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

21 from pygeodesy.basics import isscalar, _reverange, _xinstanceof

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, Ellipsoid

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

27 from pygeodesy.elliptic import Elliptic as _Ef

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

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

30 from pygeodesy.fsums import Fsum, _sum

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

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

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

34 from pygeodesy.props import Property, Property_RO, _update_all

35 from pygeodesy.units import Degrees, Meter

36# from pygeodesy.utily import _passarg # _MODS

37

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

39 try:

40 from math import exp2 as _exp2

41 except ImportError: # Python 3.11-

42

43 def _exp2(x):

44 return pow(_2_0, x)

45

46__all__ = ()

47__version__ = '23.08.27'

48

49_TRIPS = 1024 # XXX 2 or 3?

50

51

52 class AuxLat(AuxAngle):

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

54

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

56 maintain precision close to the poles, I{Authalic} latitude C{Xi},

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

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

59

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

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

62 '''

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

64 _E = _WGS84.ellipsoid

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

66

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

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

69

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

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

72 L{Ellipsoid2} or L{a_f2Tuple}).

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

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

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

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

77 is not scalar.

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

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

80 C{ALorder}.

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

82 '''

83 if a_earth is not _WGS84:

84 n = name or AuxLat.__name__

85 try:

86 if b is f is None:

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

88 elif isscalar(a_earth):

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

90 else:

91 raise ValueError()

92 except Exception as x:

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

94 self._E = E

95

96 if name:

97 self.name = name

98 if ALorder:

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

100

101 @Property_RO

102 def a(self):

103 '''Get the equatorial radius (C{Meter}, conventionally).

104 '''

105 return self.ellipsoid.a

106

107 equatoradius = a

108

109 @Property

110 def ALorder(self):

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

112 '''

113 return self._mAL

114

115 @ALorder.setter # PYCHOK setter!

116 def ALorder(self, order):

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

118 '''

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

120 if self._mAL != m:

121 _update_all(self)

122 self._mAL = m

123

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

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

126 f = self.f

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

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

129 e = self._e

130 s = (atan(e * s) if f < 0 else asinh(self._e1 * s)) / e

131 return s

132

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

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

135

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

137

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

139 '''

140 _xinstanceof(AuxAngle, Phi=Phi)

141 # assert Phi._AUX == Aux.PHI

142 tphi = fabs(Phi.tan)

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

144 y, x = Phi._yx_normalized

145 q = self._q

146 qv = self._qf(tphi)

147 Dq2 = self._Dq(tphi)

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

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

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

151

152 if _xkwds_get(diff_name, diff=False):

153 if isnan(tphi):

154 d = self._e2m1_sq2

155 else:

156 c = self.Parametric(Phi)._x_normalized

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

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

159 Xi._diff = d

160 else:

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

162 # assert Xi._AUX == Aux.XI

163 return Xi

164

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

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

167

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

169 the I{Taylor} series.

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

171

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

173 units as the ellipsoid axes).

174

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

176 exceeds C{f_max}.

177 '''

178 f = self.f

179 if exact or not f:

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

181 elif fabs(f) < f_max:

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

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

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

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

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

187 m = self.ALorder

188 c2 = _MODS.karney._polynomial(self._n, _AR2Coeffs[m], 0, m)

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

190 else:

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

192 return c2 * _0_5

193

194 @Property_RO

195 def b(self):

196 '''Get the polar radius (C{Meter}, conventionally).

197 '''

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

199

200 polaradius = b

201

202 def _coeffs(self, auxout, auxin):

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

204 aL = self.ALorder # aka Lmax

205 k = Aux._1d(auxout, auxin)

206 try:

207 cs = AuxLat._csc[aL][k]

208 except KeyError:

209 if auxout == auxin:

210 cs = _0_0s(aL) # not cached

211 else:

212 Cx = _CXcoeffs(aL)

213 try:

214 Cx = Cx[auxout][auxin]

215 except KeyError as x:

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

217

218 d = x = n = self._n

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

220 x = self._n2

221

222 def _m(m):

223 return m // 2

224 else:

225 def _m(m): # PYCHOK expected

226 return m

227

228 i = 0

229 cs = []

230 _p = _MODS.karney._polynomial

231 for r in _reverange(aL):

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

233 cs.append(_p(x, Cx, i, j) * d)

234 d *= n

235 i = j

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

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

238 return cs

239

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

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

242

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

244

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

246 '''

247 _xinstanceof(AuxAngle, Phi=Phi)

248 # assert Phi._AUX == Aux.PHI

249 tphi = tchi = fabs(Phi.tan)

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

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

252 scsig = _sc(sig)

253 scphi = _sc(tphi)

254 if self.f > 0:

255 # The general expression for tchi is

256 # tphi * scsig - sig * scphi

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

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

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

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

261 sigtphi = sig / tphi

262 if sig < (tphi * _0_5):

263 t = tphi - sig

264 else:

265 def _asinh_2(x):

266 return asinh(x) * _0_5

267 # Still possibly dangerous cancellation in tphi - sig.

