LAZY, _pairs # from .errors, .fsums

58# from pygeodesy.named import _NamedBase # from .karney

59 from pygeodesy.namedTuples import Forward4Tuple, Reverse4Tuple

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

61 _update_all

62# from pygeodesy.streprs import pairs as _pairs # from .lazily

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

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

65

66 from cmath import polar

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

68

69__all__ = _ALL_LAZY.ktm

70__version__ = '23.12.01'

71

72

73 class KTMError(_ValueError):

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

75 '''

76 pass

77

78

79 class KTransverseMercator(_NamedBase):

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

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

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

83

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

85 projection and its inverse in terms of a series.

86

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

88 90}, about +/- 82.6 degrees for WGS84, where I{e} is the eccentricity. Beyond

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

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

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

92 degrees for the WGS84 ellipsoid.

93

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

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

96 over the entire ellipsoid.

97

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

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

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

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

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

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

104 are can be simply included by the calling function.

105

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

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

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

109 from true North.

110 '''

111 _datum = _WGS84

112 _k0 = _K0_UTM # central scale factor

113 _lat0 = _0_0 # central parallel

114 _lon0 = _0_0 # central meridian

115 _mTM = 6

116 _raiser = False # throw Error

117

118 def __init__(self, a_earth=_EWGS84, f=None, lon0=0, k0=_K0_UTM, name=NN,

119 raiser=False, **TMorder):

120 '''New L{KTransverseMercator}.

121

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

123 L{a_f2Tuple}, L{Datum}, 2-tuple (C{a, f})) or the

124 equatorial radius (C{scalar}, C{meter}).

125 @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is

126 a C{scalar}, ignored otherwise.

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

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

129 @kwarg name: Optional name (C{str}).

130 @kwarg raiser: If C{True}, throw a L{KTMError} for C{forward}

131 singularities (C{bool}).

132 @kwarg TMorder: Keyword argument B{C{TMorder}}, see property C{TMorder}.

133

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

135 '''

136 if f is not None:

137 self.ellipsoid = a_earth, f

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

139 pass

140 elif isinstance(a_earth, Datum):

141 self.datum = a_earth

142 else:

143 self.ellipsoid = a_earth

144 self.lon0 = lon0

145 self.k0 = k0

146 if name: # PYCHOK no cover

147 self.name = name

148 if raiser:

149 self.raiser = True

150 if TMorder:

151 self.TMorder = _xkwds_get(TMorder, TMorder=self._mTM)

152

153 @Property_RO

154 def _Alp(self):

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

156

157 @Property_RO

158 def _b1(self):

159 n = self.ellipsoid.n

160 if n: # isEllipsoidal

161 m = self.TMorder // 2

162 B1 = _B1Coeffs[m]

163 m += 1

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

165 else: # isSpherical

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

167 return b1

168

169 @Property_RO

170 def _Bet(self):

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

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

173

174 @property

175 def datum(self):

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

177 '''

178 return self._datum

179

180 @datum.setter # PYCHOK setter!

181 def datum(self, datum):

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

183 '''

184 _xinstanceof(Datum, datum=datum)

185 if self._datum != datum:

186 _update_all(self)

187 self._datum = datum

188

189 @Property

190 def ellipsoid(self):

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

192 '''

193 return self.datum.ellipsoid

194

195 @ellipsoid.setter # PYCHOK setter!

196 def ellipsoid(self, a_earth_f):

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

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

199 '''

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

201

202 @Property_RO

203 def equatoradius(self):

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

205 '''

206 return self.ellipsoid.a

207

208 a = equatoradius

209

210 @Property_RO

211 def flattening(self):

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

213 '''

214 return self.ellipsoid.f

215

216 f = flattening

217

218 def forward(self, lat, lon, lon0=None, name=NN):

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

220

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

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

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

224 @kwarg name: Optional name (C{str}).

225

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

227 with C{easting} and C{northing} in C{meter}, unfalsed, the

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

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

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

231

232 @raise KTMError: For singularities, iff property C{raiser} is

233 C{True}.

234 '''

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

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

237 lon, _lon = _unsigned2(lon)

238 backside = lon > 90

239 if backside: # PYCHOK no cover

240 lon = _loneg(lon)

241 if lat == 0:

242 _lat = True

243

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

245 E = self.ellipsoid

246 if cphi and lat != 90:

247 t = sphi / cphi

248 tp = E.es_taupf(t)

249 h = hypot(tp, clam)

250 if h:

251 xip = atan2(tp, clam)

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

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

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

255 elif self.raiser:

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

257 else: # PYCHOK no cover

258 xip, etap = _0_0, _copysignINF(slam)

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

260 else: # PYCHOK no cover

261 xip, etap = PI_2, _0_0

262 g, k = lon, E.es_c

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

264 g -= d

265 k *= t * self._k0_b1

266

267 if backside: # PYCHOK no cover

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

269 y *= self._k0_a1

270 x *= self._k0_a1

271 if _lat:

272 y, g = neg_(y, g)

273 if _lon:

274 x, g = neg_(x, g)

275 return Forward4Tuple(x, y, _norm180(g), k, name=name or self.name)

276

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

278 def k0(self):

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

280 '''

281 return self._k0 # aka scale0

282

283 @k0.setter # PYCHOK setter!

