1

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

3

4 u'''A pure Python version of I{Karney}'s C++ class U{GeodesicExact

5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}.

6

7 Class L{GeodesicExact} follows the naming, methods and return values

8 of class C{Geodesic} from I{Karney}'s Python U{geographiclib

9<https://GitHub.com/geographiclib/geographiclib-python>}.

10

11 Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2023)

12 and licensed under the MIT/X11 License. For more information, see the

13 U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.

14'''

15# make sure int/int division yields float quotient

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

17

18# A copy of comments from Karney's C{GeodesicExact.cpp}:

19#

20# This is a reformulation of the geodesic problem. The

21# notation is as follows:

22# - at a general point (no suffix or 1 or 2 as suffix)

23# - phi = latitude

24# - lambda = longitude

25# - beta = latitude on auxiliary sphere

26# - omega = longitude on auxiliary sphere

27# - alpha = azimuth of great circle

28# - sigma = arc length along great circle

29# - s = distance

30# - tau = scaled distance (= sigma at multiples of PI/2)

31# - at northwards equator crossing

32# - beta = phi = 0

33# - omega = lambda = 0

34# - alpha = alpha0

35# - sigma = s = 0

36# - a 12 suffix means a difference, e.g., s12 = s2 - s1.

37# - s and c prefixes mean sin and cos

38

39 from pygeodesy.basics import _xinstanceof, _xor, unsigned0

40 from pygeodesy.constants import EPS, EPS0, EPS02, MANT_DIG, NAN, PI, _EPSqrt, \

41 _SQRT2_2, isnan, _0_0, _0_001, _0_01, _0_1, _0_5, \

42 _1_0, _N_1_0, _1_75, _2_0, _N_2_0, _2__PI, _3_0, \

43 _4_0, _6_0, _8_0, _16_0, _90_0, _180_0, _1000_0

44 from pygeodesy.datums import _earth_datum, _WGS84, _EWGS84

45# from pygeodesy.ellipsoids import _EWGS84 # from .datums

46# from pygeodesy.fmath import cbrt as _cbrt, hypot as _hypot # from .karney

47 from pygeodesy.fsums import fsumf_, fsum1f_

48 from pygeodesy.geodesicx.gxbases import _cosSeries, _GeodesicBase, \

49 _sincos12, _sin1cos2, _xnC4

50 from pygeodesy.geodesicx.gxline import _GeodesicLineExact, _TINY, _update_glXs

51 from pygeodesy.interns import NN, _COMMASPACE_, _DOT_, _UNDER_

52 from pygeodesy.karney import _around, _atan2d, Caps, _cbrt, _copysign, _diff182, \

53 _fix90, GDict, GeodesicError, _hypot, _K_2_0, _norm2, \

54 _norm180, _polynomial, _signBit, _sincos2, _sincos2d, \

55 _sincos2de, _unsigned2

56 from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS

57 from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple

58 from pygeodesy.props import deprecated_Property, Property, Property_RO, property_RO

59 from pygeodesy.streprs import Fmt, pairs

60 from pygeodesy.utily import atan2d as _atan2d_reverse, _unrollon, _Wrap, wrap360

61

62 from math import atan2, copysign, cos, degrees, fabs, radians, sqrt

63

64__all__ = ()

65__version__ = '23.12.09'

66

67_MAXIT1 = 20

68_MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits

69

70# increased multiplier in defn of _TOL1 from 100 to 200 to fix Inverse

71# case 52.784459512564 0 -52.784459512563990912 179.634407464943777557

72# which otherwise failed for Visual Studio 10 (Release and Debug)

73_TOL0 = EPS

74_TOL1 = _TOL0 * -200 # negative

75_TOL2 = _EPSqrt # == sqrt(_TOL0)

76_TOL3 = _TOL2 * _0_1

77_TOLb = _TOL2 * _TOL0 # Check on bisection interval

78_THR1 = _TOL2 * _1000_0 + _1_0

79

80_TINY3 = _TINY * _3_0

81_TOL08 = _TOL0 * _8_0

82_TOL016 = _TOL0 * _16_0

83

84

85 def _atan12(*sincos12, **sineg0):

86 '''(INTERNAL) Return C{ang12} in C{radians}.

87 '''

88 return atan2(*_sincos12(*sincos12, **sineg0))

89

90

91 def _eTOL2(f):

92 # Using the auxiliary sphere solution with dnm computed at

93 # (bet1 + bet2) / 2, the relative error in the azimuth

94 # consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2.

95 # (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000.

96

97 # For a given f and sig12, the max error occurs for lines

98 # near the pole. If the old rule for computing dnm = (dn1

99 # + dn2)/2 is used, then the error increases by a factor of

100 # 2.) Setting this equal to epsilon gives sig12 = etol2.

101

102 # Here 0.1 is a safety factor (error decreased by 100) and

103 # max(0.001, abs(f)) stops etol2 getting too large in the

104 # nearly spherical case.

105 t = min(_1_0, _1_0 - f * _0_5) * max(_0_001, fabs(f)) * _0_5

106 return _TOL3 / (sqrt(t) if t > EPS02 else EPS0)

107

108

109 class _PDict(GDict):

110 '''(INTERNAL) Parameters passed around in C{._GDictInverse} and

111 optionally returned when C{GeodesicExact.debug} is C{True}.

112 '''

113 def set_sigs(self, ssig1, csig1, ssig2, csig2):

114 '''Update the C{sig1} and C{sig2} parameters.

115 '''

116 self.set_(ssig1=ssig1, csig1=csig1, sncndn1=(ssig1, csig1, self.dn1), # PYCHOK dn1

117 ssig2=ssig2, csig2=csig2, sncndn2=(ssig2, csig2, self.dn2)) # PYCHOK dn2

118

119 def toGDict(self): # PYCHOK no cover

120 '''Return as C{GDict} without attrs C{sncndn1} and C{sncndn2}.

121 '''

122 def _rest(sncndn1=None, sncndn2=None, **rest): # PYCHOK sncndn* not used

123 return GDict(rest)

124

125 return _rest(**self)

126

127

128 class GeodesicExact(_GeodesicBase):

129 '''A pure Python version of I{Karney}'s C++ class U{GeodesicExact

130 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>},

131 modeled after I{Karney}'s Python class U{geodesic.Geodesic<https://GitHub.com/

132 geographiclib/geographiclib-python>}.

133 '''

134 _datum = _WGS84

135 _nC4 = 30 # default C4order

136

137 def __init__(self, a_ellipsoid=_EWGS84, f=None, name=NN, C4order=None,

138 C4Order=None): # for backward compatibility

139 '''New L{GeodesicExact} instance.

140

141 @arg a_ellipsoid: An ellipsoid (L{Ellipsoid}) or datum (L{Datum}) or

142 the equatorial radius of the ellipsoid (C{scalar},

143 conventionally in C{meter}), see B{C{f}}.

144 @arg f: The flattening of the ellipsoid (C{scalar}) if B{C{a_ellipsoid}}

145 is specified as C{scalar}.

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

147 @kwarg C4order: Optional series expansion order (C{int}), see property

148 L{C4order}, default C{30}.

149 @kwarg C4Order: DEPRECATED, use keyword argument B{C{C4order}}.

150

151 @raise GeodesicError: Invalid B{C{C4order}}.

152 '''

153 _earth_datum(self, a_ellipsoid, f=f, name=name)

154 if name:

155 self.name = name

156 if C4order: # XXX private copy, always?

157 self.C4order = C4order

158 elif C4Order: # for backward compatibility

159 self.C4Order = C4Order

160

161 @Property_RO

162 def a(self):

163 '''Get the I{equatorial} radius, semi-axis (C{meter}).

164 '''

165 return self.ellipsoid.a

166

167 def ArcDirect(self, lat1, lon1, azi1, a12, outmask=Caps.STANDARD):

168 '''Solve the I{Direct} geodesic problem in terms of (spherical) arc length.

169

170 @arg lat1: Latitude of the first point (C{degrees}).

