Coverage for pygeodesy/auxilats/auxLat.py: 90%

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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:Charles@Karney.com>} (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, _2cos2x, _sc, _sn, _Ufloats 

21from pygeodesy.basics import isscalar, _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 

29from pygeodesy.fmath import cbrt, Fsum 

30# from pygeodesy.fsums import Fsum # from .fmath 

31from pygeodesy.interns import NN, _DOT_, _UNDER_ # _earth_ 

32from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS 

33from pygeodesy.props import Property, Property_RO, _update_all 

34from pygeodesy.units import Degrees, Meter 

35 

36from math import asinh, atan, atan2, copysign, cosh, fabs, sin, sinh, sqrt 

37try: 

38 from math import exp2 as _exp2 

39except ImportError: # Python 3.11- 

40 

41 def _exp2(x): 

42 return pow(_2_0, x) 

43 

44__all__ = () 

45__version__ = '23.08.09' 

46 

47_TRIPS = 1024 # XXX 2 or 3? 

48 

49 

50def _passarg(arg): # PYCHOK to .utily 

51 return arg 

52 

53 

54class AuxLat(AuxAngle): 

55 '''Accurate conversion between I{Auxiliary} latitudes on an ellipsoid. 

56 

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

58 maintain precision close to 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 _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 isscalar(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 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 = (atan(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 = _MODS.karney._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 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 k = Aux._1d(auxout, auxin) 

208 try: 

209 cs = AuxLat._csc[aL][k] 

210 except KeyError: 

211 if auxout == auxin: 

212 cs = _0_0s(aL) # not cached 

213 else: 

214 Cx = _CXcoeffs(aL) 

215 try: 

216 Cx = Cx[auxout][auxin] 

217 except KeyError as x: 

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

219 

220 d = x = n = self._n 

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

222 x = self._n2 

223 

224 def _m(m): 

225 return m // 2 

226 else: 

227 def _m(m): # PYCHOK expected 

228 return m 

229 

230 i = 0 

231 cs = [] 

232 _p = _MODS.karney._polynomial 

233 for r in _reverange(aL): 

234 j = i + _m(r) + 1 # order m = j - i - 1 

235 cs.append(_p(x, Cx, i, j) * d) 

236 d *= n 

237 i = j 

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

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

240 return cs 

241 

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

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

244 

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

246 

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

248 ''' 

249 _xinstanceof(AuxAngle, Phi=Phi) 

250 # assert Phi._AUX == Aux.PHI 

251 tphi = tchi = fabs(Phi.tan) 

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

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

254 scsig = _sc(sig) 

255 scphi = _sc(tphi) 

256 if self.f > 0: 

257 # The general expression for tchi is 

258 # tphi * scsig - sig * scphi 

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

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

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

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

263 sigtphi = sig / tphi 

264 if sig < (tphi * _0_5): 

265 tphimsig = tphi - sig 

266 else: 

267 def _asinh_2(x): 

268 return asinh(x) * _0_5 

269 # Still possibly dangerous cancellation in tphi - sig. 

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

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

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

273 # Note sinh(atanh(sphi)) = tphi 

274 # Turn the crank on divided differences, substitute 

275 # sphi = tphi/_sc(tphi) 

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

277 e = self._e 

278 em1 = self._e2m1 / (_1_0 + e) 

279 # assert em1 != 0 

280 scb = _sc(self._fm1 * tphi) # sec(beta) 

281 scphib = _sc(tphi) / scb # sec(phi) / sec(beta) 

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

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

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

285 try: 

286 Dg = Fsum(atphi, atphib, t, e * t) 

287 except ValueError: # Fsum(NAN) exception 

288 Dg = sum((atphi, atphib, t, e * t)) 

289 e *= atphib 

290 t = atphi - e 

291 Dg *= cosh(atphi + e) * _over(sinh(t), t) 

292 tphimsig = float(Dg) * em1 # tphi - sig 

293 tchi = _over(tphimsig * (_1_0 + sigtphi), 

294 scsig + scphi * sigtphi) 

295 else: 

296 tchi = tphi * scsig - sig * scphi 

297 

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

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

300 

301 if _xkwds_get(diff_name, diff=False): 

302 if isinf(tphi): # PYCHOK np cover 

303 d = self._conformal_diff 

304 else: 

305 x = self.Parametric(Phi)._x_normalized 

306 d = (self._e2m1 * _over(x, Chi._x_normalized) 

307 * _over(x, Phi._x_normalized)) if x else _0_0 

308 Chi._diff = d 

309 # assrt Chi._AUX == Aux.CHI 

310 return Chi 

311 

312 @Property_RO 

313 def _conformal_diff(self): 

