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 _reverange, _xinstanceof, _passarg 

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 

32# from pygeodesy.internals import _passarg # from .basics 

33from pygeodesy.interns import NN, _DOT_, _UNDER_ # _earth_ 

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

35from pygeodesy.props import Property, Property_RO, _update_all 

36from pygeodesy.units import _isDegrees, _isRadius, Degrees, Meter 

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

38 

39from math import asinh, atan2, copysign, cosh, fabs, sin, sinh, sqrt 

40try: 

41 from math import exp2 as _exp2 

42except ImportError: # Python 3.11- 

43 

44 def _exp2(x): 

45 return pow(_2_0, x) 

46 

47__all__ = () 

48__version__ = '24.04.14' 

49 

50_TRIPS = 1024 # XXX 2 or 3? 

51 

52 

53class AuxLat(AuxAngle): 

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

55 on an ellipsoid. 

56 

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

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

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

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

61 

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

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

64 ''' 

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

66 _E = _WGS84.ellipsoid 

67# _Lmax = 0 # overwritten below 

68 _mAL = 6 # 4, 6 or 8 aka Lmax 

69 

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

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

72 

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

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

75 L{Ellipsoid2} or L{a_f2Tuple}). 

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

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

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

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

80 is not scalar. 

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

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

83 C{ALorder}. 

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

85 ''' 

86 if a_earth is not _WGS84: 

87 n = name or AuxLat.__name__ 

88 try: 

89 if b is f is None: 

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

91 elif _isRadius(a_earth): 

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

93 else: 

94 raise ValueError() 

95 except Exception as x: 

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

97 self._E = E 

98 

99 if name: 

100 self.name = name 

101 if ALorder: 

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

103 

104 @Property_RO 

105 def a(self): 

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

107 ''' 

108 return self.ellipsoid.a 

109 

110 equatoradius = a 

111 

112 @Property 

113 def ALorder(self): 

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

115 ''' 

116 return self._mAL 

117 

118 @ALorder.setter # PYCHOK setter! 

119 def ALorder(self, order): 

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

121 ''' 

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

123 if self._mAL != m: 

124 _update_all(self) 

125 self._mAL = m 

126 

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

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

129 f = self.f 

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

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

132 e = self._e 

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

134 return s 

135 

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

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

138 

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

140 

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

142 ''' 

143 _xinstanceof(AuxAngle, Phi=Phi) 

144 # assert Phi._AUX == Aux.PHI 

145 tphi = fabs(Phi.tan) 

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

147 y, x = Phi._yx_normalized 

148 q = self._q 

149 qv = self._qf(tphi) 

150 Dq2 = self._Dq(tphi) 

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

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

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

154 

155 if _xkwds_get(diff_name, diff=False): 

156 if isnan(tphi): 

157 d = self._e2m1_sq2 

158 else: 

159 c = self.Parametric(Phi)._x_normalized 

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

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

162 Xi._diff = d 

163 else: 

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

165 # assert Xi._AUX == Aux.XI 

166 return Xi 

167 

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

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

170 

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

172 the I{Taylor} series. 

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

174 

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

176 units as the ellipsoid axes). 

177 

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

179 exceeds C{f_max}. 

180 ''' 

181 f = self.f 

182 if exact or not f: 

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

184 elif fabs(f) < f_max: 

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

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

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

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

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

190 m = self.ALorder 

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

192 c2 *= self.a * (self.a + self.b) 

193 else: 

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

195 return c2 * _0_5 

196 

197 @Property_RO 

198 def b(self): 

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

200 ''' 

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

202 

203 polaradius = b 

204 

205 def _coeffs(self, auxout, auxin): 

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

207 aL = self.ALorder # aka Lmax 

208 if auxout == auxin: 

209 return _0_0s(aL) # uncached 

210 

211 k = Aux._1d(auxout, auxin) 

212 try: # cached 

213 return AuxLat._csc[aL][k] 

214 except KeyError: 

215 pass 

216 

217 Cx = _CXcoeffs(aL) 

218 try: 

