Coverage for pygeodesy/auxilats/auxLat.py: 93%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Class L{AuxLat} transcoded to Python from I{Karney}'s C++ class U{AuxLatitude 

5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>} 

6in I{GeographicLib version 2.2+}. 

7 

8Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2022-2023) and licensed 

9under the MIT/X11 License. For more information, see the U{GeographicLib 

10<https://GeographicLib.SourceForge.io>} documentation. 

11 

12@see: U{Auxiliary latitudes<https:#GeographicLib.SourceForge.io/C++/doc/auxlat.html>} 

13 U{On auxiliary latitudes<https:#ArXiv.org/abs/2212.05818>}. 

14''' 

15# make sure int/int division yields float quotient, see .basics 

16from __future__ import division as _; del _ # PYCHOK semicolon 

17 

18from pygeodesy.auxilats.auxAngle import AuxAngle, AuxBeta, AuxChi, _AuxClass, \ 

19 AuxMu, AuxPhi, AuxTheta, AuxXi 

20from pygeodesy.auxilats.auxily import Aux, _sc, _sn, _Ufloats, atan1 

21from pygeodesy.basics import _passarg, _reverange, _xinstanceof 

22from pygeodesy.constants import INF, MAX_EXP, MIN_EXP, NAN, PI_2, PI_4, _EPSqrt, \ 

23 _0_0, _0_0s, _0_1, _0_25, _0_5, _1_0, _2_0, _3_0, \ 

24 _360_0, isfinite, isinf, isnan, _log2, _over 

25from pygeodesy.datums import _ellipsoidal_datum, _WGS84, Ellipsoid 

26# from pygeodesy.ellipsoids import Ellipsoid # from .datums 

27from pygeodesy.elliptic import Elliptic as _Ef 

28from pygeodesy.errors import AuxError, _xkwds, _xkwds_get, _Xorder 

29# from pygeodesy.fmath import cbrt # from .karney 

30from pygeodesy.fsums import Fsum, _Fsumf_, _sum 

31from pygeodesy.karney import _2cos2x, _polynomial, _ALL_DOCS, cbrt, _MODS 

32from pygeodesy.interns import NN, _DOT_, _UNDER_ # _earth_ 

33# from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS # from .karney 

34from pygeodesy.props import Property, Property_RO, _update_all 

35from pygeodesy.units import _isDegrees, _isRadius, Degrees, Meter 

36# from pygeodesy.utily import atan1 # from .auxily 

37 

38from math import asinh, atan2, copysign, cosh, fabs, sin, sinh, sqrt 

39try: 

40 from math import exp2 as _exp2 

41except ImportError: # Python 3.11- 

42 

43 def _exp2(x): 

44 return pow(_2_0, x) 

45 

46__all__ = () 

47__version__ = '24.04.14' 

48 

49_TRIPS = 1024 # XXX 2 or 3? 

50 

51 

52class AuxLat(AuxAngle): 

53 '''Base class for accurate conversion between I{Auxiliary} latitudes 

54 on an ellipsoid. 

55 

56 Latitudes are represented by L{AuxAngle} instances in order to 

57 maintain precision near the poles, I{Authalic} latitude C{Xi}, 

58 I{Conformal} C{Chi}, I{Geocentric} C{Theta}, I{Geographic} C{Phi}, 

59 I{Parametric} C{Beta} and I{Rectifying} C{Mu}. 

60 

61 @see: I{Karney}'s C++ class U{AuxLatitude 

62 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1AuxLatitude.html>}. 

63 ''' 

64 _csc = dict() # global coeffs cache: [aL][k], upto max(k) * (4 + 6 + 8) floats 

65 _E = _WGS84.ellipsoid 

66# _Lmax = 0 # overwritten below 

67 _mAL = 6 # 4, 6 or 8 aka Lmax 

68 

69 def __init__(self, a_earth=_WGS84, f=None, b=None, name=NN, **ALorder): 

70 '''New L{AuxLat} instance on an ellipsoid or datum. 

71 

72 @arg a_earth: Equatorial radius, semi-axis (C{meter}) or an 

73 ellipsoid or datum (L{Datum}, L{Ellipsoid}, 

74 L{Ellipsoid2} or L{a_f2Tuple}). 

75 @kwarg f: Flattening: M{(a - b) / a} (C{float}, near zero for 

76 spherical), ignored if B{C{a_earth}} is not scalar. 

77 @kwarg b: Optional polar radius, semi-axis (C{meter}, same 

78 units as B{C{a_earth}}), ignored if B{C{a_earth}} 

79 is not scalar. 