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

315 ''' 

316 e = self._e 

317 if self.f > 0: 

318 ss = sinh(e * asinh(self._e1)) 

319 d = _over(_1_0, _sc(ss) + ss) 

320 else: 

321 ss = sinh(-atan(e) * e) 

322 d = _sc(ss) - ss 

323 return d 

324 

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

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

327 Z = d = Zeta_d 

328 if isinstance(Z, AuxAngle): 

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

330 if auxin == auxout: 

331 pass 

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

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

334 elif exact: 

335 p = Aux.power(auxout, auxin) 

336 if p is None: 

337 P = self._fromAux(Z) # Phi 

338 Z = self._toAux(auxout, P) 

339 Z._iter = P.iteration 

340 else: 

341 y, x = Z._yx 

342 if p: 

343 y *= pow(self._fm1, p) 

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

345 else: 

346 cs = self._coeffs(auxout, auxin) 

347 yx = Z._yx_normalized 

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

349 # assert Z._yx == yx 

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

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

352 # assert Z._AUX == auxout 

353 return Z 

354 

355 elif isinstance(d, Degrees) or isscalar(d): 

356 Z = AuxPhi.fromDegrees(d) 

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

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

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

360 

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

362 

363 def _Dq(self, tphi): 

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

365 sphi = _sn(tphi) 

366 if tphi > 0: 

367 scphi = _sc(tphi) 

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

369 if d: 

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

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

372 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1); 

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

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

375 e2m1, ed = self._e2m1, self._e * d 

376 if e2m1 and ed: 

377 e2 = self._e2 

378 if e2 > 0: # assert self.f > 0 

379 scb = _sc(self._fm1 * tphi) 

380 q = scphi / scb 

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

382 q += asinh(self._e1 * d * scphi / scb) / ed 

383 else: 

384 s2 = sphi * e2 

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

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

387 else: 

388 q = INF 

389 else: 

390 q = self._2_e2m12 

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

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

393 return q 

394 

395 @Property_RO 

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

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

398 

399 @Property_RO 

400 def _e1(self): 

401 return sqrt(fabs(self._e12)) 

402 

403 @Property_RO 

404 def _e12(self): 

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

406 

407 @Property_RO 

408 def _e12p1(self): 

409 return _1_0 / self._e2m1 

410 

411 @Property_RO 

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

413 return self.ellipsoid.e2 

414 

415 @Property_RO 

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

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

418 

419 @Property_RO 

420 def _e2m1_sq2(self): 

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

422 

423 @Property_RO 

424 def _2_e2m12(self): 

425 return _2_0 / self._e2m1**2 

426 

427 @Property_RO 

428 def _Ef_fRG_a2b2_PI_4(self): 

429 E = self.ellipsoid 

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

431 

432 @Property_RO 

433 def ellipsoid(self): 

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

435 ''' 

436 return self._E 

437 

438 @Property_RO 

439 def f(self): 

440 '''Get the flattening (C{float}). 

441 ''' 

442 return self.ellipsoid.f 

443 

444 flattening = f 

445 

446 @Property_RO 

447 def _fm1(self): # 1 - flattening 

448 return self.ellipsoid.f1 

449 

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

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

452 

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

454 

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

456 ''' 

457 _xinstanceof(AuxAngle, Zeta=Zeta) 

458 aux = Zeta._AUX 

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

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

461 if f is None: 

462 Phi = AuxPhi(NAN, name=n) 

463 elif callable(f): 

464 t = self._fm1 

465 t *= f(t) 

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

467 else: # assert isscalar(f) 

468 y, x = Zeta._yx 

469 Phi = AuxPhi(y / f, x, name=n) 

470 # assert Phi._AUX == Aux.PHI 

471 return Phi 

472 

473 @Property_RO 

474 def _fromAuxCase(self): 

475 '''(INTERNAL) switch(auxin): ... 