219 Cx = Cx[auxout][auxin] 

220 except KeyError as x: 

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

222 

223 d = x = n = self._n 

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

225 x = self._n2 

226 

227 def _m(aL): 

228 for m in _reverange(aL): 

229 yield m // 2 

230 else: 

231 _m = _reverange # PYCHOK expected 

232 

233 i = 0 

234 cs = [] 

235 _c = cs.append 

236 _p = _polynomial 

237 for m in _m(aL): 

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

239 _c(_p(x, Cx, i, j) * d) 

240 d *= n 

241 i = j 

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

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

244 return cs 

245 

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

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

248 

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

250 

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

252 ''' 

253 _xinstanceof(AuxAngle, Phi=Phi) 

254 # assert Phi._AUX == Aux.PHI 

255 tphi = tchi = fabs(Phi.tan) 

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

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

258 scsig = _sc(sig) 

259 scphi = _sc(tphi) 

260 if self.f > 0: 

261 # The general expression for tchi is 

262 # tphi * scsig - sig * scphi 

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

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

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

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

267 sigtphi = sig / tphi 

268 if sig < (tphi * _0_5): 

269 t = tphi - sig 

270 else: 

271 def _asinh_2(x): 

272 return asinh(x) * _0_5 

273 # Still possibly dangerous cancellation in tphi - sig. 

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

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

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

277 # Note sinh(atanh(sphi)) = tphi 

278 # Turn the crank on divided differences, substitute 

279 # sphi = tphi / _sc(tphi) 

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

281 e = self._e 

282 em1 = self._e2m1 / (_1_0 + e) 

283 # assert em1 != 0 

284 scb = self._scbeta(tphi) 

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

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

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

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

289 try: 

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

291 except ValueError: # Fsum(NAN) exception 

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

293 e *= atphib 

294 t = atphi - e 

295 if t: # sinh(0) == 0 

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

297 t = float(Dg) # tphi - sig 

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

299 scsig + scphi * sigtphi) if t else _0_0 

300 else: 

301 tchi = tphi * scsig - sig * scphi 

302 

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

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

305 

306 if _xkwds_get(diff_name, diff=False): 

307 if isinf(tphi): # PYCHOK np cover 

308 d = self._conformal_diff 

309 else: 

310 d = self.Parametric(Phi)._x_normalized 

311 if d: 

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

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

314 Chi._diff = d 

315 # assrt Chi._AUX == Aux.CHI 

316 return Chi 

317 

318 @Property_RO 

319 def _conformal_diff(self): # PYCHOK no cover 

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

321 ''' 

322 e = self._e 

323 if self.f > 0: 

324 ss = sinh(asinh(self._e1) * e) 

325 d = _over(_1_0, _sc(ss) + ss) 

326 elif e: 

327 ss = sinh(-atan1(e) * e) 

328 d = _sc(ss) - ss 

329 else: 

330 d = _1_0 

331 return d 

332 

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

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

335 Z = d = Zeta_d 

336 if isinstance(Z, AuxAngle): 

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

338 if auxin == auxout: 

339 pass 

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

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

342 elif exact: 

343 p = Aux.power(auxout, auxin) 

344 if p is None: 

345 P = self._fromAux(Z) # Phi 

346 Z = self._toAux(auxout, P) 

347 Z._iter = P.iteration 

348 else: 

349 y, x = Z._yx 

350 if p: 

351 y *= pow(self._fm1, p) 

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

353 else: 

354 cs = self._coeffs(auxout, auxin) 

355 yx = Z._yx_normalized 

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

357 # assert Z._yx == yx 

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

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

360 # assert Z._AUX == auxout 

361 return Z 

362 

363 elif _isDegrees(d): 

364 Z = AuxPhi.fromDegrees(d) 

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

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

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

368 

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

370 

371 def _Dq(self, tphi): 

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

373 sphi = _sn(tphi) 

374 if tphi > 0: 

375 scphi = _sc(tphi) 