284 def k0(self, k0):

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

286

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

288 '''

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

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

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

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

293 self._k0 = k0

294

295 @Property_RO

296 def _k0_a1(self):

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

298 '''

299 return self._k0_b1 * self.equatoradius

300

301 @Property_RO

302 def _k0_b1(self):

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

304 '''

305 return self.k0 * self._b1

306

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

308 def lon0(self):

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

310 '''

311 return self._lon0

312

313 @lon0.setter # PYCHOK setter!

314 def lon0(self, lon0):

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

316

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

318 '''

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

320

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

322 def raiser(self):

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

324 '''

325 return self._raiser

326

327 @raiser.setter # PYCHOK setter!

328 def raiser(self, raiser):

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

330 for C{forward} singularities.

331 '''

332 self._raiser = bool(raiser)

333

334 def reset(self, lat0, lon0):

335 '''Set the central parallel and meridian.

336

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

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

339

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

341 parallel and meridian.

342

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

344 '''

345 t = self._lat0, self.lon0

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

347 self. lon0 = lon0

348 return t

349

350 def reverse(self, x, y, lon0=None, name=NN):

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

352

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

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

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

356

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

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

359 '''

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

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

362 backside = xi > PI_2

363 if backside: # PYCHOK no cover

364 xi = PI - xi

365

366 E = self.ellipsoid

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

368 t = self._k0_b1

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

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

371 if c > 0:

372 r = hypot(h, c)

373 else: # PYCHOK no cover

374 r = fabs(h)

375 c = _0_0

376 if r:

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

378 s = sin(xip) # Newton for tau

379 t = E.es_tauf(s / r)

380 lat = atan1d(t)

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

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

383 else: # PYCHOK no cover

384 lat = _90_0

385 lon = _0_0

386 k *= E.es_c

387

388 if backside: # PYCHOK no cover

389 lon, g = _loneg(lon), _loneg(g)

390 if _lat:

391 lat, g = neg_(lat, g)

392 if _lon:

393 lon, g = neg_(lon, g)

394 lat += self._lat0

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

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

397 name=name or self.name)

398

399 @Property

400 def TMorder(self):

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

402 '''

403 return self._mTM

404

405 @TMorder.setter # PYCHOK setter!

406 def TMorder(self, order):

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

408 '''

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

410 if self._mTM != m:

411 _update_all(self)

412 self._mTM = m

413

414 def toStr(self, **kwds):

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

416

417 @arg kwds: Optional, overriding keyword arguments.

418 '''

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

420 if self.name: # PYCHOK no cover

421 d.update(name=self.name)

422 return _COMMASPACE_.join(_pairs(d, **kwds))

423

424

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

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

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

428

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

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

431

432 @note: Python function C{cmath.fsum} is no longer available,

433 but stil mentioned in Note 4 of the comments before

434 CPython function U{math_fsum<https://GitHub.com/python/

435 cpython/blob/main/Modules/mathmodule.c>}

436 '''

437 r = fsum1f_(a.real * b0.real, -a.imag * b0.imag, -b1.real, Cn)

438 j = fsum1f_(a.real * b0.imag, a.imag * b0.real, -b1.imag)

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 _c = _cma

464 while n > 0:

465 Cn = C[n]

466 y1 = _c(a, y0, y1, Cn)

467 z1 = _c(a, z0, z1, Cn * (n * 2))

468 n -= 1

469 Cn = C[n]

470 y0 = _c(a, y1, y0, Cn)

471 z0 = _c(a, z1, z0, Cn * (n * 2))

472 n -= 1

473 # assert n == 0

474 x = _c(s, y0, -x, _0_0)

475 c = _c(c, z0, z1, _1_0)

476

477 # Gauss-Schreiber to Gauss-Krueger TM

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

479 k, g = polar(c)

480 g = degrees(g)

481 else: # E.isSpherical

482 g, k = _0_0, _1_0

483

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

485

486

487 def _sinhcosh2(x):

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

489 '''