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

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

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

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

272 # Turn the crank on divided differences, substitute

273 # sphi = tphi / _sc(tphi)

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

275 e = self._e

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

277 # assert em1 != 0

278 scb = self._scbeta(tphi)

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

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

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

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

283 try:

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

285 except ValueError: # Fsum(NAN) exception

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

287 e *= atphib

288 t = atphi - e

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

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

291 t = float(Dg) # tphi - sig

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

293 scsig + scphi * sigtphi) # if t else _0_0

294 else:

295 tchi = tphi * scsig - sig * scphi

296

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

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

299

300 if _xkwds_get(diff_name, diff=False):

301 if isinf(tphi): # PYCHOK np cover

302 d = self._conformal_diff

303 else:

304 d = self.Parametric(Phi)._x_normalized

305 if d:

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

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

308 Chi._diff = d

309 # assrt Chi._AUX == Aux.CHI

310 return Chi

311

312 @Property_RO

313 def _conformal_diff(self): # PYCHOK no cover

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

315 '''

316 e = self._e

317 if self.f > 0:

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

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

320 elif e:

321 ss = sinh(-atan(e) * e)

322 d = _sc(ss) - ss

323 else:

324 d = _1_0

325 return d

326

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

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

329 Z = d = Zeta_d

330 if isinstance(Z, AuxAngle):

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

332 if auxin == auxout:

333 pass

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

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

336 elif exact:

337 p = Aux.power(auxout, auxin)

338 if p is None:

339 P = self._fromAux(Z) # Phi

340 Z = self._toAux(auxout, P)

341 Z._iter = P.iteration

342 else:

343 y, x = Z._yx

344 if p:

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

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

347 else:

348 cs = self._coeffs(auxout, auxin)

349 yx = Z._yx_normalized

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

351 # assert Z._yx == yx

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

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

354 # assert Z._AUX == auxout

355 return Z

356

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

358 Z = AuxPhi.fromDegrees(d)

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

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

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

362

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

364

365 def _Dq(self, tphi):

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

367 sphi = _sn(tphi)

368 if tphi > 0:

369 scphi = _sc(tphi)

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

371 if d:

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

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

374 # (1 + e2 * sphi) / ((1 - e2 * sphi * sphi) * e2m1)

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

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

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

378 if e2m1 and ed:

379 e2 = self._e2

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

381 scb = self._scbeta(tphi)

382 q = scphib = scphi / scb

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

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

385 else:

386 s2 = sphi * e2

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

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

389 else: # PYCHOK no cover

390 q = INF

391 else: # PYCHOK no cover

392 q = self._2_e2m12

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

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

395 return q

396

397 @Property_RO

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

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

400

401 @Property_RO

402 def _e1(self):

403 return sqrt(fabs(self._e12))

404

405 @Property_RO

406 def _e12(self):

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

408

409 @Property_RO

410 def _e12p1(self):

411 return _1_0 / self._e2m1

412

413 @Property_RO

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

415 return self.ellipsoid.e2

416

417 @Property_RO

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

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

420

421 @Property_RO

422 def _e2m1_sq2(self):

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

424

425 @Property_RO

426 def _2_e2m12(self):

427 return _2_0 / self._e2m1**2

428

429 @Property_RO

430 def _Ef_fRG_a2b2_PI_4(self):

431 E = self.ellipsoid

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

433

434 @Property_RO

435 def ellipsoid(self):

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

437 '''

438 return self._E

439

440 @Property_RO

441 def f(self):

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

443 '''

444 return self.ellipsoid.f

445

446 flattening = f

447

448 @Property_RO

449 def _fm1(self): # 1 - flattening

450 return self.ellipsoid.f1

451

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

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

454

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

456

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

458 '''

459 _xinstanceof(AuxAngle, Zeta=Zeta)

460 aux = Zeta._AUX

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

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

463 if f is None:

464 Phi = AuxPhi(NAN, name=n)

465 elif callable(f):

466 t = self._fm1

467 t *= f(t)

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

469 else: # assert isscalar(f)

470 y, x = Zeta._yx

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

472 # assert Phi._AUX == Aux.PHI

473 return Phi

474

475 @Property_RO

476 def _fromAuxCase(self):

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

478 '''

479 return {Aux.AUTHALIC: cbrt,

480 Aux.CONFORMAL: _MODS.utily._passarg,

481 Aux.GEOCENTRIC: self._e2m1,

482 Aux.GEOGRAPHIC: _1_0,

483 Aux.PARAMETRIC: self._fm1,

484 Aux.RECTIFYING: sqrt}

485

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

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

488

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

490

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

492 '''

493 _xinstanceof(AuxAngle, Phi=Phi)

494 # assert Phi._AUX == Aux.PHI

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

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

497 if _xkwds_get(diff_name, diff=False):

498 Theta._diff = self._e2m1

499 return Theta

500

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

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

503

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

505

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

507 '''

508 _xinstanceof(AuxAngle, Phi=Phi)

509 # assert Phi._AUX == Aux.PHI

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

511

512 @Property_RO

513 def _n(self): # 3rd flattening

514 return self.ellipsoid.n

515

516 @Property_RO

517 def _n2(self):

518 return self._n**2

519

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

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

522

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

524

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

526 '''

527 _xinstanceof(AuxAngle, Phi=Phi)

528 # assert Phi._AUX == Aux.PHI

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

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

531 if _xkwds_get(diff_name, diff=False):

532 Beta._diff = self._fm1

533 return Beta

534

535 Reduced = Parametric

536

537 @Property_RO

538 def _q(self): # constant _q

539 q, f = self._e12p1, self.f

540 if f:

541 e = self._e

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

543 else:

544 q += _1_0

545 return q

546

547 def _qf(self, tphi):

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

549 scb = self._scbeta(tphi)

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

551

552 def _qIntegrand(self, beta):

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

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

555 # = (1-f) * (cos(chi) - cos(xi)) / cos(phi) *

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

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

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

559 # then the integral is

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

561 # where

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

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

564 Beta = AuxBeta.fromRadians(beta)

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

566 Ax, _cv = Aux, self.convert

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

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

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

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

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

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

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

574 r = _0_0 # XXX 0 avoids NAN summation exceptions

575 return r

576

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

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

579

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

581

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

583 '''

584 Beta = self.Parametric(Phi)

585 # assert Beta._AUX == Aux.BETA

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

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

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

589 if self.f < 0:

590 a, b = b, a

591 ka, kb = kb, ka

592 ka1, kb1 = kb1, ka1

593 sb, cb = cb, sb

594 # now a, b = larger/smaller semiaxis

595 # Beta measured from larger semiaxis

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

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

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

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

600 # 1 - ka * (1 - sb2) = 1 - ka + ka*sb2

601 sb2 = sb**2

602 cb2 = cb**2

603 da2 = ka1 + ka * sb2

604 db2 = _1_0 - kb * sb2

605 # DLMF Eq. 19.25.9

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

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

608 # DLMF Eq. 19.25.10 with complementary angles

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

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

611 ka * sb / sqrt(da2))

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

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

614 if self.f < 0:

615 a, b = b, a

616 my, mx = mx, my

617 mr = (my + mx) / PI_2

618 if mr:

619 my = sin(my / mr)

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

621 else: # zero Mu

622 my, mx = _0_0, _1_0

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

624 Mu = AuxMu(my, mx, # normalized

625 name=n).copyquadrant(Phi)

626

627 if _xkwds_get(diff_name, diff=False):

628 d, x = _0_0, Beta._x_normalized

629 if x and mr:

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

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

632 * (x / Phi._x_normalized)

633 else:

634 d = mr / a

635 Mu._diff = self._fm1 * d

636 return Mu

637

638 def RectifyingRadius(self, exact=False):

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

640

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

642 otherwise the I{Taylor} series.

643

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

645 as the ellipsoid axes).

646 '''

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

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

649

650 @Property_RO

651 def _RectifyingR(self):

652 m = self.ALorder

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

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

655

656 def _scbeta(self, tphi):

657 return _sc(self._fm1 * tphi)

658

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

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

661

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

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

664

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

666 '''

667 _xinstanceof(AuxAngle, Phi=Phi)

668 # assert Phi._AUX == Aux.PHI

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

670 m = _toAuxCase.get(auxout, None)

671 if m: # callable

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

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

674 A = AuxPhi(Phi, name=n)

675 else: # auxout?