476 ''' 

477 return {Aux.AUTHALIC: cbrt, 

478 Aux.CONFORMAL: _passarg, 

479 Aux.GEOCENTRIC: self._e2m1, 

480 Aux.GEOGRAPHIC: _1_0, 

481 Aux.PARAMETRIC: self._fm1, 

482 Aux.RECTIFYING: sqrt} 

483 

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

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

486 

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

488 

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

490 ''' 

491 _xinstanceof(AuxAngle, Phi=Phi) 

492 # assert Phi._AUX == Aux.PHI 

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

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

495 if _xkwds_get(diff_name, diff=False): 

496 Theta._diff = self._e2m1 

497 return Theta 

498 

499 def Geodetic(self, Phi, **diff_name): 

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

501 

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

503 

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

505 ''' 

506 _xinstanceof(AuxAngle, Phi=Phi) 

507 # assert Phi._AUX == Aux.PHI 

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

509 

510 @Property_RO 

511 def _n(self): # 3rd flattening 

512 return self.ellipsoid.n 

513 

514 @Property_RO 

515 def _n2(self): 

516 return self._n**2 

517 

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

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

520 

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

522 

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

524 ''' 

525 _xinstanceof(AuxAngle, Phi=Phi) 

526 # assert Phi._AUX == Aux.PHI 

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

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

529 if _xkwds_get(diff_name, diff=False): 

530 Beta._diff = self._fm1 

531 return Beta 

532 

533 Reduced = Parametric 

534 

535 @Property_RO 

536 def _q(self): # constant _q 

537 q, f = self._e12p1, self.f 

538 if f: 

539 e = self._e 

540 q += (asinh(self._e1) if f > 0 else atan(e)) / e 

541 else: 

542 q += _1_0 

543 return q 

544 

545 def _qf(self, tphi): 

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

547 scb = _sc(self._fm1 * tphi) 

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

549 

550 def _qIntegrand(self, beta): 

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

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

553 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) * 

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

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

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

557 # then the integral is 

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

559 # where 

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

561 # d[K-1] = -1/(4*K) * c[K-1] 

562 Beta = AuxBeta.fromRadians(beta) 

563 if Beta.x: # and self._fm1: 

564 Ax, _cv = Aux, self.convert 

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

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

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

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

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

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

571 else: # beta == PI_2, PI3_2, ... 

572 r = _0_0 # XXX 0 avoids NAN summation exceptions 

573 return r 

574 

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

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

577 

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

579 

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

581 ''' 

582 Beta = self.Parametric(Phi) 

583 # assert Beta._AUX == Aux.BETA 

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

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

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

587 if self.f < 0: 

588 a, b = b, a 

589 ka, kb = kb, ka 

590 ka1, kb1 = kb1, ka1 

591 sb, cb = cb, sb 

592 # now a, b = larger/smaller semiaxis 

593 # Beta measured from larger semiaxis 

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

595 # NB kb <= 0; 0 <= ka <= 1 

596 # sa = b*E(Beta, sqrt(kb)) 

597 # sb = a*E(Beta',sqrt(ka)) 

598 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2 

599 sb2 = sb**2 

600 cb2 = cb**2 

601 da2 = ka1 + ka * sb2 

602 db2 = _1_0 - kb * sb2 

603 # DLMF Eq. 19.25.9 

604 my = b * sb * (_Ef.fRF(cb2, db2, _1_0) - 

605 kb * sb2 * _Ef.fRD(cb2, db2, _1_0, _3_0)) 

606 # DLMF Eq. 19.25.10 with complementary angles 

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

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

609 ka * sb / sqrt(da2)) 

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

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

612 if self.f < 0: 

613 a, b = b, a 

614 my, mx = mx, my 

615 mr = (my + mx) / PI_2 

616 if mr: 

617 my = sin(my / mr) 

618 mx = sin(mx / mr) # XXX zero? 

619 else: # zero Mu 

620 my, mx = _0_0, _1_0 

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

622 Mu = AuxMu(my, mx, # normalized 

623 name=n).copyquadrant(Phi) 

624 

625 if _xkwds_get(diff_name, diff=False): 

626 d, x = _0_0, Beta._x_normalized 

627 if x and mr: 

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

629 d = b / mr * (x / Mu.x)**2 \ 

630 * (x / Phi._x_normalized) 

631 else: 

632 d = mr / a 

633 Mu._diff = self._fm1 * d 

634 return Mu 

635 

636 def RectifyingRadius(self, exact=False): 

637 '''Get the I{Rectifying} radius. 

638 

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

640 otherwise the I{Taylor} series. 

641 

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

643 as the ellipsoid axes). 

644 ''' 

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

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

647 

648 @Property_RO 

649 def _RectifyingR(self): 

650 m = self.ALorder 

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

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

653 

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

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

656 

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

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

659 

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

661 ''' 

662 _xinstanceof(AuxAngle, Phi=Phi) 

663 # assert Phi._AUX == Aux.PHI 

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

665 m = _toAuxCase.get(auxout, None) 

666 if m: # callable 

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

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

669 A = AuxPhi(Phi, name=n) 

670 else: # auxout? 