80 @kwarg ALorder: Optional keyword arguments B{C{ALorder}} to 

81 set L{AuxLat}'s C{order}, see property 

82 C{ALorder}. 

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

84 ''' 

85 if a_earth is not _WGS84: 

86 n = name or AuxLat.__name__ 

87 try: 

88 if b is f is None: 

89 E = _ellipsoidal_datum(a_earth, name=n).ellipsoid # XXX raiser=_earth_ 

90 elif _isRadius(a_earth): 

91 E = Ellipsoid(a_earth, f=f, b=b, name=_UNDER_(n)) 

92 else: 

93 raise ValueError() 

94 except Exception as x: 

95 raise AuxError(a_earth=a_earth, f=f, b=b, cause=x) 

96 self._E = E 

97 

98 if name: 

99 self.name = name 

100 if ALorder: 

101 self.ALorder = _xkwds_get(ALorder, ALorder=AuxLat._mAL) 

102 

103 @Property_RO 

104 def a(self): 

105 '''Get the C{ellipsoid}'s equatorial radius (C{meter}, conventionally). 

106 ''' 

107 return self.ellipsoid.a 

108 

109 equatoradius = a 

110 

111 @Property 

112 def ALorder(self): 

113 '''Get the I{AuxLat} order (C{int}, 4, 6 or 8). 

114 ''' 

115 return self._mAL 

116 

117 @ALorder.setter # PYCHOK setter! 

118 def ALorder(self, order): 

119 '''Set the I{AuxLat} order (C{int}, 4, 6 or 8). 

120 ''' 

121 m = _Xorder(_AR2Coeffs, AuxError, ALorder=order) 

122 if self._mAL != m: 

123 _update_all(self) 

124 self._mAL = m 

125 

126 def _atanhee(self, tphi): # see Ellipsoid._es_atanh, .albers._atanhee 

127 # atanh(e * sphi) = asinh(e' * sbeta) 

128 f = self.f 

129 s = _sn(self._fm1 * tphi) if f > 0 else _sn(tphi) 

130 if f: # atanh(e * sphi) = asinh(e' * sbeta) 

131 e = self._e 

132 s = _over(atan1(e * s) if f < 0 else asinh(self._e1 * s), e) 

133 return s 

134 

135 def Authalic(self, Phi, **diff_name): 

136 '''Convert I{Geographic} to I{Aunthalic} latitude. 

137 

138 @arg Phi: Geographic latitude (L{AuxAngle}). 

139 

140 @return: Parametric latitude, C{Beta} (L{AuxAngle}). 

141 ''' 

142 _xinstanceof(AuxAngle, Phi=Phi) 

143 # assert Phi._AUX == Aux.PHI 

144 tphi = fabs(Phi.tan) 

145 if isfinite(tphi) and tphi and self.f: 

146 y, x = Phi._yx_normalized 

147 q = self._q 

148 qv = self._qf(tphi) 

149 Dq2 = self._Dq(tphi) 

150 Dq2 *= (q + qv) / (fabs(y) + _1_0) # _Dq(-tphi) 

151 Xi = AuxXi(copysign(qv, Phi.y), x * sqrt(Dq2), 

152 name=_xkwds_get(diff_name, name=Phi.name)) 

153 

154 if _xkwds_get(diff_name, diff=False): 

155 if isnan(tphi): 

156 d = self._e2m1_sq2 

157 else: 

158 c = self.Parametric(Phi)._x_normalized 

159 d = _over(c, Xi._x_normalized)**3 

160 d *= _over(c, x) * _over(_2_0, q) 

161 Xi._diff = d 

162 else: 

163 Xi = AuxXi(*Phi._yx) # diff default 

164 # assert Xi._AUX == Aux.XI 

165 return Xi 

166 

167 def AuthalicRadius2(self, exact=False, f_max=_0_1): 

168 '''Get the I{Authalic} radius I{squared}. 

169 

170 @kwarg exact: If C{True}, use the exact expression, otherwise 

171 the I{Taylor} series. 

172 @kwarg f_max: C{Flattening} not to exceed (C{float}). 

173 

174 @return: Authalic radius I{squared} (C{meter} I{squared}, same 

175 units as the ellipsoid axes). 

176 

177 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)} 

178 exceeds C{f_max}. 

179 ''' 

180 f = self.f 

181 if exact or not f: 

182 c2 = self.ellipsoid.b2 * self._q # == ellipsoid.c2x * 2 

183 elif fabs(f) < f_max: 

184 # Using a * (a + b) / 2 as the multiplying factor leads to a rapidly 

185 # converging series in n. Of course, using this series isn't really 

186 # necessary, since the exact expression is simple to evaluate. However, 

187 # we do it for consistency with RectifyingRadius and, presumably, the 

188 # roundoff error is smaller compared to that for the exact expression. 