171 @arg lon1: Longitude of the first point (C{degrees}).

172 @arg azi1: Azimuth at the first point (compass C{degrees}).

173 @arg a12: Arc length between the points (C{degrees}), can be negative.

174 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying

175 the quantities to be returned.

176

177 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,

178 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},

179 C{lon1}, C{azi1} and arc length C{a12} always included.

180

181 @see: C++ U{GeodesicExact.ArcDirect

182 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}

183 and Python U{Geodesic.ArcDirect<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.

184 '''

185 return self._GDictDirect(lat1, lon1, azi1, True, a12, outmask)

186

187 def ArcDirectLine(self, lat1, lon1, azi1, a12, caps=Caps.ALL, name=NN):

188 '''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as arc length.

189

190 @arg lat1: Latitude of the first point (C{degrees}).

191 @arg lon1: Longitude of the first point (C{degrees}).

192 @arg azi1: Azimuth at the first point (compass C{degrees}).

193 @arg a12: Arc length between the points (C{degrees}), can be negative.

194 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying

195 the capabilities the L{GeodesicLineExact} instance

196 should possess, i.e., which quantities can be

197 returned by calls to L{GeodesicLineExact.Position}

198 and L{GeodesicLineExact.ArcPosition}.

199

200 @return: A L{GeodesicLineExact} instance.

201

202 @note: The third point of the L{GeodesicLineExact} is set to correspond

203 to the second point of the I{Inverse} geodesic problem.

204

205 @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}.

206

207 @see: C++ U{GeodesicExact.ArcDirectLine

208 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and

209 Python U{Geodesic.ArcDirectLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.

210 '''

211 return self._GenDirectLine(lat1, lon1, azi1, True, a12, caps, name=name)

212

213 def Area(self, polyline=False, name=NN):

214 '''Set up a L{GeodesicAreaExact} to compute area and

215 perimeter of a polygon.

216

217 @kwarg polyline: If C{True} perimeter only, otherwise

218 area and perimeter (C{bool}).

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

220

221 @return: A L{GeodesicAreaExact} instance.

222

223 @note: The B{C{debug}} setting is passed as C{verbose}

224 to the returned L{GeodesicAreaExact} instance.

225 '''

226 gaX = _MODS.geodesicx.GeodesicAreaExact(self, polyline=polyline,

227 name=name or self.name)

228 if self.debug:

229 gaX.verbose = True

230 return gaX

231

232 @Property_RO

233 def b(self):

234 '''Get the ellipsoid's I{polar} radius, semi-axis (C{meter}).

235 '''

236 return self.ellipsoid.b

237

238 @Property_RO

239 def c2x(self):

240 '''Get the ellipsoid's I{authalic} earth radius I{squared} (C{meter} I{squared}).

241 '''

242 # The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2))

243 # in the definition of _c2. The latter is more accurate for very

244 # oblate ellipsoids (which the Geodesic class does not handle). Of

245 # course, the area calculation in GeodesicExact is still based on a

246 # series and only holds for moderately oblate (or prolate) ellipsoids.

247 return self.ellipsoid.c2x

248

249 c2 = c2x # in this particular case

250

251 def C4f(self, eps):

252 '''Evaluate the C{C4x} coefficients for B{C{eps}}.

253

254 @arg eps: Polynomial factor (C{float}).

255

256 @return: C{C4order}-Tuple of C{C4x(B{eps})} coefficients.

257 '''

258 def _c4(nC4, C4x):

259 i, x, e = 0, _1_0, eps

260 _p = _polynomial

261 for r in range(nC4, 0, -1):

262 j = i + r

263 yield _p(e, C4x, i, j) * x

264 x *= e

265 i = j

266 # assert i == (nC4 * (nC4 + 1)) // 2

267

268 return tuple(_c4(self._nC4, self._C4x))

269

270 def _C4f_k2(self, k2): # in ._GDictInverse and gxline._GeodesicLineExact._C4a

271 '''(INTERNAL) Compute C{eps} from B{C{k2}} and invoke C{C4f}.

272 '''

273 return self.C4f(k2 / fsumf_(_2_0, sqrt(k2 + _1_0) * _2_0, k2))

274

275 @Property

276 def C4order(self):

277 '''Get the series expansion order (C{int}, 24, 27 or 30).

278 '''

279 return self._nC4

280

281 @C4order.setter # PYCHOK .setter!

282 def C4order(self, order):

283 '''Set the series expansion order (C{int}, 24, 27 or 30).

284

285 @raise GeodesicError: Invalid B{C{order}}.

286 '''

287 _xnC4(C4order=order)

288 if self._nC4 != order:

289 GeodesicExact._C4x._update(self)

290 _update_glXs(self) # zap cached _GeodesicLineExact attrs _B41, _C4a

291 self._nC4 = order

292

293 @deprecated_Property

294 def C4Order(self):

295 '''DEPRECATED, use property C{C4order}.

296 '''

297 return self.C4order

298

299 @C4Order.setter # PYCHOK .setter!

300 def C4Order(self, order):

301 '''DEPRECATED, use property C{C4order}.

302 '''

303 _xnC4(C4Order=order)

304 self.C4order = order

305

306 @Property_RO

307 def _C4x(self):

308 '''Get this ellipsoid's C{C4} coefficients, I{cached} tuple.

309

310 @see: Property L{C4order}.

311 '''

312 # see C4coeff() in GeographicLib.src.GeodesicExactC4.cpp

313 def _C4(nC4):

314 i, n, cs = 0, self.n, _C4coeffs(nC4)

315 _p = _polynomial

316 for r in range(nC4 + 1, 1, -1):

317 for j in range(1, r):

318 j = j + i # (j - i - 1) order of polynomial

319 yield _p(n, cs, i, j) / cs[j]

320 i = j + 1

321 # assert i == (nC4 * (nC4 + 1) * (nC4 + 5)) // 6

322

323 return tuple(_C4(self._nC4)) # 3rd flattening

324

325 @property_RO

326 def datum(self):

327 '''Get the datum (C{Datum}).

328 '''

329 return self._datum

330

331 def Direct(self, lat1, lon1, azi1, s12=0, outmask=Caps.STANDARD):

332 '''Solve the I{Direct} geodesic problem

333

334 @arg lat1: Latitude of the first point (C{degrees}).

335 @arg lon1: Longitude of the first point (C{degrees}).

336 @arg azi1: Azimuth at the first point (compass C{degrees}).

337 @arg s12: Distance between the points (C{meter}), can be negative.

338 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying

339 the quantities to be returned.

340

341 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,

342 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},

343 C{lon1}, C{azi1} and distance C{s12} always included.

344

345 @see: C++ U{GeodesicExact.Direct

346 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}

347 and Python U{Geodesic.Direct<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.

348 '''

349 return self._GDictDirect(lat1, lon1, azi1, False, s12, outmask)

350

351 def Direct3(self, lat1, lon1, azi1, s12): # PYCHOK outmask

352 '''Return the destination lat, lon and reverse azimuth

353 (final bearing) in C{degrees}.

354

355 @return: L{Destination3Tuple}C{(lat, lon, final)}.

356 '''

357 r = self._GDictDirect(lat1, lon1, azi1, False, s12, Caps._AZIMUTH_LATITUDE_LONGITUDE)

358 return Destination3Tuple(r.lat2, r.lon2, r.azi2) # no iteration

359

360 def _DirectLine(self, ll1, azi12, s12=0, **caps_name):

361 '''(INTERNAL) Short-cut version.

362 '''

363 return self.DirectLine(ll1.lat, ll1.lon, azi12, s12, **caps_name)

364

365 def DirectLine(self, lat1, lon1, azi1, s12, caps=Caps.STANDARD, name=NN):

366 '''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as distance.

367

368 @arg lat1: Latitude of the first point (C{degrees}).

369 @arg lon1: Longitude of the first point (C{degrees}).

370 @arg azi1: Azimuth at the first point (compass C{degrees}).

371 @arg s12: Distance between the points (C{meter}), can be negative.