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

377 if d: 

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

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

380 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1) 

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

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

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

384 if e2m1 and ed: 

385 e2 = self._e2 

386 if e2 > 0: # assert self.f > 0 

387 scb = self._scbeta(tphi) 

388 q = scphib = scphi / scb 

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

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

391 else: 

392 s2 = sphi * e2 

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

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

395 else: # PYCHOK no cover 

396 q = INF 

397 else: # PYCHOK no cover 

398 q = self._2_e2m12 

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

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

401 return q 

402 

403 @Property_RO 

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

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

406 

407 @Property_RO 

408 def _e1(self): 

409 return sqrt(fabs(self._e12)) 

410 

411 @Property_RO 

412 def _e12(self): 

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

414 

415 @Property_RO 

416 def _e12p1(self): 

417 return _1_0 / self._e2m1 

418 

419 @Property_RO 

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

421 return self.ellipsoid.e2 

422 

423 @Property_RO 

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

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

426 

427 @Property_RO 

428 def _e2m1_sq2(self): 

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

430 

431 @Property_RO 

432 def _2_e2m12(self): 

433 return _2_0 / self._e2m1**2 

434 

435 @Property_RO 

436 def _Ef_fRG_a2b2_PI_4(self): 

437 E = self.ellipsoid 

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

439 

440 @Property_RO 

441 def ellipsoid(self): 

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

443 ''' 

444 return self._E 

445 

446 @Property_RO 

447 def f(self): 

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

449 ''' 

450 return self.ellipsoid.f 

451 

452 flattening = f 

453 

454 @Property_RO 

455 def _fm1(self): # 1 - flattening 

456 return self.ellipsoid.f1 

457 

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

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

460 

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

462 

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

464 ''' 

465 _xinstanceof(AuxAngle, Zeta=Zeta) 

466 aux = Zeta._AUX 

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

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

469 if f is None: 

470 Phi = AuxPhi(NAN, name=n) 

471 elif callable(f): 

472 t = self._fm1 

473 t *= f(t) 

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

475 else: # assert isscalar(f) 

476 y, x = Zeta._yx 

477 Phi = AuxPhi(y / f, x, name=n) 

478 # assert Phi._AUX == Aux.PHI 

479 return Phi 

480 

481 @Property_RO 

482 def _fromAuxCase(self): 

483 '''(INTERNAL) switch(auxin): ... 

484 ''' 

485 return {Aux.AUTHALIC: cbrt, 

486 Aux.CONFORMAL: _passarg, 

487 Aux.GEOCENTRIC: self._e2m1, 

488 Aux.GEOGRAPHIC: _1_0, 

489 Aux.PARAMETRIC: self._fm1, 

490 Aux.RECTIFYING: sqrt} 

491 

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

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

494 

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

496 

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

498 ''' 

499 _xinstanceof(AuxAngle, Phi=Phi) 

500 # assert Phi._AUX == Aux.PHI 

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

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

503 if _xkwds_get(diff_name, diff=False): 

504 Theta._diff = self._e2m1 

505 return Theta 

506 

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

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

509 

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

511 

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

513 ''' 

514 _xinstanceof(AuxAngle, Phi=Phi) 

515 # assert Phi._AUX == Aux.PHI 

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

517 

518 @Property_RO 

519 def _n(self): # 3rd flattening 

520 return self.ellipsoid.n 

521 

522 @Property_RO 

523 def _n2(self): 

524 return self._n**2 

525 

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

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

528 

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

530 

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

532 ''' 

533 _xinstanceof(AuxAngle, Phi=Phi) 

534 # assert Phi._AUX == Aux.PHI 

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

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

537 if _xkwds_get(diff_name, diff=False): 

538 Beta._diff = self._fm1 

539 return Beta 

540 

541 Reduced = Parametric 

542 

543 @Property_RO 

544 def _q(self): # constant _q 

545 q, f = self._e12p1, self.f 

546 if f: 

547 e = self._e 

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

549 else: 

550 q += _1_0 

551 return q 

552 

553 def _qf(self, tphi): 

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

555 scb = self._scbeta(tphi) 

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

557 

558 def _qIntegrand(self, beta): 

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

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

561 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) * 

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

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

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

565 # then the integral is 

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

567 # where 

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

569 # d[K-1] = -1/(4*K) * c[K-1] 

570 Beta = AuxBeta.fromRadians(beta) 

571 if Beta.x: # and self._fm1: 

572 Ax, _cv = Aux, self.convert 

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

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

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

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

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

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

579 else: # beta == PI_2, PI3_2, ... 