189 m = self.ALorder 

190 c2 = _polynomial(self._n, _AR2Coeffs[m], 0, m) 

191 c2 *= self.a * (self.a + self.b) 

192 else: 

193 raise AuxError(exact=exact, f=f, f_max=f_max) 

194 return c2 * _0_5 

195 

196 @Property_RO 

197 def b(self): 

198 '''Get the C{ellipsoid}'s polar radius (C{meter}, conventionally). 

199 ''' 

200 return self.ellipsoid.b # a * (_1_0 - f) 

201 

202 polaradius = b 

203 

204 def _coeffs(self, auxout, auxin): 

205 # Get the polynomial coefficients as 4-, 6- or 8-tuple 

206 aL = self.ALorder # aka Lmax 

207 if auxout == auxin: 

208 return _0_0s(aL) # uncached 

209 

210 k = Aux._1d(auxout, auxin) 

211 try: # cached 

212 return AuxLat._csc[aL][k] 

213 except KeyError: 

214 pass 

215 

216 Cx = _CXcoeffs(aL) 

217 try: 

218 Cx = Cx[auxout][auxin] 

219 except KeyError as x: 

220 raise AuxError(auxout=auxout, auxin=auxin, cause=x) 

221 

222 d = x = n = self._n 

223 if Aux.use_n2(auxin) and Aux.use_n2(auxout): 

224 x = self._n2 

225 

226 def _m(aL): 

227 for m in _reverange(aL): 

228 yield m // 2 

229 else: 

230 _m = _reverange # PYCHOK expected 

231 

232 i = 0 

233 cs = [] 

234 _c = cs.append 

235 _p = _polynomial 

236 for m in _m(aL): 

237 j = i + m + 1 # order m = j - i - 1 

238 _c(_p(x, Cx, i, j) * d) 

239 d *= n 

240 i = j 

241 # assert i == len(Cx) and len(cs) == aL 

242 AuxLat._csc.setdefault(aL, {})[k] = cs = tuple(cs) 

243 return cs 

244 

245 def Conformal(self, Phi, **diff_name): 

246 '''Convert I{Geographic} to I{Conformal} latitude. 

247 

248 @arg Phi: Geographic latitude (L{AuxAngle}). 

249 

250 @return: Conformal latitude, C{Chi} (L{AuxAngle}). 

251 ''' 

252 _xinstanceof(AuxAngle, Phi=Phi) 

253 # assert Phi._AUX == Aux.PHI 

254 tphi = tchi = fabs(Phi.tan) 

255 if isfinite(tphi) and tphi and self.f: 

256 sig = sinh(self._atanhee(tphi) * self._e2) 

257 scsig = _sc(sig) 

258 scphi = _sc(tphi) 

259 if self.f > 0: 

260 # The general expression for tchi is 

261 # tphi * scsig - sig * scphi 

262 # This involves cancellation if f > 0, so change to 

263 # (tphi - sig) * (tphi + sig) / (tphi * scsig + sig * scphi) 

264 # To control overflow, write as (sigtphi = sig / tphi) 

265 # (tphi - sig) * (1 + sigtphi) / (scsig + sigtphi * scphi) 

266 sigtphi = sig / tphi 

267 if sig < (tphi * _0_5): 

268 t = tphi - sig 

269 else: 

270 def _asinh_2(x): 

271 return asinh(x) * _0_5 

272 # Still possibly dangerous cancellation in tphi - sig. 

273 # Write tphi - sig = (1 - e) * Dg(1, e) 

274 # Dg(x, y) = (g(x) - g(y)) / (x - y) 

275 # g(x) = sinh(x * atanh(sphi * x)) 

276 # Note sinh(atanh(sphi)) = tphi 

277 # Turn the crank on divided differences, substitute 

278 # sphi = tphi / _sc(tphi) 

279 # atanh(x) = asinh(x / sqrt(1 - x^2)) 

280 e = self._e 

281 em1 = self._e2m1 / (_1_0 + e) 

282 # assert em1 != 0 

283 scb = self._scbeta(tphi) 

284 scphib = scphi / scb # sec(phi) / sec(beta) 

285 atphib = _asinh_2(tphi * e / scb) # atanh(e * sphi) 

286 atphi = _asinh_2(tphi) # atanh(sphi) 

287 t = _asinh_2(em1 * (tphi * scphib)) / em1 

288 try: 

289 Dg = _Fsumf_(atphi, atphib, t, e * t) 

290 except ValueError: # Fsum(NAN) exception 

291 Dg = _sum((atphi, atphib, t, e * t)) 

292 e *= atphib 

293 t = atphi - e 

294 if t: # sinh(0) == 0 

295 Dg *= sinh(t) / t * cosh(atphi + e) * em1 

296 t = float(Dg) # tphi - sig 

297 tchi = _over(t * (_1_0 + sigtphi), 

298 scsig + scphi * sigtphi) if t else _0_0 

299 else: 

