1

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

3

4 u'''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>}

6 in I{GeographicLib version 2.2+}.

7

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

9 under 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

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

17

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

19 AuxMu, AuxPhi, AuxTheta, AuxXi

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

21 from pygeodesy.basics import _reverange, _xinstanceof, _passarg

22 from 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

25 from pygeodesy.datums import _ellipsoidal_datum, _WGS84, \

26 Ellipsoid, _name__, _EWGS84

27# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums

28 from pygeodesy.elliptic import Elliptic as _Ef

29 from pygeodesy.errors import AuxError, _xkwds_not, _xkwds_pop2, _Xorder

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

31 from pygeodesy.fsums import Fsum, _Fsumf_, _sum

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

33 from pygeodesy.interns import NN, _DOT_, _not_scalar_, _UNDER_

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

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

36# from pygeodesy.named import _name__ # from .datums

37 from pygeodesy.props import Property, Property_RO, _update_all

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

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

40

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

42 try:

43 from math import exp2 as _exp2

44 except ImportError: # Python 3.11-

45

46 def _exp2(x):

47 return pow(_2_0, x)

48

49__all__ = ()

50__version__ = '24.06.16'

51

52_TRIPS = 1024 # XXX 2 or 3?

53

54

55 class AuxLat(AuxAngle):

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

57 on an ellipsoid.

58

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

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

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

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

63

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

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

66 '''

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

68 _E = _EWGS84

69# _Lmax = 0 # overwritten below

70 _mAL = 6 # 4, 6 or 8 aka Lmax

71

72 def __init__(self, a_earth=_EWGS84, f=None, b=None, **ALorder_name):

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

74

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

76 datum (L{Datum}, L{Ellipsoid}, L{Ellipsoid2} or L{a_f2Tuple}).

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

78 required if B{C{a_earth}} is C{scalar} and C{B{b}=None}.

79 @kwarg b: Optional polar radius, semi-axis (C{meter}, required if B{C{a_earth}}

80 is C{scalar} and C{B{f}=None}.

81 @kwarg ALorder_name: Optional C{B{name}=NN} (C{str}) and optional order of

82 this L{AuxLat} C{B{ALorder}=6}, see property C{ALorder}.

83 '''

84 if ALorder_name:

85 M = self._mAL

86 m, name = _xkwds_pop2(ALorder_name, ALorder=M)

87 if m != M:

88 self.ALorder = m

89 else:

90 name = NN

91 try:

92 if a_earth not in (_EWGS84, _WGS84):

93 n = _name__(name, name__=AuxLat)

94 if b is f is None:

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

96 elif _isRadius(a_earth):

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

98 else:

99 raise ValueError(_not_scalar_)

100 self._E = E

101 elif not (b is f is None):

102 # turn _UnexpectedError into AuxError

103 name = _name__(name, **_xkwds_not(None, b=b, f=f))

104

105 if name:

106 self.name = name

107 except Exception as x:

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

109

110 @Property_RO

111 def a(self):

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

113 '''

114 return self.ellipsoid.a

115

116 equatoradius = a

117

118 @Property

119 def ALorder(self):

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

121 '''

122 return self._mAL

123

124 @ALorder.setter # PYCHOK setter!

125 def ALorder(self, order):

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

127 '''

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

129 if self._mAL != m:

130 _update_all(self)

131 self._mAL = m

132

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

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

135 f = self.f

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

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

138 e = self._e

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

140 return s

141

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

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

144

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

146 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}

147 optional C{B{name}=NN}.

148

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

150 '''

151 _xinstanceof(AuxAngle, Phi=Phi)

152 # assert Phi._AUX == Aux.PHI

153 tphi = fabs(Phi.tan)

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

155 y, x = Phi._yx_normalized

156 q = self._q

157 qv = self._qf(tphi)

158 Dq2 = self._Dq(tphi)

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

160 d, n = _diff_name2(Phi, **diff_name)

161 Xi = AuxXi(copysign(qv, Phi.y), x * sqrt(Dq2), name=n)

162

163 if d:

164 if isnan(tphi):

165 d = self._e2m1_sq2

166 else:

167 c = self.Parametric(Phi)._x_normalized

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

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

170 Xi._diff = d

171 else:

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

173 # assert Xi._AUX == Aux.XI

174 return Xi

175

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

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

178

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

180 the I{Taylor} series.