372 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying

373 the capabilities the L{GeodesicLineExact} instance

374 should possess, i.e., which quantities can be

375 returned by calls to L{GeodesicLineExact.Position}.

376

377 @return: A L{GeodesicLineExact} instance.

378

379 @note: The third point of the L{GeodesicLineExact} is set to correspond

380 to the second point of the I{Inverse} geodesic problem.

381

382 @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}.

383

384 @see: C++ U{GeodesicExact.DirectLine

385 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and

386 Python U{Geodesic.DirectLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.

387 '''

388 return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps, name=name)

389

390 def _dn(self, sbet, cbet): # in gxline._GeodesicLineExact.__init__

391 '''(INTERNAL) Helper.

392 '''

393 if self.f < 0: # PYCHOK no cover

394 dn = sqrt(_1_0 - cbet**2 * self.e2) / self.f1

395 else:

396 dn = sqrt(_1_0 + sbet**2 * self.ep2)

397 return dn

398

399 @Property_RO

400 def e2(self):

401 '''Get the ellipsoid's I{(1st) eccentricity squared} (C{float}), M{f * (2 - f)}.

402 '''

403 return self.ellipsoid.e2

404

405 @Property_RO

406 def _e2a2(self):

407 '''(INTERNAL) Cache M{E.e2 * E.a2}.

408 '''

409 return self.e2 * self.ellipsoid.a2

410

411 @Property_RO

412 def _e2_f1(self):

413 '''(INTERNAL) Cache M{E.e2 * E.f1}.

414 '''

415 return self.e2 / self.f1

416

417 @Property_RO

418 def _eF(self):

419 '''(INTERNAL) Get the elliptic function, aka C{.E}.

420 '''

421 return _MODS.elliptic.Elliptic(k2=-self.ep2)

422

423 def _eF_reset_cHe2_f1(self, x, y):

424 '''(INTERNAL) Reset elliptic function and return M{cH * e2 / f1 * ...}.

425 '''

426 self._eF_reset_k2(x)

427 return y * self._eF.cH * self._e2_f1

428

429 def _eF_reset_k2(self, x):

430 '''(INTERNAL) Reset elliptic function and return C{k2}.

431 '''

432 ep2 = self.ep2

433 k2 = x**2 * ep2 # see .gxline._GeodesicLineExact._eF

434 self._eF.reset(k2=-k2, alpha2=-ep2) # kp2, alphap2 defaults

435 _update_glXs(self) # zap cached/memoized _GeodesicLineExact attrs

436 return k2

437

438 @Property_RO

439 def ellipsoid(self):

440 '''Get the ellipsoid (C{Ellipsoid}).

441 '''

442 return self.datum.ellipsoid

443

444 @Property_RO

445 def ep2(self):

446 '''Get the ellipsoid's I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)}.

447 '''

448 return self.ellipsoid.e22 # == self.e2 / self.f1**2

449

450 e22 = ep2 # for ellipsoid compatibility

451

452 @Property_RO

453 def _eTOL2(self):

454 '''(INTERNAL) The si12 threshold for "really short".

455 '''

456 return _eTOL2(self.f)

457

458 @Property_RO

459 def flattening(self):

460 '''Get the C{ellipsoid}'s I{flattening} (C{scalar}), M{(a - b) / a}, C{0} for spherical, negative for prolate.

461 '''

462 return self.ellipsoid.f

463

464 f = flattening

465

466 @Property_RO

467 def f1(self): # in .css.CassiniSoldner.reset

468 '''Get the C{ellipsoid}'s I{1 - flattening} (C{float}).

469 '''

470 return self.ellipsoid.f1

471

472 @Property_RO

473 def _f180(self):

474 '''(INTERNAL) Cached/memoized.

475 '''

476 return self.f * _180_0

477

478 def _GDictDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask=Caps.STANDARD):

479 '''(INTERNAL) As C{_GenDirect}, but returning a L{GDict}.

480

481 @return: A L{GDict} ...

482 '''

483 C = outmask if arcmode else (outmask | Caps.DISTANCE_IN)

484 glX = self.Line(lat1, lon1, azi1, C | Caps.LINE_OFF)

485 return glX._GDictPosition(arcmode, s12_a12, outmask)

486

487 def _GDictInverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): # MCCABE 33, 41 vars

488 '''(INTERNAL) As C{_GenInverse}, but returning a L{GDict}.

489

490 @return: A L{GDict} ...

491 '''

492 Cs = Caps

493 if self._debug: # PYCHOK no cover

494 outmask |= Cs._DEBUG_INVERSE & self._debug

495 outmask &= Cs._OUT_MASK # incl. _SALPs_CALPs and _DEBUG_

496 # compute longitude difference carefully (with _diff182):

497 # result is in [-180, +180] but -180 is only for west-going

498 # geodesics, +180 is for east-going and meridional geodesics

499 lon12, lon12s = _diff182(lon1, lon2)

500 # see C{result} from geographiclib.geodesic.Inverse

501 if (outmask & Cs.LONG_UNROLL): # == (lon1 + lon12) + lon12s

502 r = GDict(lon1=lon1, lon2=fsumf_(lon1, lon12, lon12s))

503 else:

504 r = GDict(lon1=_norm180(lon1), lon2=_norm180(lon2))

505 if _K_2_0: # GeographicLib 2.0

506 # make longitude difference positive

507 lon12, lon_ = _unsigned2(lon12)

508 if lon_:

509 lon12s = -lon12s

510 lam12 = radians(lon12)

511 # calculate sincosd(_around(lon12 + correction))

512 slam12, clam12 = _sincos2de(lon12, lon12s)

513 # supplementary longitude difference

514 lon12s = fsumf_(_180_0, -lon12, -lon12s)

515 else: # GeographicLib 1.52

516 # make longitude difference positive and if very close

517 # to being on the same half-meridian, then make it so.

518 if lon12 < 0: # _signBit(lon12)

519 lon_, lon12 = True, -_around(lon12)

520 lon12s = _around(fsumf_(_180_0, -lon12, lon12s))

521 else:

522 lon_, lon12 = False, _around(lon12)

523 lon12s = _around(fsumf_(_180_0, -lon12, -lon12s))

524 lam12 = radians(lon12)

525 if lon12 > _90_0:

526 slam12, clam12 = _sincos2d(lon12s)

527 clam12 = -clam12

528 else:

529 slam12, clam12 = _sincos2(lam12)

530 # If really close to the equator, treat as on equator.

531 lat1 = _around(_fix90(lat1))

532 lat2 = _around(_fix90(lat2))

533 r.set_(lat1=lat1, lat2=lat2)

534 # Swap points so that point with higher (abs) latitude is

535 # point 1. If one latitude is a NAN, then it becomes lat1.

536 swap_ = fabs(lat1) < fabs(lat2) or isnan(lat2)

537 if swap_:

538 lat1, lat2 = lat2, lat1

539 lon_ = not lon_

540 if _signBit(lat1):

541 lat_ = False # note, False

542 else: # make lat1 <= -0

543 lat_ = True # note, True

544 lat1, lat2 = -lat1, -lat2

545 # Now 0 <= lon12 <= 180, -90 <= lat1 <= -0 and lat1 <= lat2 <= -lat1

546 # and lat_, lon_, swap_ register the transformation to bring the

547 # coordinates to this canonical form, where False means no change

548 # made. We make these transformations so that there are few cases

549 # to check, e.g., on verifying quadrants in atan2. In addition,

550 # this enforces some symmetries in the results returned.

551

552 # Initialize for the meridian. No longitude calculation is

553 # done in this case to let the parameter default to 0.