300 tchi = tphi * scsig - sig * scphi 

301 

302 n = _xkwds_get(diff_name, name=Phi.name) 

303 Chi = AuxChi(tchi, name=n).copyquadrant(Phi) 

304 

305 if _xkwds_get(diff_name, diff=False): 

306 if isinf(tphi): # PYCHOK np cover 

307 d = self._conformal_diff 

308 else: 

309 d = self.Parametric(Phi)._x_normalized 

310 if d: 

311 d = _over(d, Chi._x_normalized) * \ 

312 _over(d, Phi._x_normalized) * self._e2m1 

313 Chi._diff = d 

314 # assrt Chi._AUX == Aux.CHI 

315 return Chi 

316 

317 @Property_RO 

318 def _conformal_diff(self): # PYCHOK no cover 

319 '''(INTERNAL) Constant I{Conformal} diff. 

320 ''' 

321 e = self._e 

322 if self.f > 0: 

323 ss = sinh(asinh(self._e1) * e) 

324 d = _over(_1_0, _sc(ss) + ss) 

325 elif e: 

326 ss = sinh(-atan1(e) * e) 

327 d = _sc(ss) - ss 

328 else: 

329 d = _1_0 

330 return d 

331 

332 def convert(self, auxout, Zeta_d, exact=False): 

333 # Convert I{Auxiliary} or I{scalar} latitude 

334 Z = d = Zeta_d 

335 if isinstance(Z, AuxAngle): 

336 A, auxin = _AuxClass(auxout), Z._AUX 

337 if auxin == auxout: 

338 pass 

339 elif not (isfinite(Z.tan) and Z.tan): # XXX 

340 Z = A(*Z._yx, aux=auxout, name=Z.name) 

341 elif exact: 

342 p = Aux.power(auxout, auxin) 

343 if p is None: 

344 P = self._fromAux(Z) # Phi 

345 Z = self._toAux(auxout, P) 

346 Z._iter = P.iteration 

347 else: 

348 y, x = Z._yx 

349 if p: 

350 y *= pow(self._fm1, p) 

351 Z = A(y, x, aux=auxout, name=Z.name) 

352 else: 

353 cs = self._coeffs(auxout, auxin) 

354 yx = Z._yx_normalized 

355 Z = A(*yx, aux=auxout, name=Z.name) 

356 # assert Z._yx == yx 

357 r = _Clenshaw(True, Z, cs, self.ALorder) 

358 Z += AuxAngle.fromRadians(r, aux=auxout) 

359 # assert Z._AUX == auxout 

360 return Z 

361 

362 elif _isDegrees(d): 

363 Z = AuxPhi.fromDegrees(d) 

364 d = round((d - Z.toDegrees) / _360_0) * _360_0 

365 d += self.convert(auxout, Z, exact).toDegrees 

366 return Degrees(d, name=Aux.Greek(auxout)) 

367 

368 raise AuxError(auxout=auxout, Zeta_d=Zeta_d, exact=exact) 

369 

370 def _Dq(self, tphi): 

371 # I{Divided Difference} of (q(1) - q(sphi)) / (1 - sphi). 

372 sphi = _sn(tphi) 

373 if tphi > 0: 

374 scphi = _sc(tphi) 

375 d = _1_0 / (scphi**2 * (_1_0 + sphi)) # XXX - sphi 

376 if d: 

377 # General expression for _Dq(1, sphi) is 

378 # atanh(e * d / (1 - e2 * sphi)) / (e * d) + 

379 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1) 

380 # with atanh(e * d / (1 - e2 * sphi)) = 

381 # atanh(e * d * scphi / (scphi - e2 * tphi)) 

382 e2m1, ed = self._e2m1, (self._e * d) 

383 if e2m1 and ed: 

384 e2 = self._e2 

385 if e2 > 0: # assert self.f > 0 

386 scb = self._scbeta(tphi) 

387 q = scphib = scphi / scb 

388 q *= (scphi + tphi * e2) / (e2m1 * scb) 

389 q += asinh(self._e1 * d * scphib) / ed 

390 else: 

391 s2 = sphi * e2 

392 q = (_1_0 + s2) / ((_1_0 - sphi * s2) * e2m1) 

393 q += (atan2(ed, _1_0 - s2) / ed) if e2 < 0 else _1_0 

394 else: # PYCHOK no cover 

395 q = INF 

396 else: # PYCHOK no cover 

397 q = self._2_e2m12 

398 else: # not reached, open-coded in .Authalic 

399 q = _over(self._q - self._qf(tphi), _1_0 - sphi) 

400 return q 

401 

402 @Property_RO 

403 def _e(self): # unsigned, (1st) eccentricity 

404 return self.ellipsoid.e # sqrt(fabs(self._e2)) 