580 r = _0_0 # XXX 0 avoids NAN summation exceptions 

581 return r 

582 

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

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

585 

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

587 

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

589 ''' 

590 Beta = self.Parametric(Phi) 

591 # assert Beta._AUX == Aux.BETA 

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

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

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

595 if self.f < 0: 

596 a, b = b, a 

597 ka, kb = kb, ka 

598 ka1, kb1 = kb1, ka1 

599 sb, cb = cb, sb 

600 # now a, b = larger/smaller semiaxis 

601 # Beta measured from larger semiaxis 

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

603 # NB kb <= 0; 0 <= ka <= 1 

604 # sa = b*E(Beta, sqrt(kb)) 

605 # sb = a*E(Beta',sqrt(ka)) 

606 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2 

607 sb2 = sb**2 

608 cb2 = cb**2 

609 da2 = ka1 + ka * sb2 

610 db2 = _1_0 - kb * sb2 

611 # DLMF Eq. 19.25.9 

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

613 # DLMF Eq. 19.25.10 with complementary angles 

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

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

616 ka * sb / sqrt(da2)) 

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

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

619 if self.f < 0: 

620 a, b = b, a 

621 my, mx = mx, my 

622 mr = (my + mx) / PI_2 

623 if mr: 

624 my = sin(my / mr) 

625 mx = sin(mx / mr) # XXX zero? 

626 else: # zero Mu 

627 my, mx = _0_0, _1_0 

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

629 Mu = AuxMu(my, mx, # normalized 

630 name=n).copyquadrant(Phi) 

631 

632 if _xkwds_get(diff_name, diff=False): 

633 d, x = _0_0, Beta._x_normalized 

634 if x and mr: 

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

636 d = b / mr * (x / Mu.x)**2 \ 

637 * (x / Phi._x_normalized) 

638 else: 

639 d = mr / a 

640 Mu._diff = self._fm1 * d 

641 return Mu 

642 

643 def RectifyingRadius(self, exact=False): 

644 '''Get the I{Rectifying} radius. 

645 

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

647 otherwise the I{Taylor} series. 

648 

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

650 as the ellipsoid axes). 

651 ''' 

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

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

654 

655 @Property_RO 

656 def _RectifyingR(self): 

657 m = self.ALorder 

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

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

660 

661 def _scbeta(self, tphi): 

662 return _sc(self._fm1 * tphi) 

663 

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

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

666 

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

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

669 

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

671 ''' 

672 _xinstanceof(AuxAngle, Phi=Phi) 

673 # assert Phi._AUX == Aux.PHI 

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

675 m = _toAuxCase.get(auxout, None) 

676 if m: # callable 

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

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

679 A = AuxPhi(Phi, name=n) 

680 else: # auxout? 

681 A = AuxPhi(NAN, name=n) 

682 # assert A._AUX == auxout 

683 return A 

684 

685 def _toZeta(self, zetaux): 

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

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

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

689 ''' 

690 class _AuxPhy(AuxPhi): 

691 # lean C{AuxPhi} instance. 

692 # _diff = _1_0 

693 # _x = _1_0 

694 

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

696 self._y = tphi 

697 

698 m = _toAuxCase.get(zetaux, None) 

699 if m: # callable 

700 

701 def _toZeta(tphi): 

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

703 

704 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC 

705 _toZeta = _AuxPhy 

706 

707 else: # zetaux? 