490 return sinh(x), cosh(x)

491

492

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

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

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

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

497 '''

498 Cs = _Coeffs[m]

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

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

501 n = n_ = E.n

502 if n: # isEllipsoidal

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

504 # constant, it cancels when evaluating a definite

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

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

507 _X, _p = X.append, _polynomial

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

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

510 j = i + r + 1

511 _X(_p(n, Cs, i, j) * n_ / Cs[j])

512 i = j + 1

513 n_ *= n

514 X = tuple(X)

515 else: # isSpherical

516 X = _0_0s(m + 1)

517 return X

518

519

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

521_AlpCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00

522 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4

523 164, 225, -480, 360, 720, # Alp[1]/n^1, polynomial(n), order 3

524 557, -864, 390, 1440, # Alp[2]/n^2, polynomial(n), order 2

525 -1236, 427, 1680, # PYCHOK Alp[3]/n^3, polynomial(n), order 1

526 49561, 161280), # Alp[4]/n^4, polynomial(n), order 0, count = 14

527 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5

528 -635, 328, 450, -960, 720, 1440, # Alp[1]/n^1, polynomial(n), order 4

529 4496, 3899, -6048, 2730, 10080, # PYCHOK Alp[2]/n^2, polynomial(n), order 3

530 15061, -19776, 6832, 26880, # PYCHOK Alp[3]/n^3, polynomial(n), order 2

531 -171840, 49561, 161280, # Alp[4]/n^4, polynomial(n), order 1

532 34729, 80640), # PYCHOK Alp[5]/n^5, polynomial(n), order 0, count = 20

533 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6

534 31564, -66675, 34440, 47250, -100800, 75600, 151200, # Alp[1]/n^1, polynomial(n), order 5

535 -1983433, 863232, 748608, -1161216, 524160, 1935360, # PYCHOK Alp[2]/n^2, polynomial(n), order 4

536 670412, 406647, -533952, 184464, 725760, # Alp[3]/n^3, polynomial(n), order 3

537 6601661, -7732800, 2230245, 7257600, # Alp[4]/n^4, polynomial(n), order 2

538 -13675556, 3438171, 7983360, # PYCHOK Alp[5]/n^5, polynomial(n), order 1

539 212378941, 319334400), # Alp[6]/n^6, polynomial(n), order 0, count = 27

540 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7

541 1804025, 2020096, -4267200, 2204160, 3024000, -6451200, 4838400, 9676800, # Alp[1]/n^1, polynomial(n), order 6

542 4626384, -9917165, 4316160, 3743040, -5806080, 2620800, 9676800, # Alp[2]/n^2, polynomial(n), order 5

543 -67102379, 26816480, 16265880, -21358080, 7378560, 29030400, # PYCHOK Alp[3]/n^3, polynomial(n), order 4

544 155912000, 72618271, -85060800, 24532695, 79833600, # Alp[4]/n^4, polynomial(n), order 3

545 102508609, -109404448, 27505368, 63866880, # Alp[5]/n^5, polynomial(n), order 2

546 -12282192400, 2760926233, 4151347200, # PYCHOK Alp[6]/n^6, polynomial(n), order 1

547 1522256789, 1383782400), # Alp[7]/n^7, polynomial(n), order 0, count = 35

548 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8

549 -75900428, 37884525, 42422016, -89611200, 46287360, 63504000, -135475200, 101606400, 203212800, # Alp[1]/n^1, polynomial(n), order 7

550 148003883, 83274912, -178508970, 77690880, 67374720, -104509440, 47174400, 174182400, # PYCHOK Alp[2]/n^2, polynomial(n), order 6

551 318729724, -738126169, 294981280, 178924680, -234938880, 81164160, 319334400, # PYCHOK Alp[3]/n^3, polynomial(n), order 5

552 -40176129013, 14967552000, 6971354016, -8165836800, 2355138720, 7664025600, # Alp[4]/n^4, polynomial(n), order 4

553 10421654396, 3997835751, -4266773472, 1072709352, 2490808320, # PYCHOK Alp[5]/n^5, polynomial(n), order 3

554 175214326799, -171950693600, 38652967262, 58118860800, # PYCHOK Alp[6]/n^6, polynomial(n), order 2

555 -67039739596, 13700311101, 12454041600, # PYCHOK Alp[7]/n^7, polynomial(n), order 1

556 1424729850961, 743921418240) # PYCHOK Alp[8]/n^8, polynomial(n), order 0, count = 44