554 sbet1, cbet1 = self._sinf1cos2d(lat1)

555 sbet2, cbet2 = self._sinf1cos2d(lat2)

556 # If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure

557 # of the |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1),

558 # abs(sbet2) + sbet1 is a better measure. This logic is used

559 # in assigning calp2 in _Lambda6. Sometimes these quantities

560 # vanish and in that case we force bet2 = +/- bet1 exactly. An

561 # example where is is necessary is the inverse problem

562 # 48.522876735459 0 -48.52287673545898293 179.599720456223079643

563 # which failed with Visual Studio 10 (Release and Debug)

564 if cbet1 < -sbet1:

565 if cbet2 == cbet1:

566 sbet2 = copysign(sbet1, sbet2)

567 elif fabs(sbet2) == -sbet1:

568 cbet2 = cbet1

569

570 p = _PDict(sbet1=sbet1, cbet1=cbet1, dn1=self._dn(sbet1, cbet1),

571 sbet2=sbet2, cbet2=cbet2, dn2=self._dn(sbet2, cbet2))

572

573 _meridian = _b = True # i.e. not meridian, not b

574 if lat1 == -90 or slam12 == 0:

575 # Endpoints are on a single full meridian,

576 # so the geodesic might lie on a meridian.

577 salp1, calp1 = slam12, clam12 # head to target lon

578 salp2, calp2 = _0_0, _1_0 # then head north

579 # tan(bet) = tan(sig) * cos(alp)

580 p.set_sigs(sbet1, calp1 * cbet1, sbet2, calp2 * cbet2)

581 # sig12 = sig2 - sig1

582 sig12 = _atan12(sbet1, p.csig1, sbet2, p.csig2, sineg0=True) # PYCHOK csig*

583 s12x, m12x, _, \

584 M12, M21 = self._Length5(sig12, outmask | Cs.REDUCEDLENGTH, p)

585 # Add the check for sig12 since zero length geodesics

586 # might yield m12 < 0. Test case was

587 # echo 20.001 0 20.001 0 | GeodSolve -i

588 # In fact, we will have sig12 > PI/2 for meridional

589 # geodesic which is not a shortest path.

590 if m12x >= 0 or sig12 < _1_0:

591 # Need at least 2 to handle 90 0 90 180

592 # Prevent negative s12 or m12 from geographiclib 1.52

593 if sig12 < _TINY3 or (sig12 < _TOL0 and (s12x < 0 or m12x < 0)):

594 sig12 = m12x = s12x = _0_0

595 else:

596 _b = False # apply .b to s12x, m12x

597 _meridian = False

598 C = 1

599 # else: # m12 < 0, prolate and too close to anti-podal

600 # _meridian = True

601 a12 = _0_0 # if _b else degrees(sig12)

602

603 if _meridian:

604 _b = sbet1 == 0 and (self.f <= 0 or lon12s >= self._f180) # and sbet2 == 0

605 if _b: # Geodesic runs along equator

606 calp1 = calp2 = _0_0

607 salp1 = salp2 = _1_0

608 sig12 = lam12 / self.f1 # == omg12

609 somg12, comg12 = _sincos2(sig12)

610 m12x = self.b * somg12

611 s12x = self.a * lam12

612 M12 = M21 = comg12

613 a12 = lon12 / self.f1

614 C = 2

615 else:

616 # Now point1 and point2 belong within a hemisphere bounded by a

617 # meridian and geodesic is neither meridional or equatorial.

618 p.set_(slam12=slam12, clam12=clam12)

619 # Figure a starting point for Newton's method

620 sig12, salp1, calp1, \

621 salp2, calp2, dnm = self._InverseStart6(lam12, p)

622 if sig12 is None: # use Newton's method

623 # pre-compute the constant _Lambda6 term, once

624 p.set_(bet12=None if cbet2 == cbet1 and fabs(sbet2) == -sbet1 else

625 (((cbet1 + cbet2) * (cbet2 - cbet1)) if cbet1 < -sbet1 else

626 ((sbet1 + sbet2) * (sbet1 - sbet2))))

627 sig12, salp1, calp1, \

628 salp2, calp2, domg12 = self._Newton6(salp1, calp1, p)

629 s12x, m12x, _, M12, M21 = self._Length5(sig12, outmask, p)

630 if (outmask & Cs.AREA):

631 # omg12 = lam12 - domg12

632 s, c = _sincos2(domg12)

633 somg12, comg12 = _sincos12(s, c, slam12, clam12)

634 C = 3 # Newton

635 else: # from _InverseStart6: dnm, salp*, calp*

636 C = 4 # Short lines

637 s, c = _sincos2(sig12 / dnm)

638 m12x = dnm**2 * s

639 s12x = dnm * sig12

640 M12 = M21 = c

641 if (outmask & Cs.AREA):

642 somg12, comg12 = _sincos2(lam12 / (self.f1 * dnm))

643

644 else: # _meridian is False

645 somg12 = comg12 = NAN

646

647 r.set_(a12=a12 if _b else degrees(sig12)) # in [0, 180]

648

649 if (outmask & Cs.DISTANCE):

650 r.set_(s12=unsigned0(s12x if _b else (self.b * s12x)))

651

652 if (outmask & Cs.REDUCEDLENGTH):

653 r.set_(m12=unsigned0(m12x if _b else (self.b * m12x)))

654

655 if (outmask & Cs.GEODESICSCALE):

656 if swap_:

657 M12, M21 = M21, M12

658 r.set_(M12=unsigned0(M12),

659 M21=unsigned0(M21))

660

661 if (outmask & Cs.AREA):

662 S12 = self._InverseArea(_meridian, salp1, calp1,

663 salp2, calp2,

664 somg12, comg12, p)

665 if _xor(swap_, lat_, lon_):

666 S12 = -S12

667 r.set_(S12=unsigned0(S12))

668

669 if (outmask & (Cs.AZIMUTH | Cs._SALPs_CALPs)):

670 if swap_:

671 salp1, salp2 = salp2, salp1

672 calp1, calp2 = calp2, calp1

673 if _xor(swap_, lon_):

674 salp1, salp2 = -salp1, -salp2

675 if _xor(swap_, lat_):

676 calp1, calp2 = -calp1, -calp2

677

678 if (outmask & Cs.AZIMUTH):

679 r.set_(azi1=_atan2d(salp1, calp1),

680 azi2=_atan2d_reverse(salp2, calp2, reverse=outmask & Cs.REVERSE2))

681 if (outmask & Cs._SALPs_CALPs):

682 r.set_(salp1=salp1, calp1=calp1,

683 salp2=salp2, calp2=calp2)

684

685 if (outmask & Cs._DEBUG_INVERSE): # PYCHOK no cover

686 E, eF = self.ellipsoid, self._eF

687 p.set_(C=C, a=self.a, f=self.f, f1=self.f1,

688 e=E.e, e2=self.e2, ep2=self.ep2,

689 c2=E.c2, c2x=self.c2x,

690 eFcD=eF.cD, eFcE=eF.cE, eFcH=eF.cH,

691 eFk2=eF.k2, eFa2=eF.alpha2)

692 p.update(r) # r overrides p

693 r = p.toGDict()

694 return self._iter2tion(r, p)

695

696 def _GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask=Caps.STANDARD):

697 '''(INTERNAL) The general I{Inverse} geodesic calculation.

698

699 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2,

700 s12, m12, M12, M21, S12)}.

701 '''

702 r = self._GDictDirect(lat1, lon1, azi1, arcmode, s12_a12, outmask)

703 return r.toDirect9Tuple()

704

705 def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12, caps, name=NN):

706 '''(INTERNAL) Helper for C{ArcDirectLine} and C{DirectLine}.

707

708 @return: A L{GeodesicLineExact} instance.

709 '''

710 azi1 = _norm180(azi1)

711 # guard against underflow in salp0. Also -0 is converted to +0.

712 s, c = _sincos2d(_around(azi1))

713 C = caps if arcmode else (caps | Caps.DISTANCE_IN)

714 return _GeodesicLineExact(self, lat1, lon1, azi1, C,

715 self._debug, s, c, name=name)._GenSet(arcmode, s12_a12)

716

717 def _GenInverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD):

718 '''(INTERNAL) The general I{Inverse} geodesic calculation.

719

720 @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2,

721 m12, M12, M21, S12)}.