405 

406 @Property_RO 

407 def _e1(self): 

408 return sqrt(fabs(self._e12)) 

409 

410 @Property_RO 

411 def _e12(self): 

412 return _over(self._e2, _1_0 - self._e2) 

413 

414 @Property_RO 

415 def _e12p1(self): 

416 return _1_0 / self._e2m1 

417 

418 @Property_RO 

419 def _e2(self): # signed, (1st) eccentricity squared 

420 return self.ellipsoid.e2 

421 

422 @Property_RO 

423 def _e2m1(self): # 1 less 1st eccentricity squared 

424 return self.ellipsoid.e21 # == ._fm1**2 

425 

426 @Property_RO 

427 def _e2m1_sq2(self): 

428 return self._e2m1 * sqrt(self._q * _0_5) 

429 

430 @Property_RO 

431 def _2_e2m12(self): 

432 return _2_0 / self._e2m1**2 

433 

434 @Property_RO 

435 def _Ef_fRG_a2b2_PI_4(self): 

436 E = self.ellipsoid 

437 return _Ef.fRG(E.a2, E.b2) / PI_4 

438 

439 @Property_RO 

440 def ellipsoid(self): 

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

442 ''' 

443 return self._E 

444 

445 @Property_RO 

446 def f(self): 

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

448 ''' 

449 return self.ellipsoid.f 

450 

451 flattening = f 

452 

453 @Property_RO 

454 def _fm1(self): # 1 - flattening 

455 return self.ellipsoid.f1 

456 

457 def _fromAux(self, Zeta, **name): 

458 '''Convert I{Auxiliary} to I{Geographic} latitude. 

459 

460 @arg Zeta: Auxiliary latitude (L{AuxAngle}). 

461 

462 @return: Geographic latitude, I{Phi} (L{AuxAngle}). 

463 ''' 

464 _xinstanceof(AuxAngle, Zeta=Zeta) 

465 aux = Zeta._AUX 

466 n = _xkwds_get(name, name=Zeta.name) 

467 f = self._fromAuxCase.get(aux, None) 

468 if f is None: 

469 Phi = AuxPhi(NAN, name=n) 

470 elif callable(f): 

471 t = self._fm1 

472 t *= f(t) 

473 Phi = _Newton(t, Zeta, self._toZeta(aux), name=n) 

474 else: # assert isscalar(f) 

475 y, x = Zeta._yx 

476 Phi = AuxPhi(y / f, x, name=n) 

477 # assert Phi._AUX == Aux.PHI 

478 return Phi 

479 

480 @Property_RO 

481 def _fromAuxCase(self): 

482 '''(INTERNAL) switch(auxin): ... 

483 ''' 

484 return {Aux.AUTHALIC: cbrt, 

485 Aux.CONFORMAL: _passarg, 

486 Aux.GEOCENTRIC: self._e2m1, 

487 Aux.GEOGRAPHIC: _1_0, 

488 Aux.PARAMETRIC: self._fm1, 

489 Aux.RECTIFYING: sqrt} 

490 

491 def Geocentric(self, Phi, **diff_name): 

492 '''Convert I{Geographic} to I{Geocentric} latitude. 

493 

494 @arg Phi: Geographic latitude (L{AuxAngle}). 

495 

496 @return: Geocentric latitude, C{Phi} (L{AuxAngle}). 

497 ''' 

498 _xinstanceof(AuxAngle, Phi=Phi) 

499 # assert Phi._AUX == Aux.PHI 

500 Theta = AuxTheta(Phi.y * self._e2m1, Phi.x, 

501 name=_xkwds_get(diff_name, name=Phi.name)) 

502 if _xkwds_get(diff_name, diff=False): 

503 Theta._diff = self._e2m1 

504 return Theta 

505 

506 def Geodetic(self, Phi, **diff_name): # PYCHOK no cover 

507 '''Convert I{Geographic} to I{Geodetic} latitude. 

508 

509 @arg Phi: Geographic latitude (L{AuxAngle}). 

510 

511 @return: Geodetic latitude, C{Phi} (L{AuxAngle}). 

512 ''' 

513 _xinstanceof(AuxAngle, Phi=Phi) 

514 # assert Phi._AUX == Aux.PHI 

515 return AuxPhi(Phi, name=_xkwds_get(diff_name, name=Phi.name)) 

516 

517 @Property_RO 

518 def _n(self): # 3rd flattening 

519 return self.ellipsoid.n 

520 

521 @Property_RO 

522 def _n2(self): 

523 return self._n**2 

524 

525 def Parametric(self, Phi, **diff_name): 

526 '''Convert I{Geographic} to I{Parametric} latitude. 

527 

528 @arg Phi: Geographic latitude (L{AuxAngle}). 