557}

558_B1Coeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00

559 2: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 2

560 1, 16, 64, 64), # b1 * (n + 1), polynomial(n2), order 2

561 3: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 3

562 1, 4, 64, 256, 256), # b1 * (n + 1), polynomial(n2), order 3

563 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER/2 == 4

564 25, 64, 256, 4096, 16384, 16384) # PYCHOK b1 * (n + 1), polynomial(n2), order 4

565}

566_BetCoeffs = { # Generated by Maxima on 2015-05-14 22:55:13-04:00

567 4: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 4

568 -4, 555, -960, 720, 1440, # Bet[1]/n^1, polynomial(n), order 3

569 -437, 96, 30, 1440, # Bet[2]/n^2, polynomial(n), order 2

570 -148, 119, 3360, # Bet[3]/n^3, polynomial(n), order 1

571 4397, 161280), # PYCHOK Bet[4]/n^4, polynomial(n), order 0, count = 14

572 5: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 5

573 -3645, -64, 8880, -15360, 11520, 23040, # Bet[1]/n^1, polynomial(n), order 4

574 4416, -3059, 672, 210, 10080, # PYCHOK Bet[2]/n^2, polynomial(n), order 3

575 -627, -592, 476, 13440, # Bet[3]/n^3, polynomial(n), order 2

576 -3520, 4397, 161280, # Bet[4]/n^4, polynomial(n), order 1

577 4583, 161280), # PYCHOK Bet[5]/n^5, polynomial(n), order 0, count = 20

578 6: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 6

579 384796, -382725, -6720, 932400, -1612800, 1209600, 2419200, # Bet[1]/n^1, polynomial(n), order 5

580 -1118711, 1695744, -1174656, 258048, 80640, 3870720, # PYCHOK Bet[2]/n^2, polynomial(n), order 4

581 22276, -16929, -15984, 12852, 362880, # Bet[3]/n^3, polynomial(n), order 3

582 -830251, -158400, 197865, 7257600, # PYCHOK Bet[4]/n^4, polynomial(n), order 2

583 -435388, 453717, 15966720, # PYCHOK Bet[5]/n^5, polynomial(n), order 1

584 20648693, 638668800), # Bet[6]/n^6, polynomial(n), order 0, count = 27

585 7: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 7

586 -5406467, 6156736, -6123600, -107520, 14918400, -25804800, 19353600, 38707200, # Bet[1]/n^1, polynomial(n), order 6

587 829456, -5593555, 8478720, -5873280, 1290240, 403200, 19353600, # PYCHOK Bet[2]/n^2, polynomial(n), order 5

588 9261899, 3564160, -2708640, -2557440, 2056320, 58060800, # PYCHOK Bet[3]/n^3, polynomial(n), order 4

589 14928352, -9132761, -1742400, 2176515, 79833600, # PYCHOK Bet[4]/n^4, polynomial(n), order 3

590 -8005831, -1741552, 1814868, 63866880, # Bet[5]/n^5, polynomial(n), order 2

591 -261810608, 268433009, 8302694400, # Bet[6]/n^6, polynomial(n), order 1

592 219941297, 5535129600), # PYCHOK Bet[7]/n^7, polynomial(n), order 0, count = 35

593 8: ( # GEOGRAPHICLIB_TRANSVERSEMERCATOR_ORDER == 8

594 31777436, -37845269, 43097152, -42865200, -752640, 104428800, -180633600, 135475200, 270950400, # Bet[1]/n^1, polynomial(n), order 7

595 24749483, 14930208, -100683990, 152616960, -105719040, 23224320, 7257600, 348364800, # Bet[2]/n^2, polynomial(n), order 6

596 -232468668, 101880889, 39205760, -29795040, -28131840, 22619520, 638668800, # PYCHOK Bet[3]/n^3, polynomial(n), order 5

597 324154477, 1433121792, -876745056, -167270400, 208945440, 7664025600, # Bet[4]/n^4, polynomial(n), order 4

598 457888660, -312227409, -67920528, 70779852, 2490808320, # Bet[5]/n^5, polynomial(n), order 3

599 -19841813847, -3665348512, 3758062126, 116237721600, # PYCHOK Bet[6]/n^6, polynomial(n), order 2

600 -1989295244, 1979471673, 49816166400, # PYCHOK Bet[7]/n^7, polynomial(n), order 1

601 191773887257, 3719607091200) # Bet[8]/n^8, polynomial(n), order 0, count = 44

602}

603

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

605

606 if __name__ == '__main__':

607

608 from pygeodesy.interns import _usage

609 from sys import argv, exit as _exit

610

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

612

613# **) MIT License

614#

615# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved.

616#

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

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

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

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

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

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

623#

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

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

626#

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

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

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

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

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

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

633# OTHER DEALINGS IN THE SOFTWARE.