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

182

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

184 units as the ellipsoid axes).

185

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

187 exceeds C{f_max}.

188 '''

189 f = self.f

190 if exact or not f:

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

192 elif fabs(f) < f_max:

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

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

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

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

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

198 m = self.ALorder

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

200 c2 *= self.a * (self.a + self.b)

201 else:

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

203 return c2 * _0_5

204

205 @Property_RO

206 def b(self):

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

208 '''

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

210

211 polaradius = b

212

213 def _coeffs(self, auxout, auxin):

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

215 aL = self.ALorder # aka Lmax

216 if auxout == auxin:

217 return _0_0s(aL) # uncached

218

219 k = Aux._1d(auxout, auxin)

220 try: # cached

221 return AuxLat._csc[aL][k]

222 except KeyError:

223 pass

224

225 Cx = _CXcoeffs(aL)

226 try:

227 Cx = Cx[auxout][auxin]

228 except KeyError as x:

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

230

231 d = x = n = self._n

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

233 x = self._n2

234

235 def _m(aL):

236 for m in _reverange(aL):

237 yield m // 2

238 else:

239 _m = _reverange # PYCHOK expected

240

241 i = 0

242 cs = []

243 _c = cs.append

244 _p = _polynomial

245 for m in _m(aL):

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

247 _c(_p(x, Cx, i, j) * d)

248 d *= n

249 i = j

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

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

252 return cs

253

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

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

256

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

258 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}

259 and an optional C{B{name}=NN}.

260

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

262 '''

263 _xinstanceof(AuxAngle, Phi=Phi)

264 # assert Phi._AUX == Aux.PHI

265 tphi = tchi = fabs(Phi.tan)

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

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

268 scsig = _sc(sig)

269 scphi = _sc(tphi)

270 if self.f > 0:

271 # The general expression for tchi is

272 # tphi * scsig - sig * scphi

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

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

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

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

277 sigtphi = sig / tphi

278 if sig < (tphi * _0_5):

279 t = tphi - sig

280 else:

281 def _asinh_2(x):

282 return asinh(x) * _0_5

283 # Still possibly dangerous cancellation in tphi - sig.

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

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

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

287 # Note sinh(atanh(sphi)) = tphi

288 # Turn the crank on divided differences, substitute

289 # sphi = tphi / _sc(tphi)

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

291 e = self._e

292 em1 = self._e2m1 / (_1_0 + e)

293 # assert em1 != 0

294 scb = self._scbeta(tphi)

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

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

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

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

299 try:

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

301 except ValueError: # Fsum(NAN) exception

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

303 e *= atphib

304 t = atphi - e

305 if t: # sinh(0) == 0

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

307 t = float(Dg) # tphi - sig

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

309 scsig + scphi * sigtphi) if t else _0_0

310 else:

311 tchi = tphi * scsig - sig * scphi

312

313 d, n = _diff_name2(Phi, **diff_name)

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

315

316 if d:

317 if isinf(tphi): # PYCHOK np cover

318 d = self._conformal_diff

319 else:

320 d = self.Parametric(Phi)._x_normalized

321 if d:

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

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

324 Chi._diff = d

325 # assrt Chi._AUX == Aux.CHI

326 return Chi

327

328 @Property_RO

329 def _conformal_diff(self): # PYCHOK no cover

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

331 '''

332 e = self._e

333 if self.f > 0:

334 ss = sinh(asinh(self._e1) * e)

335 d = _over(_1_0, _sc(ss) + ss)

336 elif e:

337 ss = sinh(-atan1(e) * e)

338 d = _sc(ss) - ss

339 else:

340 d = _1_0

341 return d

342

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

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

345 Z = d = Zeta_d

346 if isinstance(Z, AuxAngle):

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

348 if auxin == auxout:

349 pass

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

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

352 elif exact:

353 p = Aux.power(auxout, auxin)

354 if p is None:

355 P = self._fromAux(Z) # Phi

356 Z = self._toAux(auxout, P)

357 Z._iter = P.iteration

358 else:

359 y, x = Z._yx

360 if p:

361 y *= pow(self._fm1, p)

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

363 else:

364 cs = self._coeffs(auxout, auxin)

365 yx = Z._yx_normalized

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

367 # assert Z._yx == yx

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

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

370 # assert Z._AUX == auxout

371 return Z

372

373 elif _isDegrees(d):

374 Z = AuxPhi.fromDegrees(d)

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

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

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

378

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

380

381 def _Dq(self, tphi):

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

383 sphi = _sn(tphi)

384 if tphi > 0:

385 scphi = _sc(tphi)

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

387 if d:

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

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

390 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1)

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

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

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

394 if e2m1 and ed:

395 e2 = self._e2

396 if e2 > 0: # assert self.f > 0

397 scb = self._scbeta(tphi)

398 q = scphib = scphi / scb

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

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

401 else:

402 s2 = sphi * e2

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

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

405 else: # PYCHOK no cover

406 q = INF

407 else: # PYCHOK no cover

408 q = self._2_e2m12

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

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

411 return q

412

413 @Property_RO

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

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

416

417 @Property_RO

418 def _e1(self):

419 return sqrt(fabs(self._e12))

420

421 @Property_RO

422 def _e12(self):

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

424

425 @Property_RO

426 def _e12p1(self):

427 return _1_0 / self._e2m1

428

429 @Property_RO

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

431 return self.ellipsoid.e2

432

433 @Property_RO

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

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

436

437 @Property_RO

438 def _e2m1_sq2(self):

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

440

441 @Property_RO

442 def _2_e2m12(self):

443 return _2_0 / self._e2m1**2

444

445 @Property_RO

446 def _Ef_fRG_a2b2_PI_4(self):

447 E = self.ellipsoid

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

449

450 @Property_RO

451 def ellipsoid(self):

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

453 '''

454 return self._E

455

456 @Property_RO

457 def f(self):

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

459 '''

460 return self.ellipsoid.f

461

462 flattening = f

463

464 @Property_RO

465 def _fm1(self): # 1 - flattening

466 return self.ellipsoid.f1

467

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

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

470

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

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

473

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

475 '''

476 _xinstanceof(AuxAngle, Zeta=Zeta)

477 aux = Zeta._AUX

478 n = _name__(name, _or_nameof=Zeta)

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

480 if f is None:

481 Phi = AuxPhi(NAN, name=n)

482 elif callable(f):

483 t = self._fm1

484 t *= f(t)

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

486 else: # assert isscalar(f)

487 y, x = Zeta._yx

488 Phi = AuxPhi(y / f, x, name=n)

489 # assert Phi._AUX == Aux.PHI

490 return Phi

491

492 @Property_RO

493 def _fromAuxCase(self):

494 '''(INTERNAL) switch(auxin): ...

495 '''

496 return {Aux.AUTHALIC: cbrt,

497 Aux.CONFORMAL: _passarg,

498 Aux.GEOCENTRIC: self._e2m1,

499 Aux.GEOGRAPHIC: _1_0,

500 Aux.PARAMETRIC: self._fm1,

501 Aux.RECTIFYING: sqrt}

502

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

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

505

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

507 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}

508 and an optional C{B{name}=NN}.

509

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

511 '''

512 _xinstanceof(AuxAngle, Phi=Phi)

513 # assert Phi._AUX == Aux.PHI

514 d, n = _diff_name2(Phi, **diff_name)

515 Theta = AuxTheta(Phi.y * self._e2m1, Phi.x, name=n)

516 if d:

517 Theta._diff = self._e2m1

518 return Theta

519

520 def Geodetic(self, Phi, **name): # PYCHOK no cover

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

522

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

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

525

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

527 '''

528 _xinstanceof(AuxAngle, Phi=Phi)

529 # assert Phi._AUX == Aux.PHI

530 _, n = _diff_name2(Phi, **name)

531 return AuxPhi(Phi, name=n)

532

533 @Property_RO

534 def _n(self): # 3rd flattening

535 return self.ellipsoid.n

536

537 @Property_RO

538 def _n2(self):

539 return self._n**2

540

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

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

543

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

545 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}

546 and an optional C{B{name}=NN}.