529 

530 @return: Parametric latitude, C{Beta} (L{AuxAngle}). 

531 ''' 

532 _xinstanceof(AuxAngle, Phi=Phi) 

533 # assert Phi._AUX == Aux.PHI 

534 Beta = AuxBeta(Phi.y * self._fm1, Phi.x, 

535 name=_xkwds_get(diff_name, name=Phi.name)) 

536 if _xkwds_get(diff_name, diff=False): 

537 Beta._diff = self._fm1 

538 return Beta 

539 

540 Reduced = Parametric 

541 

542 @Property_RO 

543 def _q(self): # constant _q 

544 q, f = self._e12p1, self.f 

545 if f: 

546 e = self._e 

547 q += _over(asinh(self._e1) if f > 0 else atan1(e), e) 

548 else: 

549 q += _1_0 

550 return q 

551 

552 def _qf(self, tphi): 

553 # function _q: atanh(e * sphi) / e + sphi / (1 - (e * sphi)^2) 

554 scb = self._scbeta(tphi) 

555 return self._atanhee(tphi) + (tphi / scb) * (_sc(tphi) / scb) 

556 

557 def _qIntegrand(self, beta): 

558 # pbeta(beta) = integrate(q(beta), beta), with beta in radians 

559 # q(beta) = (1-f) * (sin(xi) - sin(chi)) / cos(phi) 

560 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) * 

561 # (cos(xi) + cos(chi)) / (sin(xi) + sin(chi)) 

562 # Fit q(beta)/cos(beta) with Fourier transform 

563 # q(beta)/cos(beta) = sum(c[k] * sin((2*k+1)*beta), k, 0, K-1) 

564 # then the integral is 

565 # pbeta = sum(d[k] * cos((2*k+2)*beta), k, 0, K-1) 

566 # where 

567 # d[k] = -1/(4*(k+1)) * (c[k] + c[k+1]) for k in 0..K-2 

568 # d[K-1] = -1/(4*K) * c[K-1] 

569 Beta = AuxBeta.fromRadians(beta) 

570 if Beta.x: # and self._fm1: 

571 Ax, _cv = Aux, self.convert 

572 Phi = _cv(Ax.PHI, Beta, exact=True) 

573 schi, cchi = _cv(Ax.CHI, Phi, exact=True)._yx_normalized 

574 sxi, cxi = _cv(Ax.XI, Phi, exact=True)._yx_normalized 

575 r = (sxi - schi) if fabs(schi) < fabs(cchi) else \ 

576 _over(_2cos2x(cchi, cxi), (sxi + schi) * _2_0) 

577 r *= _over(self._fm1, Phi._x_normalized * Beta._x_normalized) 

578 else: # beta == PI_2, PI3_2, ... 

579 r = _0_0 # XXX 0 avoids NAN summation exceptions 

580 return r 

581 

582 def Rectifying(self, Phi, **diff_name): 

583 '''Convert I{Geographic} to I{Rectifying} latitude. 

584 

585 @arg Phi: Geographic latitude (L{AuxAngle}). 

586 

587 @return: Rectifying latitude, C{Mu} (L{AuxAngle}). 

588 ''' 

589 Beta = self.Parametric(Phi) 

590 # assert Beta._AUX == Aux.BETA 

591 sb, cb = map(fabs, Beta._yx_normalized) 

592 a, ka, ka1 = _1_0, self._e2, self._e2m1 

593 b, kb, kb1 = self._fm1, -self._e12, self._e12p1 

594 if self.f < 0: 

595 a, b = b, a 

596 ka, kb = kb, ka 

597 ka1, kb1 = kb1, ka1 

598 sb, cb = cb, sb 

599 # now a, b = larger/smaller semiaxis 

600 # Beta measured from larger semiaxis 

601 # kb, ka = modulus-squared for distance from Beta = 0, pi/2 

602 # NB kb <= 0; 0 <= ka <= 1 

603 # sa = b*E(Beta, sqrt(kb)) 

604 # sb = a*E(Beta',sqrt(ka)) 

605 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2 

606 sb2 = sb**2 

607 cb2 = cb**2 

608 da2 = ka1 + ka * sb2 

609 db2 = _1_0 - kb * sb2 

610 # DLMF Eq. 19.25.9 

611 my = b * sb * _Ef._RFRD(cb2, db2, _1_0, kb * sb2) 

612 # DLMF Eq. 19.25.10 with complementary angles 

613 mx = a * cb * (_Ef.fRF(sb2, da2, _1_0) * ka1 + 

614 ka * cb2 * _Ef.fRD(sb2, _1_0, da2, _3_0) * ka1 + 

615 ka * sb / sqrt(da2)) 

616 # my + mx = 2*_Ef.fRG(a*a, b*b) = a*E(e) = b*E(i*e') 

617 # mr = a*E(e)*(2/pi) = b*E(i*e')*(2/pi) 

618 if self.f < 0: 

619 a, b = b, a 

620 my, mx = mx, my 

621 mr = (my + mx) / PI_2 

622 if mr: 

623 my = sin(my / mr) 

624 mx = sin(mx / mr) # XXX zero? 