722 '''

723 r = self._GDictInverse(lat1, lon1, lat2, lon2, outmask | Caps._SALPs_CALPs)

724 return r.toInverse10Tuple()

725

726 def _Inverse(self, ll1, ll2, wrap, **outmask):

727 '''(INTERNAL) Short-cut version, see .base.ellipsoidalDI.intersecant2.

728 '''

729 if wrap:

730 ll2 = _unrollon(ll1, _Wrap.point(ll2))

731 return self.Inverse(ll1.lat, ll1.lon, ll2.lat, ll2.lon, **outmask)

732

733 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD):

734 '''Perform the I{Inverse} geodesic calculation.

735

736 @arg lat1: Latitude of the first point (C{degrees}).

737 @arg lon1: Longitude of the first point (C{degrees}).

738 @arg lat2: Latitude of the second point (C{degrees}).

739 @arg lon2: Longitude of the second point (C{degrees}).

740 @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying

741 the quantities to be returned.

742

743 @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2,

744 lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1},

745 C{lon1}, C{azi1} and distance C{s12} always included.

746

747 @note: The third point of the L{GeodesicLineExact} is set to correspond

748 to the second point of the I{Inverse} geodesic problem.

749

750 @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}.

751

752 @see: C++ U{GeodesicExact.InverseLine

753 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and

754 Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.

755 '''

756 return self._GDictInverse(lat1, lon1, lat2, lon2, outmask)

757

758 def Inverse1(self, lat1, lon1, lat2, lon2, wrap=False):

759 '''Return the non-negative, I{angular} distance in C{degrees}.

760

761 @kwarg wrap: If C{True}, wrap or I{normalize} and unroll

762 B{C{lat2}} and B{C{lon2}} (C{bool}).

763 '''

764 # see .FrechetKarney.distance, .HausdorffKarney._distance

765 # and .HeightIDWkarney._distances

766 if wrap:

767 _, lat2, lon2 = _Wrap.latlon3(lat1, lat2, lon2, True) # _Geodesic.LONG_UNROLL

768 return fabs(self._GDictInverse(lat1, lon1, lat2, lon2, Caps._ANGLE_ONLY).a12)

769

770 def Inverse3(self, lat1, lon1, lat2, lon2): # PYCHOK outmask

771 '''Return the distance in C{meter} and the forward and

772 reverse azimuths (initial and final bearing) in C{degrees}.

773

774 @return: L{Distance3Tuple}C{(distance, initial, final)}.

775 '''

776 r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps.AZIMUTH_DISTANCE)

777 return Distance3Tuple(r.s12, wrap360(r.azi1), wrap360(r.azi2),

778 iteration=r.iteration)

779

780 def _InverseLine(self, ll1, ll2, wrap, **caps_name):

781 '''(INTERNAL) Short-cut version.

782 '''

783 if wrap:

784 ll2 = _unrollon(ll1, _Wrap.point(ll2))

785 return self.InverseLine(ll1.lat, ll1.lon, ll2.lat, ll2.lon, **caps_name)

786

787 def InverseLine(self, lat1, lon1, lat2, lon2, caps=Caps.STANDARD, name=NN):

788 '''Define a L{GeodesicLineExact} in terms of the I{Inverse} geodesic problem.

789

790 @arg lat1: Latitude of the first point (C{degrees}).

791 @arg lon1: Longitude of the first point (C{degrees}).

792 @arg lat2: Latitude of the second point (C{degrees}).

793 @arg lon2: Longitude of the second point (C{degrees}).

794 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying

795 the capabilities the L{GeodesicLineExact} instance

796 should possess, i.e., which quantities can be

797 returned by calls to L{GeodesicLineExact.Position}

798 and L{GeodesicLineExact.ArcPosition}.

799

800 @return: A L{GeodesicLineExact} instance.

801

802 @note: The third point of the L{GeodesicLineExact} is set to correspond

803 to the second point of the I{Inverse} geodesic problem.

804

805 @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}.

806

807 @see: C++ U{GeodesicExact.InverseLine

808 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} and

809 Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.

810 '''

811 Cs = Caps

812 r = self._GDictInverse(lat1, lon1, lat2, lon2, Cs._SALPs_CALPs) # No need for AZIMUTH

813 C = (caps | Cs.DISTANCE) if (caps & (Cs.DISTANCE_IN & Cs._OUT_MASK)) else caps

814 azi1 = _atan2d(r.salp1, r.calp1)

815 return _GeodesicLineExact(self, lat1, lon1, azi1, C, # ensure a12 is distance

816 self._debug, r.salp1, r.calp1, name=name)._GenSet(True, r.a12)

817

818 def _InverseArea(self, _meridian, salp1, calp1, # PYCHOK 9 args

819 salp2, calp2,

820 somg12, comg12, p):

821 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length.

822

823 @return: Area C{S12}.

824 '''

825 # from _Lambda6: sin(alp1) * cos(bet1) = sin(alp0), calp0 > 0

826 salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1)

827 A4 = calp0 * salp0

828 if A4:

829 # from _Lambda6: tan(bet) = tan(sig) * cos(alp)

830 k2 = calp0**2 * self.ep2

831 C4a = self._C4f_k2(k2)

832 B41 = _cosSeries(C4a, *_norm2(p.sbet1, calp1 * p.cbet1))

833 B42 = _cosSeries(C4a, *_norm2(p.sbet2, calp2 * p.cbet2))

834 # multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0)

835 A4 *= self._e2a2

836 S12 = A4 * (B42 - B41)

837 else: # avoid problems with indeterminate sig1, sig2 on equator

838 A4 = B41 = B42 = k2 = S12 = _0_0

839

840 if (_meridian and # omg12 < 3/4 * PI

841 comg12 > -_SQRT2_2 and # lon diff not too big

842 (p.sbet2 - p.sbet1) < _1_75): # lat diff not too big

843 # use tan(Gamma/2) = tan(omg12/2) *

844 # (tan(bet1/2) + tan(bet2/2)) /

845 # (tan(bet1/2) * tan(bet2/2) + 1))

846 # with tan(x/2) = sin(x) / (1 + cos(x))

847 dbet1 = p.cbet1 + _1_0

848 dbet2 = p.cbet2 + _1_0

849 domg12 = comg12 + _1_0

850 salp12 = (p.sbet1 * dbet2 + dbet1 * p.sbet2) * somg12

851 calp12 = (p.sbet1 * p.sbet2 + dbet1 * dbet2) * domg12

852 alp12 = atan2(salp12, calp12) * _2_0

853 else:

854 # alp12 = alp2 - alp1, used in atan2, no need to normalize

855 salp12, calp12 = _sincos12(salp1, calp1, salp2, calp2)

856 # The right thing appears to happen if alp1 = +/-180 and

857 # alp2 = 0, viz salp12 = -0 and alp12 = -180. However,

858 # this depends on the sign being attached to 0 correctly.

859 # Following ensures the correct behavior.

860 if salp12 == 0 and calp12 < 0:

861 alp12 = _copysign(PI, calp1)

862 else:

863 alp12 = atan2(salp12, calp12)

864

865 p.set_(alp12=alp12, A4=A4, B41=B41, B42=B42, k2=k2)

866 return S12 + self.c2x * alp12

867

868 def _InverseStart6(self, lam12, p):

869 '''(INTERNAL) Return a starting point for Newton's method in

870 C{salp1} and C{calp1} indicated by C{sig12=None}. If

871 Newton's method doesn't need to be used, return also

872 C{salp2}, C{calp2}, C{dnm} and C{sig12} non-C{None}.

873

874 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, dnm)}

875 and C{p.set_sigs} updated for Newton, C{sig12=None}.