547

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

549 '''

550 _xinstanceof(AuxAngle, Phi=Phi)

551 # assert Phi._AUX == Aux.PHI

552 d, n = _diff_name2(Phi, **diff_name)

553 Beta = AuxBeta(Phi.y * self._fm1, Phi.x, name=n)

554 if d:

555 Beta._diff = self._fm1

556 return Beta

557

558 Reduced = Parametric

559

560 @Property_RO

561 def _q(self): # constant _q

562 q, f = self._e12p1, self.f

563 if f:

564 e = self._e

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

566 else:

567 q += _1_0

568 return q

569

570 def _qf(self, tphi):

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

572 scb = self._scbeta(tphi)

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

574

575 def _qIntegrand(self, beta):

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

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

578 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) *

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

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

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

582 # then the integral is

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

584 # where

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

586 # d[K-1] = -1/(4*K) * c[K-1]

587 Beta = AuxBeta.fromRadians(beta)

588 if Beta.x: # and self._fm1:

589 Ax, _cv = Aux, self.convert

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

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

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

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

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

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

596 else: # beta == PI_2, PI3_2, ...

597 r = _0_0 # XXX 0 avoids NAN summation exceptions

598 return r

599

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

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

602

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

604 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}

605 and an optional C{B{name}=NN}.

606

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

608 '''

609 Beta = self.Parametric(Phi)

610 # assert Beta._AUX == Aux.BETA

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

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

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

614 if self.f < 0:

615 a, b = b, a

616 ka, kb = kb, ka

617 ka1, kb1 = kb1, ka1

618 sb, cb = cb, sb

619 # now a, b = larger/smaller semiaxis

620 # Beta measured from larger semiaxis

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

622 # NB kb <= 0; 0 <= ka <= 1

623 # sa = b*E(Beta, sqrt(kb))

624 # sb = a*E(Beta',sqrt(ka))

625 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2

626 sb2 = sb**2

627 cb2 = cb**2

628 da2 = ka1 + ka * sb2

629 db2 = _1_0 - kb * sb2

630 # DLMF Eq. 19.25.9

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

632 # DLMF Eq. 19.25.10 with complementary angles

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

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

635 ka * sb / sqrt(da2))

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

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

638 if self.f < 0:

639 a, b = b, a

640 my, mx = mx, my

641 mr = (my + mx) / PI_2

642 if mr:

643 my = sin(my / mr)

644 mx = sin(mx / mr) # XXX zero?

645 else: # zero Mu

646 my, mx = _0_0, _1_0

647 d, n = _diff_name2(Phi, **diff_name)

648 Mu = AuxMu(my, mx, # normalized

649 name=n).copyquadrant(Phi)

650 if d:

651 d, x = _0_0, Beta._x_normalized

652 if x and mr:

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

654 d = b / mr * (x / Mu.x)**2 \

655 * (x / Phi._x_normalized)

656 else:

657 d = mr / a

658 Mu._diff = self._fm1 * d

659 return Mu

660

661 def RectifyingRadius(self, exact=False):

662 '''Get the I{Rectifying} radius.

663

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

665 otherwise the I{Taylor} series.

666

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

668 as the ellipsoid axes).

669 '''

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

671 return Meter(r, name__=self.RectifyingRadius)

672

673 @Property_RO

674 def _RectifyingR(self):

675 m = self.ALorder

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

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

678

679 def _scbeta(self, tphi):

680 return _sc(self._fm1 * tphi)

681

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

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

684

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

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

687 @kwarg diff_name: Use C{B{diff}=True} to set C{diff}

688 and an optional C{B{name}=NN}.

689

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

691 '''

692 _xinstanceof(AuxAngle, Phi=Phi)

693 # assert Phi._AUX == Aux.PHI

694 d, n = _diff_name2(Phi, **diff_name)

695 m = _toAuxCase.get(auxout, None)

696 if m: # callable

697 A = m(self, Phi, diff=d, name=n)

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

699 A = AuxPhi(Phi, name=n)

700 else: # auxout?

701 A = AuxPhi(NAN, name=n)

702 # assert A._AUX == auxout

703 return A

704

705 def _toZeta(self, zetaux):

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

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

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

709 '''

710 class _AuxPhy(AuxPhi):

711 # lean C{AuxPhi} instance.