625 else: # zero Mu 

626 my, mx = _0_0, _1_0 

627 n = _xkwds_get(diff_name, name=Phi.name) 

628 Mu = AuxMu(my, mx, # normalized 

629 name=n).copyquadrant(Phi) 

630 

631 if _xkwds_get(diff_name, diff=False): 

632 d, x = _0_0, Beta._x_normalized 

633 if x and mr: 

634 if Mu.x and Phi.x and not isinf(Phi.tan): 

635 d = b / mr * (x / Mu.x)**2 \ 

636 * (x / Phi._x_normalized) 

637 else: 

638 d = mr / a 

639 Mu._diff = self._fm1 * d 

640 return Mu 

641 

642 def RectifyingRadius(self, exact=False): 

643 '''Get the I{Rectifying} radius. 

644 

645 @arg exact: If C{True}, use the exact expression, 

646 otherwise the I{Taylor} series. 

647 

648 @return: Rectifying radius (L{Meter}, same units 

649 as the ellipsoid axes). 

650 ''' 

651 r = self._Ef_fRG_a2b2_PI_4 if exact else self._RectifyingR 

652 return Meter(r, name=self.RectifyingRadius.__name__) 

653 

654 @Property_RO 

655 def _RectifyingR(self): 

656 m = self.ALorder 

657 d = _polynomial(self._n2, _RRCoeffs[m], 0, m // 2) 

658 return d * (self.a + self.b) * _0_5 

659 

660 def _scbeta(self, tphi): 

661 return _sc(self._fm1 * tphi) 

662 

663 def _toAux(self, auxout, Phi, **diff_name): 

664 '''Convert I{Geographic} to I{Auxiliary} latitude. 

665 

666 @arg auxout: I{Auxiliary} kind (C{Aux.KIND}). 

667 @arg Phi: Geographic latitude (L{AuxLat}). 

668 

669 @return: Auxiliary latitude, I{Eta} (L{AuxLat}). 

670 ''' 

671 _xinstanceof(AuxAngle, Phi=Phi) 

672 # assert Phi._AUX == Aux.PHI 

673 n = _xkwds_get(diff_name, name=Phi.name) 

674 m = _toAuxCase.get(auxout, None) 

675 if m: # callable 

676 A = m(self, Phi, **_xkwds(diff_name, name=n)) 

677 elif auxout == Aux.GEODETIC: # == GEOGRAPHIC 

678 A = AuxPhi(Phi, name=n) 

679 else: # auxout? 

680 A = AuxPhi(NAN, name=n) 

681 # assert A._AUX == auxout 

682 return A 

683 

684 def _toZeta(self, zetaux): 

685 '''Return a (lean) function to create C{AuxPhi(tphi)} and 

686 convert that into C{AuxAngle} of (fixed) kind C{zetaux} 

687 for use only inside the C{_Newton} loop. 

688 ''' 

689 class _AuxPhy(AuxPhi): 

690 # lean C{AuxPhi} instance. 

691 # _diff = _1_0 

692 # _x = _1_0 

693 

694 def __init__(self, tphi): # PYCHOK signature 

695 self._y = tphi 

696 

697 m = _toAuxCase.get(zetaux, None) 

698 if m: # callable 

699 

700 def _toZeta(tphi): 

701 return m(self, _AuxPhy(tphi), diff=True) 

702 

703 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC 

704 _toZeta = _AuxPhy 

705 

706 else: # zetaux? 

707 

708 def _toZeta(unused): # PYCHOK expected 

709 return _AuxPhy(NAN) 

710 

711 return _toZeta 

712 

713 

714# switch(auxout): ... 

715_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic, 

716 Aux.CONFORMAL: AuxLat.Conformal, 

717 Aux.GEOCENTRIC: AuxLat.Geocentric, 

718 Aux.PARAMETRIC: AuxLat.Parametric, 

719 Aux.RECTIFYING: AuxLat.Rectifying} 

720 

721 

722def _Clenshaw(sinp, Zeta, cs, K): 

723 sz, cz = Zeta._yx # isnormal 

724 # Evaluate sum(c[k] * sin((2*k+2) * Zeta)) if sinp else 

725 # sum(c[k] * cos((2*k+2) * Zeta)) 

726 x = _2cos2x(cz, sz) # 2 * cos(2*Zeta) 

727 if isfinite(x): 

728 U0, U1 = Fsum(), Fsum() 

729 # assert len(cs) == K 

730 for r in _reverange(K): 