876 '''

877 sig12 = None # use Newton

878 salp1 = calp1 = salp2 = calp2 = dnm = NAN

879

880 # bet12 = bet2 - bet1 in [0, PI)

881 sbet12, cbet12 = _sincos12(p.sbet1, p.cbet1, p.sbet2, p.cbet2)

882 shortline = cbet12 >= 0 and sbet12 < _0_5 and (p.cbet2 * lam12) < _0_5

883 if shortline:

884 # sin((bet1 + bet2)/2)^2 = (sbet1 + sbet2)^2 / (

885 # (cbet1 + cbet2)^2 + (sbet1 + sbet2)^2)

886 t = (p.sbet1 + p.sbet2)**2

887 s = t / ((p.cbet1 + p.cbet2)**2 + t)

888 dnm = sqrt(_1_0 + self.ep2 * s)

889 somg12, comg12 = _sincos2(lam12 / (self.f1 * dnm))

890 else:

891 somg12, comg12 = p.slam12, p.clam12

892

893 # bet12a = bet2 + bet1 in (-PI, 0], note -sbet1

894 sbet12a, cbet12a = _sincos12(-p.sbet1, p.cbet1, p.sbet2, p.cbet2)

895

896 c = fabs(comg12) + _1_0 # == (1 - comg12) if comg12 < 0

897 s = somg12**2 / c

898 t = p.sbet1 * p.cbet2 * s

899 salp1 = p.cbet2 * somg12

900 calp1 = (sbet12a - t) if comg12 < 0 else (sbet12 + t)

901

902 ssig12 = _hypot(salp1, calp1)

903 csig12 = p.sbet1 * p.sbet2 + p.cbet1 * p.cbet2 * comg12

904

905 if shortline and ssig12 < self._eTOL2: # really short lines

906 t = c if comg12 < 0 else s

907 salp2, calp2 = _norm2(somg12 * p.cbet1,

908 sbet12 - p.cbet1 * p.sbet2 * t)

909 sig12 = atan2(ssig12, csig12) # do not use Newton

910

911 elif (self._n_0_1 or # Skip astroid calc if too eccentric

912 csig12 >= 0 or ssig12 >= (p.cbet1**2 * self._n6PI)):

913 pass # nothing to do, 0th order spherical approximation OK

914

915 else:

916 # Scale lam12 and bet2 to x, y coordinate system where antipodal

917 # point is at origin and singular point is at y = 0, x = -1

918 lam12x = atan2(-p.slam12, -p.clam12) # lam12 - PI

919 f = self.f

920 if f < 0: # PYCHOK no cover

921 # ssig1=sbet1, csig1=-cbet1, ssig2=sbet2, csig2=cbet2

922 p.set_sigs(p.sbet1, -p.cbet1, p.sbet2, p.cbet2)

923 # if lon12 = 180, this repeats a calculation made in Inverse

924 _, m12b, m0, _, _ = self._Length5(atan2(sbet12a, cbet12a) + PI,

925 Caps.REDUCEDLENGTH, p)

926 t = p.cbet1 * PI # x = dlat, y = dlon

927 x = m12b / (t * p.cbet2 * m0) - _1_0

928 sca = (sbet12a / (x * p.cbet1)) if x < -_0_01 else (-f * t)

929 y = lam12x / sca

930 else: # f >= 0, however f == 0 does not get here

931 sca = self._eF_reset_cHe2_f1(p.sbet1, p.cbet1 * _2_0)

932 x = lam12x / sca # dlon

933 y = sbet12a / (sca * p.cbet1) # dlat

934

935 if y > _TOL1 and x > -_THR1: # strip near cut

936 if f < 0: # PYCHOK no cover

937 calp1 = max( _0_0, x) if x > _TOL1 else max(_N_1_0, x)

938 salp1 = sqrt(_1_0 - calp1**2)

939 else:

940 salp1 = min( _1_0, -x)

941 calp1 = -sqrt(_1_0 - salp1**2)

942 else:

943 # Estimate alp1, by solving the astroid problem.

944 #

945 # Could estimate alpha1 = theta + PI/2, directly, i.e.,

946 # calp1 = y/k; salp1 = -x/(1+k); for _f >= 0

947 # calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check)

948 #

949 # However, it's better to estimate omg12 from astroid and use

950 # spherical formula to compute alp1. This reduces the mean

951 # number of Newton iterations for astroid cases from 2.24

952 # (min 0, max 6) to 2.12 (min 0, max 5). The changes in the

953 # number of iterations are as follows:

954 #

955 # change percent

956 # 1 5

957 # 0 78

958 # -1 16

959 # -2 0.6

960 # -3 0.04

961 # -4 0.002

962 #

963 # The histogram of iterations is (m = number of iterations

964 # estimating alp1 directly, n = number of iterations

965 # estimating via omg12, total number of trials = 148605):

966 #

967 # iter m n

968 # 0 148 186

969 # 1 13046 13845

970 # 2 93315 102225

971 # 3 36189 32341

972 # 4 5396 7

973 # 5 455 1

974 # 6 56 0

975 #

976 # omg12 is near PI, estimate work with omg12a = PI - omg12

977 k = _Astroid(x, y)

978 sca *= (y * (k + _1_0) / k) if f < 0 else \

979 (x * k / (k + _1_0))

980 s, c = _sincos2(-sca) # omg12a

981 # update spherical estimate of alp1 using omg12 instead of lam12

982 salp1 = p.cbet2 * s

983 calp1 = sbet12a - s * salp1 * p.sbet1 / (c + _1_0) # c = -c

984

985 # sanity check on starting guess. Backwards check allows NaN through.

986 salp1, calp1 = _norm2(salp1, calp1) if salp1 > 0 else (_1_0, _0_0)

987

988 return sig12, salp1, calp1, salp2, calp2, dnm

989

990 def _Lambda6(self, salp1, calp1, diffp, p):

991 '''(INTERNAL) Helper.

992

993 @return: 6-Tuple C{(lam12, sig12, salp2, calp2, domg12,

994 dlam12} and C{p.set_sigs} updated.

995 '''

996 eF = self._eF

997 f1 = self.f1

998

999 if p.sbet1 == calp1 == 0: # PYCHOK no cover

1000 # Break degeneracy of equatorial line

1001 calp1 = -_TINY

1002

1003 # sin(alp1) * cos(bet1) = sin(alp0), # calp0 > 0

1004 salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1)

1005 # tan(bet1) = tan(sig1) * cos(alp1)

1006 # tan(omg1) = sin(alp0) * tan(sig1)

1007 # = sin(bet1) * tan(alp1)

1008 somg1 = salp0 * p.sbet1

1009 comg1 = calp1 * p.cbet1

1010 ssig1, csig1 = _norm2(p.sbet1, comg1)

1011 # Without normalization we have schi1 = somg1

1012 cchi1 = f1 * p.dn1 * comg1

1013

1014 # Enforce symmetries in the case abs(bet2) = -bet1.

1015 # Need to be careful about this case, since this can

1016 # yield singularities in the Newton iteration.

1017 # sin(alp2) * cos(bet2) = sin(alp0)

1018 salp2 = (salp0 / p.cbet2) if p.cbet2 != p.cbet1 else salp1

1019 # calp2 = sqrt(1 - sq(salp2))

1020 # = sqrt(sq(calp0) - sq(sbet2)) / cbet2

1021 # and subst for calp0 and rearrange to give (choose

1022 # positive sqrt to give alp2 in [0, PI/2]).

1023 calp2 = fabs(calp1) if p.bet12 is None else (

1024 sqrt((calp1 * p.cbet1)**2 + p.bet12) / p.cbet2)

1025 # tan(bet2) = tan(sig2) * cos(alp2)

1026 # tan(omg2) = sin(alp0) * tan(sig2).

1027 somg2 = salp0 * p.sbet2

1028 comg2 = calp2 * p.cbet2

1029 ssig2, csig2 = _norm2(p.sbet2, comg2)

1030 # without normalization we have schi2 = somg2

1031 cchi2 = f1 * p.dn2 * comg2

1032

1033 # omg12 = omg2 - omg1, limit to [0, PI]

1034 somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, sineg0=True)

1035 # chi12 = chi2 - chi1, limit to [0, PI]

1036 schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, sineg0=True)

1037

1038 p.set_sigs(ssig1, csig1, ssig2, csig2)

1039 # sig12 = sig2 - sig1, limit to [0, PI]

1040 sig12 = _atan12(ssig1, csig1, ssig2, csig2, sineg0=True)

1041

1042 eta12 = self._eF_reset_cHe2_f1(calp0, salp0) * _2__PI # then ...