712 # _diff = _1_0

713 # _x = _1_0

714

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

716 self._y = tphi

717

718 m = _toAuxCase.get(zetaux, None)

719 if m: # callable

720

721 def _toZeta(tphi):

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

723

724 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC

725 _toZeta = _AuxPhy

726

727 else: # zetaux?

728

729 def _toZeta(unused): # PYCHOK expected

730 return _AuxPhy(NAN)

731

732 return _toZeta

733

734

735# switch(auxout): ...

736_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic,

737 Aux.CONFORMAL: AuxLat.Conformal,

738 Aux.GEOCENTRIC: AuxLat.Geocentric,

739 Aux.PARAMETRIC: AuxLat.Parametric,

740 Aux.RECTIFYING: AuxLat.Rectifying}

741

742

743 def _Clenshaw(sinp, Zeta, cs, K):

744 sz, cz = Zeta._yx # isnormal

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

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

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

748 if isfinite(x):

749 U0, U1 = Fsum(), Fsum()

750 # assert len(cs) == K

751 for r in _reverange(K):

752 U1 -= U0 * x + cs[r]

753 U1, U0 = U0, -U1

754 # u0*f0(Zeta) - u1*fm1(Zeta)

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

756 # fm1 = 0 if sinp else 1

757 if sinp:

758 U0 *= sz * cz * _2_0

759 else:

760 U0 *= x * _0_5

761 U0 -= U1

762 x = float(U0)

763 return x

764

765

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

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

768 '''

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

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

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

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

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

774 # assert _coeffs.ALorder == aL

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

776 return _coeffs

777

778

779 def _diff_name2(Phi, diff=False, **name):

780 '''(INTERNAL) Get C{{Bdiff}=False} and C{B{name}=NN}.

781 '''

782 n = _name__(name, _or_nameof=Phi) # if name else Phi.name

783 return diff, n

784

785

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

787 # Newton's method from AuxLat._fromAux

788 try:

789 _lg2 = _log2

790 _abs = fabs

791 tz = _abs(Zeta.tan)

792 tphi = tz / tphi # **)

793 ltz = _lg2(tz) # **)

794 ltphi = _lg2(tphi) # **)

795 ltmin = min(ltphi, MIN_EXP)

796 ltmax = max(ltphi, MAX_EXP)

797# auxin = Zeta._AUX

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

799 _TOL, _xp2 = _EPSqrt, _exp2

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

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

802 Z = _toZeta(tphi)

803 # assert Z._AUX == auxin

804 t, s_ = Z.tan, s

805 if t > tz:

806 ltmax, s = ltphi, +1

807 elif t < tz:

808 ltmin, s = ltphi, -1

809 else:

810 break

811 # derivative dtan(Z)/dtan(Phi)

812 # to dlog(tan(Z))/dlog(tan(Phi))

813 d = (ltz - _lg2(t)) * t # **)

814 if d:

815 d = d / (Z.diff * tphi) # **)

816 ltphi += d

817 tphi = _xp2(ltphi)

818 if _abs(d) < _TOL:

819 i += 1

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

821 Z = _toZeta(tphi)

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

823 break

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

825 s, n = 0, (i + 2)

826 ltphi = (ltmin + ltmax) * __2

827 tphi = _xp2(ltphi)

828 else:

829 i = _TRIPS

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

831 Phi._iter = i

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

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

834 Phi._iter = 0

835 # assert Phi._AUX == Aux.PHI

836 return Phi

837

838

839_f, _u = float, _Ufloats()

840_1__f3 = -1 / _f(3) # XXX +1 / _f(3)

841_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3),

842 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105),

843 4 / _f(15), _1__f3),

844 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693),

845 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)}

846_RRCoeffs = {4: _u(1 / _f(64), _0_25),

847 6: _u(1 / _f(256), 1 / _f(64), _0_25),

848 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used!

849 del _f, _u, _Ufloats, _1__f3

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

851

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

853

854__all__ += _ALL_DOCS(AuxLat)

855

856# **) MIT License

857#

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

859#

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

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

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

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

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

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

866#

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

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

869#

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

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

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

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

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

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

876# OTHER DEALINGS IN THE SOFTWARE.