731 U1 -= U0 * x + cs[r] 

732 U1, U0 = U0, -U1 

733 # u0*f0(Zeta) - u1*fm1(Zeta) 

734 # f0 = sin(2*Zeta) if sinp else cos(2*Zeta) 

735 # fm1 = 0 if sinp else 1 

736 if sinp: 

737 U0 *= sz * cz * _2_0 

738 else: 

739 U0 *= x * _0_5 

740 U0 -= U1 

741 x = float(U0) 

742 return x 

743 

744 

745def _CXcoeffs(aL): # PYCHOK in .auxilats.__main__ 

746 '''(INTERNAL) Get the C{CX_4}, C{_6} or C{_8} coefficients. 

747 ''' 

748 try: # from pygeodesy.auxilats._CX_x import _coeffs_x as _coeffs 

749 _CX_x = _DOT_(_MODS.auxilats.__name__, _UNDER_('_CX', aL)) 

750 _coeffs = _MODS.getattr(_CX_x, _UNDER_('_coeffs', aL)) 

751 except (AttributeError, ImportError, KeyError, TypeError) as x: 

752 raise AuxError(ALorder=aL, cause=x) 

753 # assert _coeffs.ALorder == aL 

754 # assert _coeffs.n == Aux.len(aL) 

755 return _coeffs 

756 

757 

758def _Newton(tphi, Zeta, _toZeta, **name): 

759 # Newton's method fro AuxLat._fromAux 

760 try: 

761 _lg2 = _log2 

762 _abs = fabs 

763 tz = _abs(Zeta.tan) 

764 tphi = tz / tphi # **) 

765 ltz = _lg2(tz) # **) 

766 ltphi = _lg2(tphi) # **) 

767 ltmin = min(ltphi, MIN_EXP) 

768 ltmax = max(ltphi, MAX_EXP) 

769# auxin = Zeta._AUX 

770 s, n, __2 = 0, 3, _0_5 # n = i + 2 

771 _TOL, _xp2 = _EPSqrt, _exp2 

772 for i in range(1, _TRIPS): # up to 1 Ki! 

773 # _toAux(auxin, AuxPhi(tphi), diff=True) 

774 Z = _toZeta(tphi) 

775 # assert Z._AUX == auxin 

776 t, s_ = Z.tan, s 

777 if t > tz: 

778 ltmax, s = ltphi, +1 

779 elif t < tz: 

780 ltmin, s = ltphi, -1 

781 else: 

782 break 

783 # derivative dtan(Z)/dtan(Phi) 

784 # to dlog(tan(Z))/dlog(tan(Phi)) 

785 d = (ltz - _lg2(t)) * t # **) 

786 if d: 

787 d = d / (Z.diff * tphi) # **) 

788 ltphi += d 

789 tphi = _xp2(ltphi) 

790 if _abs(d) < _TOL: 

791 i += 1 

792 # _toAux(auxin, AuxPhi(tphi), diff=True) 

793 Z = _toZeta(tphi) 

794 tphi -= _over(Z.tan - tz, Z.diff) 

795 break 

796 if (i > n and (s * s_) < 0) or not ltmin < ltphi < ltmax: 

797 s, n = 0, (i + 2) 

798 ltphi = (ltmin + ltmax) * __2 

799 tphi = _xp2(ltphi) 

800 else: 

801 i = _TRIPS 

802 Phi = AuxPhi(tphi, **name).copyquadrant(Zeta) 

803 Phi._iter = i 

804 except (ValueError, ZeroDivisionError): # **) zero t, tphi, tz or Z.diff 

805 Phi = AuxPhi(Zeta, **name) # diff as-as 

806 Phi._iter = 0 

807 # assert Phi._AUX == Aux.PHI 

808 return Phi 

809 

810 

811_f, _u = float, _Ufloats() 

812_1__f3 = -1 / _f(3) # XXX +1 / _f(3) 

813_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3), 

814 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105), 

815 4 / _f(15), _1__f3), 

816 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693), 

817 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)} 

818_RRCoeffs = {4: _u(1 / _f(64), _0_25), 

819 6: _u(1 / _f(256), 1 / _f(64), _0_25), 

820 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used! 

821del _f, _u, _Ufloats, _1__f3 

822# assert set(_AR2Coeffs.keys()) == set(_RRCoeffs.keys()) 

823 

824# AuxLat._Lmax = max(_AR2Coeffs.keys()) # == max(ALorder) 

825 

826__all__ += _ALL_DOCS(AuxLat) 

827 

828# **) MIT License 

829# 

830# Copyright (C) 2023-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

831# 

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

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

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

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

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

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

838# 

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

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

841# 

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

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

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

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

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

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

848# OTHER DEALINGS IN THE SOFTWARE.