708 

709 def _toZeta(unused): # PYCHOK expected 

710 return _AuxPhy(NAN) 

711 

712 return _toZeta 

713 

714 

715# switch(auxout): ... 

716_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic, 

717 Aux.CONFORMAL: AuxLat.Conformal, 

718 Aux.GEOCENTRIC: AuxLat.Geocentric, 

719 Aux.PARAMETRIC: AuxLat.Parametric, 

720 Aux.RECTIFYING: AuxLat.Rectifying} 

721 

722 

723def _Clenshaw(sinp, Zeta, cs, K): 

724 sz, cz = Zeta._yx # isnormal 

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

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

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

728 if isfinite(x): 

729 U0, U1 = Fsum(), Fsum() 

730 # assert len(cs) == K 

731 for r in _reverange(K): 

732 U1 -= U0 * x + cs[r] 

733 U1, U0 = U0, -U1 

734 # u0*f0(Zeta) - u1*fm1(Zeta) 

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

736 # fm1 = 0 if sinp else 1 

737 if sinp: 

738 U0 *= sz * cz * _2_0 

739 else: 

740 U0 *= x * _0_5 

741 U0 -= U1 

742 x = float(U0) 

743 return x 

744 

745 

746def _CXcoeffs(aL): # PYCHOK in .auxilats.__main__ 

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

748 ''' 

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

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

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

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

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

754 # assert _coeffs.ALorder == aL 

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

756 return _coeffs 

757 

758 

759def _Newton(tphi, Zeta, _toZeta, **name): 

760 # Newton's method fro AuxLat._fromAux 

761 try: 

762 _lg2 = _log2 

763 _abs = fabs 

764 tz = _abs(Zeta.tan) 

765 tphi = tz / tphi # **) 

766 ltz = _lg2(tz) # **) 

767 ltphi = _lg2(tphi) # **) 

768 ltmin = min(ltphi, MIN_EXP) 

769 ltmax = max(ltphi, MAX_EXP) 

770# auxin = Zeta._AUX 

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

772 _TOL, _xp2 = _EPSqrt, _exp2 

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

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

775 Z = _toZeta(tphi) 

776 # assert Z._AUX == auxin 

777 t, s_ = Z.tan, s 

778 if t > tz: 

779 ltmax, s = ltphi, +1 

780 elif t < tz: 

781 ltmin, s = ltphi, -1 

782 else: 

783 break 

784 # derivative dtan(Z)/dtan(Phi) 

785 # to dlog(tan(Z))/dlog(tan(Phi)) 

786 d = (ltz - _lg2(t)) * t # **) 

787 if d: 

788 d = d / (Z.diff * tphi) # **) 

789 ltphi += d 

790 tphi = _xp2(ltphi) 

791 if _abs(d) < _TOL: 

792 i += 1 

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

794 Z = _toZeta(tphi) 

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

796 break 

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

798 s, n = 0, (i + 2) 

799 ltphi = (ltmin + ltmax) * __2 

800 tphi = _xp2(ltphi) 

801 else: 

802 i = _TRIPS 

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

804 Phi._iter = i 

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

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

807 Phi._iter = 0 

808 # assert Phi._AUX == Aux.PHI 

809 return Phi 

810 

811 

812_f, _u = float, _Ufloats() 

813_1__f3 = -1 / _f(3) # XXX +1 / _f(3) 

814_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3), 

815 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105), 

816 4 / _f(15), _1__f3), 

817 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693), 

818 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)} 

819_RRCoeffs = {4: _u(1 / _f(64), _0_25), 

820 6: _u(1 / _f(256), 1 / _f(64), _0_25), 

821 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used! 

822del _f, _u, _Ufloats, _1__f3 

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

824 

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

826 

827__all__ += _ALL_DOCS(AuxLat) 

828 

829# **) MIT License 

830# 

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

832# 

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

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

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

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

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

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

839# 

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

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

842# 

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

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

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

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

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

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

849# OTHER DEALINGS IN THE SOFTWARE.