676 A = AuxPhi(NAN, name=n)

677 # assert A._AUX == auxout

678 return A

679

680 def _toZeta(self, zetaux):

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

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

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

684 '''

685 class _AuxPhy(AuxPhi):

686 # lean C{AuxPhi} instance.

687 # _diff = _1_0

688 # _x = _1_0

689

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

691 self._y = tphi

692

693 m = _toAuxCase.get(zetaux, None)

694 if m: # callable

695

696 def _toZeta(tphi):

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

698

699 elif zetaux == Aux.GEODETIC: # GEOGRAPHIC

700 _toZeta = _AuxPhy

701

702 else: # zetaux?

703

704 def _toZeta(unused): # PYCHOK expected

705 return _AuxPhy(NAN)

706

707 return _toZeta

708

709

710# switch(auxout): ...

711_toAuxCase = {Aux.AUTHALIC: AuxLat.Authalic,

712 Aux.CONFORMAL: AuxLat.Conformal,

713 Aux.GEOCENTRIC: AuxLat.Geocentric,

714 Aux.PARAMETRIC: AuxLat.Parametric,

715 Aux.RECTIFYING: AuxLat.Rectifying}

716

717

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

719 sz, cz = Zeta._yx # isnormal

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

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

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

723 if isfinite(x):

724 U0, U1 = Fsum(), Fsum()

725 # assert len(cs) == K

726 for r in _reverange(K):

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

728 U1, U0 = U0, -U1

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

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

731 # fm1 = 0 if sinp else 1

732 if sinp:

733 U0 *= sz * cz * _2_0

734 else:

735 U0 *= x * _0_5

736 U0 -= U1

737 x = float(U0)

738 return x

739

740

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

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

743 '''

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

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

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

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

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

749 # assert _coeffs.ALorder == aL

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

751 return _coeffs

752

753

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

755 # Newton's method fro AuxLat._fromAux

756 try:

757 _lg2 = _log2

758 _abs = fabs

759 tz = _abs(Zeta.tan)

760 tphi = tz / tphi # **)

761 ltz = _lg2(tz) # **)

762 ltphi = _lg2(tphi) # **)

763 ltmin = min(ltphi, MIN_EXP)

764 ltmax = max(ltphi, MAX_EXP)

765# auxin = Zeta._AUX

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

767 _TOL, _xp2 = _EPSqrt, _exp2

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

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

770 Z = _toZeta(tphi)

771 # assert Z._AUX == auxin

772 t, s_ = Z.tan, s

773 if t > tz:

774 ltmax, s = ltphi, +1

775 elif t < tz:

776 ltmin, s = ltphi, -1

777 else:

778 break

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

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

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

782 if d:

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

784 ltphi += d

785 tphi = _xp2(ltphi)

786 if _abs(d) < _TOL:

787 i += 1

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

789 Z = _toZeta(tphi)

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

791 break

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

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

794 ltphi = (ltmin + ltmax) * __2

795 tphi = _xp2(ltphi)

796 else:

797 i = _TRIPS

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

799 Phi._iter = i

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

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

802 Phi._iter = 0

803 # assert Phi._AUX == Aux.PHI

804 return Phi

805

806

807_f, _u = float, _Ufloats()

808_1__f3 = -1 / _f(3) # XXX +1 / _f(3)

809_AR2Coeffs = {4: _u(4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3),

810 6: _u(4 / _f(1287), 4 / _f(693), 4 / _f(315), 4 / _f(105),

811 4 / _f(15), _1__f3),

812 8: _u(4 / _f(3315), 4 / _f(2145), 4 / _f(1287), 4 / _f(693),

813 4 / _f(315), 4 / _f(105), 4 / _f(15), _1__f3)}

814_RRCoeffs = {4: _u(1 / _f(64), _0_25),

815 6: _u(1 / _f(256), 1 / _f(64), _0_25),

816 8: _u(25 / _f(16384), 1 / _f(256), 1 / _f(64), _0_25)} # PYCHOK used!

817 del _f, _u, _Ufloats, _1__f3

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

819

820__all__ += _ALL_DOCS(AuxLat)

821

822# **) MIT License

823#

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

825#

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

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

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

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

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

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

832#

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

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

835#

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

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

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

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

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

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

842# OTHER DEALINGS IN THE SOFTWARE.