671 A = AuxPhi(NAN, name=n) 

672 # assert A._AUX == auxout 

673 return A 

674 

675 def _toZeta(self, zetaux): 

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

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

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

679 ''' 

680 class _AuxPhy(AuxPhi): 

681 # lean C{AuxPhi} instance. 

682 # _diff = _1_0 

683 # _x = _1_0 

684 

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

686 self._y = tphi 

687 

688 m = _toAuxCase.get(zetaux, None) 

689 if m: # callable 

690 

691 def _toZeta(tphi): 

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

693 

694 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC 

695 _toZeta = _AuxPhy 

696 

697 else: # zetaux? 

698 

699 def _toZeta(unused): # PYCHOK expected 

700 return _AuxPhy(NAN) 

701 

702 return _toZeta 

703 

704 

705# switch(auxout): ... 

706_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic, 

707 Aux.CONFORMAL: AuxLat.Conformal, 

708 Aux.GEOCENTRIC: AuxLat.Geocentric, 

709 Aux.PARAMETRIC: AuxLat.Parametric, 

710 Aux.RECTIFYING: AuxLat.Rectifying} 

711 

712 

713def _Clenshaw(sinp, Zeta, cs, K): 

714 sz, cz = Zeta._yx # isnormal 

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

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

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

718 if isfinite(x): 

719 U0, U1 = Fsum(), Fsum() 

720 # assert len(cs) == K 

721 for r in _reverange(K): 

722 U1 -= U0 * x + cs[r] 

723 U1, U0 = U0, -U1 

724 # u0*f0(Zeta) - u1*fm1(Zeta) 

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

726 # fm1 = 0 if sinp else 1 

727 if sinp: 

728 U0 *= sz * cz * _2_0 

729 else: 

730 U0 *= x * _0_5 

731 U0 -= U1 

732 x = float(U0) 

733 return x 

734 

735 

736def _CXcoeffs(aL): # PYCHOK in .auxilats.__main__ 

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

738 ''' 

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

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

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

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

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

744 # assert _coeffs.ALorder == aL 

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

746 return _coeffs 

747 

748 

749def _Newton(tphi, Zeta, _toZeta, **name): 

750 # Newton's method fro AuxLat._fromAux 

751 try: 

752 _lg2 = _log2 

753 _abs = fabs 

754 tz = _abs(Zeta.tan) 

755 tphi = tz / tphi # **) 

756 ltz = _lg2(tz) # **) 

757 ltphi = _lg2(tphi) # **) 

758 ltmin = min(ltphi, MIN_EXP) 

759 ltmax = max(ltphi, MAX_EXP) 

760# auxin = Zeta._AUX 

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

762 _TOL, _xp2 = _EPSqrt, _exp2 

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

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

765 Z = _toZeta(tphi) 

766 # assert Z._AUX == auxin 

767 t, s_ = Z.tan, s 

768 if t > tz: 

769 ltmax, s = ltphi, +1 

770 elif t < tz: 

771 ltmin, s = ltphi, -1 

772 else: 

773 break 

774 # derivative dtan(Z)/dtan(Phi) 

775 # to dlog(tan(Z))/dlog(tan(Phi)) 

776 d = (ltz - _lg2(t)) * t # **) 

777 if d: 

778 d = d / (Z.diff * tphi) # **) 

779 ltphi += d 

780 tphi = _xp2(ltphi) 

781 if _abs(d) < _TOL: 

782 i += 1 

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

784 Z = _toZeta(tphi) 

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

786 break 

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

788 s, n = 0, (i + 2) 

789 ltphi = (ltmin + ltmax) * __2 

790 tphi = _xp2(ltphi) 

791 else: 

792 i = _TRIPS 

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

794 Phi._iter = i 

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

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

797 Phi._iter = 0 

798 # assert Phi._AUX == Aux.PHI 

799 return Phi 

800 

801 

802_f, _u = float, _Ufloats() 

803_1__f3 = -1 / _f(3) # XXX +1 / _f(3) 

804_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3), 

805 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105), 

806 4 / _f(15), _1__f3), 

807 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693), 

808 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)} 

809_RRCoeffs = {4: _u(1 / _f(64), _0_25), 

810 6: _u(1 / _f(256), 1 / _f(64), _0_25), 

811 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used! 

812del _f, _u, _Ufloats, _1__f3 

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

814 

815__all__ += _ALL_DOCS(AuxLat) 

816 

817# **) MIT License 

818# 

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

820# 

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

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

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

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

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

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

827# 

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

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

830# 

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

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

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

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

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

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

837# OTHER DEALINGS IN THE SOFTWARE.