1043 eta12 *= fsum1f_(eF.deltaH(*p.sncndn2),

1044 -eF.deltaH(*p.sncndn1), sig12)

1045 # eta = chi12 - lam12

1046 lam12 = _atan12(p.slam12, p.clam12, schi12, cchi12) - eta12

1047 # domg12 = chi12 - omg12 - deta12

1048 domg12 = _atan12( somg12, comg12, schi12, cchi12) - eta12

1049

1050 dlam12 = NAN # dv > 0 in ._Newton6

1051 if diffp:

1052 d = calp2 * p.cbet2

1053 if d:

1054 _, dlam12, _, _, _ = self._Length5(sig12, Caps.REDUCEDLENGTH, p)

1055 dlam12 *= f1 / d

1056 elif p.sbet1:

1057 dlam12 = -f1 * p.dn1 * _2_0 / p.sbet1

1058

1059 # p.set_(deta12=-eta12, lam12=lam12)

1060 return lam12, sig12, salp2, calp2, domg12, dlam12

1061

1062 def _Length5(self, sig12, outmask, p):

1063 '''(INTERNAL) Return M{m12b = (reduced length) / self.b} and

1064 calculate M{s12b = distance / self.b} and M{m0}, the

1065 coefficient of secular term in expression for reduced

1066 length and the geodesic scales C{M12} and C{M21}.

1067

1068 @return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}.

1069 '''

1070 s12b = m12b = m0 = M12 = M21 = NAN

1071

1072 Cs = Caps

1073 eF = self._eF

1074

1075 # outmask &= Cs._OUT_MASK

1076 if (outmask & Cs.DISTANCE):

1077 # Missing a factor of self.b

1078 s12b = eF.cE * _2__PI * fsum1f_(eF.deltaE(*p.sncndn2),

1079 -eF.deltaE(*p.sncndn1), sig12)

1080

1081 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE):

1082 m0x = -eF.k2 * eF.cD * _2__PI

1083 J12 = -m0x * fsum1f_(eF.deltaD(*p.sncndn2),

1084 -eF.deltaD(*p.sncndn1), sig12)

1085 if (outmask & Cs.REDUCEDLENGTH):

1086 m0 = m0x

1087 # Missing a factor of self.b. Add parens around

1088 # (csig1 * ssig2) and (ssig1 * csig2) to ensure

1089 # accurate cancellation for coincident points.

1090 m12b = fsum1f_(p.dn2 * (p.csig1 * p.ssig2),

1091 -p.dn1 * (p.ssig1 * p.csig2),

1092 J12 * (p.csig1 * p.csig2))

1093 if (outmask & Cs.GEODESICSCALE):

1094 M12 = M21 = p.ssig1 * p.ssig2 + \

1095 p.csig1 * p.csig2

1096 t = (p.cbet1 - p.cbet2) * self.ep2 * \

1097 (p.cbet1 + p.cbet2) / (p.dn1 + p.dn2)

1098 M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1

1099 M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2

1100

1101 return s12b, m12b, m0, M12, M21

1102

1103 def Line(self, lat1, lon1, azi1, caps=Caps.ALL, name=NN):

1104 '''Set up a L{GeodesicLineExact} to compute several points

1105 on a single geodesic.

1106

1107 @arg lat1: Latitude of the first point (C{degrees}).

1108 @arg lon1: Longitude of the first point (C{degrees}).

1109 @arg azi1: Azimuth at the first point (compass C{degrees}).

1110 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying

1111 the capabilities the L{GeodesicLineExact} instance

1112 should possess, i.e., which quantities can be

1113 returnedby calls to L{GeodesicLineExact.Position}

1114 and L{GeodesicLineExact.ArcPosition}.

1115

1116 @return: A L{GeodesicLineExact} instance.

1117

1118 @note: If the point is at a pole, the azimuth is defined by keeping

1119 B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking

1120 the limit C{ε → 0+}.

1121

1122 @see: C++ U{GeodesicExact.Line

1123 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}

1124 and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.

1125 '''

1126 return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug, name=name)

1127

1128 @Property_RO

1129 def n(self):

1130 '''Get the C{ellipsoid}'s I{3rd flattening} (C{scalar}), M{f / (2 - f) == (a - b) / (a + b)}.

1131 '''

1132 return self.ellipsoid.n

1133

1134 @Property_RO

1135 def _n_0_1(self):

1136 '''(INTERNAL) Cached once.

1137 '''

1138 return fabs(self.n) > _0_1

1139

1140 @Property_RO

1141 def _n6PI(self):

1142 '''(INTERNAL) Cached once.

1143 '''

1144 return fabs(self.n) * _6_0 * PI

1145

1146 def _Newton6(self, salp1, calp1, p):

1147 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length.

1148

1149 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12)}

1150 and C{p.iter} and C{p.trip} updated.

1151 '''

1152 _abs = fabs

1153 # This is a straightforward solution of f(alp1) = lambda12(alp1) -

1154 # lam12 = 0 with one wrinkle. f(alp) has exactly one root in the

1155 # interval (0, PI) and its derivative is positive at the root.

1156 # Thus f(alp) is positive for alp > alp1 and negative for alp < alp1.

1157 # During the course of the iteration, a range (alp1a, alp1b) is

1158 # maintained which brackets the root and with each evaluation of

1159 # f(alp) the range is shrunk, if possible. Newton's method is

1160 # restarted whenever the derivative of f is negative (because the

1161 # new value of alp1 is then further from the solution) or if the

1162 # new estimate of alp1 lies outside (0,PI); in this case, the new

1163 # starting guess is taken to be (alp1a + alp1b) / 2.

1164 salp1a = salp1b = _TINY

1165 calp1a, calp1b = _1_0, _N_1_0

1166 MAXIT1, TOL0 = _MAXIT1, _TOL0

1167 HALF, TOLb = _0_5, _TOLb

1168 tripb, TOLv = False, TOL0

1169 for i in range(_MAXIT2):

1170 # 1/4 meridian = 10e6 meter and random input,

1171 # estimated max error in nm (nano meter, by

1172 # checking Inverse problem by Direct).

1173 #

1174 # max iterations

1175 # log2(b/a) error mean sd

1176 # -7 387 5.33 3.68

1177 # -6 345 5.19 3.43

1178 # -5 269 5.00 3.05

1179 # -4 210 4.76 2.44

1180 # -3 115 4.55 1.87

1181 # -2 69 4.35 1.38

1182 # -1 36 4.05 1.03

1183 # 0 15 0.01 0.13

1184 # 1 25 5.10 1.53

1185 # 2 96 5.61 2.09

1186 # 3 318 6.02 2.74

1187 # 4 985 6.24 3.22

1188 # 5 2352 6.32 3.44

1189 # 6 6008 6.30 3.45

1190 # 7 19024 6.19 3.30

1191 v, sig12, salp2, calp2, \

1192 domg12, dv = self._Lambda6(salp1, calp1, i < MAXIT1, p)

1193

1194 # 2 * _TOL0 is approximately 1 ulp [0, PI]

1195 # reversed test to allow escape with NaNs

1196 if tripb or _abs(v) < TOLv:

1197 break

1198 # update bracketing values

1199 if v > 0 and (i > MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)):

1200 salp1b, calp1b = salp1, calp1

1201 elif v < 0 and (i > MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)):

1202 salp1a, calp1a = salp1, calp1

1203

1204 if i < MAXIT1 and dv > 0:

1205 dalp1 = -v / dv

1206 if _abs(dalp1) < PI:

1207 s, c = _sincos2(dalp1)

1208 # nalp1 = alp1 + dalp1

1209 s, c = _sincos12(-s, c, salp1, calp1)

1210 if s > 0:

1211 salp1, calp1 = _norm2(s, c)

1212 # in some regimes we don't get quadratic convergence

1213 # because slope -> 0. So use convergence conditions

1214 # based on epsilon instead of sqrt(epsilon)

1215 TOLv = TOL0 if _abs(v) > _TOL016 else _TOL08

1216 continue

1217 TOLv = TOL0

1218 # Either dv was not positive or updated value was outside

1219 # legal range. Use the midpoint of the bracket as the next

1220 # estimate. This mechanism is not needed for the WGS84

1221 # ellipsoid, but it does catch problems with more eccentric

1222 # ellipsoids. Its efficacy is such for the WGS84 test set

1223 # with the starting guess set to alp1 = 90 deg: the WGS84

1224 # test set: mean = 5.21, stdev = 3.93, max = 24 and WGS84

1225 # with random input: mean = 4.74, stdev = 0.99

1226 salp1, calp1 = _norm2((salp1a + salp1b) * HALF,

1227 (calp1a + calp1b) * HALF)

1228 tripb = fsum1f_(calp1a, -calp1, _abs(salp1a - salp1)) < TOLb or \

1229 fsum1f_(calp1b, -calp1, _abs(salp1b - salp1)) < TOLb

1230 else:

1231 raise GeodesicError(Fmt.no_convergence(v, TOLv), txt=repr(self)) # self.toRepr()

1232

1233 p.set_(iter=i, trip=tripb) # like .geodsolve._GDictInvoke: iter NOT iteration!

1234 return sig12, salp1, calp1, salp2, calp2, domg12

1235

1236 Polygon = Area # for C{geographiclib} compatibility

1237

1238 def _sinf1cos2d(self, lat):

1239 '''(INTERNAL) Helper, also for C{_G_GeodesicLineExact}.

1240 '''

1241 sbet, cbet = _sincos2d(lat)

1242 # ensure cbet1 = +epsilon at poles; doing the fix on beta means

1243 # that sig12 will be <= 2*tiny for two points at the same pole

1244 sbet, cbet = _norm2(sbet * self.f1, cbet)

1245 return sbet, (cbet if fabs(cbet) > _TINY else _TINY)

1246

1247 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature

1248 '''Return this C{GeodesicExact} as string.

1249

1250 @kwarg prec: The C{float} precision, number of decimal digits (0..9).

1251 Trailing zero decimals are stripped for B{C{prec}} values

1252 of 1 and above, but kept for negative B{C{prec}} values.

1253 @kwarg sep: Separator to join (C{str}).

1254

1255 @return: Tuple items (C{str}).

1256 '''

1257 d = dict(ellipsoid=self.ellipsoid, C4order=self.C4order)

1258 return sep.join(pairs(d, prec=prec))

1259

1260

1261 class GeodesicLineExact(_GeodesicLineExact):

1262 '''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact

1263 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicLineExact.html>},

1264 modeled after I{Karney}'s Python class U{geodesicline.GeodesicLine<https://GitHub.com/

1265 geographiclib/geographiclib-python>}.

1266 '''

1267

1268 def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, name=NN):

1269 '''New L{GeodesicLineExact} instance, allowing points to be found along

1270 a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}.

1271

1272 @arg geodesic: The geodesic to use (L{GeodesicExact}).

1273 @arg lat1: Latitude of the first point (C{degrees}).

1274 @arg lon1: Longitude of the first point (C{degrees}).

1275 @arg azi1: Azimuth at the first points (compass C{degrees}).

1276 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying

1277 the capabilities the L{GeodesicLineExact} instance

1278 should possess, i.e., which quantities can be

1279 returned by calls to L{GeodesicLineExact.Position}

1280 and L{GeodesicLineExact.ArcPosition}.

1281 @kwarg name: Optional name (C{str}).

1282

1283 @raise TypeError: Invalid B{C{geodesic}}.

1284 '''

1285 _xinstanceof(GeodesicExact, geodesic=geodesic)

1286 if (caps & Caps.LINE_OFF): # copy to avoid updates

1287 geodesic = geodesic.copy(deep=False, name=NN(_UNDER_, geodesic.name))

1288# _update_all(geodesic)

1289 _GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, 0, name=name)

1290

1291

1292 def _Astroid(x, y):

1293 '''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1)

1294 * k^2 - (2 * k + 1) * y^2 = 0} for positive root k.

1295 '''

1296 p = x**2

1297 q = y**2

1298 r = fsumf_(_1_0, q, p, _N_2_0)

1299 if r > 0 or q:

1300 # avoid possible division by zero when r = 0

1301 # by multiplying s and t by r^3 and r, resp.

1302 S = p * q / _4_0 # S = r^3 * s

1303 if r:

1304 r = r / _6_0 # /= chokes PyChecker

1305 r3 = r**3

1306 T3 = r3 + S

1307 # discriminant of the quadratic equation for T3 is

1308 # zero on the evolute curve p^(1/3) + q^(1/3) = 1

1309 d = (r3 + T3) * S

1310 if d < 0:

1311 # T is complex, but u is defined for a real result

1312 a = atan2(sqrt(-d), -T3) / _3_0

1313 # There are 3 possible cube roots, choose the one which

1314 # avoids cancellation. Note d < 0 implies that r < 0.

1315 u = (cos(a) * _2_0 + _1_0) * r

1316 else:

1317 # pick the sign on the sqrt to maximize abs(T3) to

1318 # minimize loss of precision due to cancellation.

1319 if d:

1320 T3 += _copysign(sqrt(d), T3) # T3 = (r * t)^3

1321 # _cbrt always returns the real root, _cbrt(-8) = -2

1322 u = _cbrt(T3) # T = r * t

1323 if u: # T can be zero; but then r2 / T -> 0

1324 u += r**2 / u

1325 u += r

1326 elif S: # d == T3**2 == S**2: sqrt(d) == abs(S) == abs(T3)

1327 u = _cbrt(S * _2_0) # == T3 + _copysign(abs(S), T3)

1328 else:

1329 u = _0_0

1330 v = _hypot(u, y) # sqrt(u**2 + q)

1331 # avoid loss of accuracy when u < 0

1332 u = (q / (v - u)) if u < 0 else (v + u)

1333 w = (u - q) / (v + v) # positive?

1334 # rearrange expression for k to avoid loss of accuracy due to

1335 # subtraction, division by 0 impossible because u > 0, w >= 0

1336 k = u / (sqrt(w**2 + u) + w) # guaranteed positive

1337

1338 else: # q == 0 && r <= 0

1339 # y = 0 with |x| <= 1. Handle this case directly, for

1340 # y small, positive root is k = abs(y) / sqrt(1 - x^2)

1341 k = _0_0

1342

1343 return k

1344

1345

1346 def _C4coeffs(nC4): # in .geodesicx.__main__

1347 '''(INTERNAL) Get the C{C4_24}, C{_27} or C{_30} series coefficients.

1348 '''

1349 try: # from pygeodesy.geodesicx._C4_xx import _coeffs_xx as _coeffs

1350 _C4_xx = _DOT_(_MODS.geodesicx.__name__, _UNDER_('_C4', nC4))

1351 _coeffs = _MODS.getattr(_C4_xx, _UNDER_('_coeffs', nC4))

1352 except (AttributeError, ImportError, TypeError) as x:

1353 raise GeodesicError(nC4=nC4, cause=x)

1354 n = _xnC4(nC4=nC4)

1355 if len(_coeffs) != n: # double check

1356 raise GeodesicError(_coeffs=len(_coeffs), _xnC4=n, nC4=nC4)

1357 return _coeffs

1358

1359

1360__all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact)

1361

1362# **) MIT License

1363#

1364# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1365#

1366# Permission is hereby granted, free of charge, to any person obtaining a

1367# copy of this software and associated documentation files (the "Software"),

1368# to deal in the Software without restriction, including without limitation

1369# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1370# and/or sell copies of the Software, and to permit persons to whom the

1371# Software is furnished to do so, subject to the following conditions:

1372#

1373# The above copyright notice and this permission notice shall be included

1374# in all copies or substantial portions of the Software.

1375#

1376# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1377# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1378# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1379# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1380# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1381# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1382# OTHER DEALINGS IN THE SOFTWARE.