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>} (2008-2024)

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 _copysign, _xinstanceof, _xor, unsigned0, typename

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.errors import GeodesicError, _xkwds_pop2

47 from pygeodesy.fmath import hypot as _hypot, Fmt

48 from pygeodesy.fsums import fsumf_, fsum1f_

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

50 _sincos12, _sin1cos2, _sinf1cos2d, \

51 _TINY, _xnC4

52 from pygeodesy.geodesicx.gxline import _GeodesicLineExact, _update_glXs

53# from pygeodesy.internals import typename # from .basics

54 from pygeodesy.interns import NN, _DOT_, _UNDER_

55 from pygeodesy.karney import GDict, _around, _atan2d, Caps, _cbrt, _diff182, \

56 _fix90, _K_2_0, _norm2, _norm180, _polynomial, \

57 _signBit, _sincos2, _sincos2d, _sincos2de, _unsigned2

58 from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS

59 from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple

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

61# from pygeodesy.streprs import Fmt # from .fmath

62 from pygeodesy.utily import atan2, atan2d as _atan2d_reverse, _unrollon, \

63 _Wrap, wrap360

64

65 from math import copysign, cos, degrees, fabs, radians, sqrt

66

67__all__ = ()

68__version__ = '25.04.14'

69

70_MAXIT1 = 20

71_MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits

72

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

74# case 52.784459512564 0 -52.784459512563990912 179.634407464943777557

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

76_TOL0 = EPS

77_TOL1 = _TOL0 * -200 # negative

78_TOL2 = _EPSqrt # == sqrt(_TOL0)

79_TOL3 = _TOL2 * _0_1

80_TOLb = _TOL2 * _TOL0 # Check on bisection interval

81_THR1 = _TOL2 * _1000_0 + _1_0

82

83_TINY3 = _TINY * _3_0

84_TOL08 = _TOL0 * _8_0

85_TOL016 = _TOL0 * _16_0

86

87

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

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

90 '''

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

92

93

94 def _eTOL2(f):

95 # Using the auxiliary sphere solution with dnm computed at

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

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

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

99

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

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

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

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

104

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

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

107 # nearly spherical case.

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

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

110

111

112 class _PDict(GDict):

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

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

115 '''

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

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

118 '''

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

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

121

122 def toGDict(self): # PYCHOK no cover

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

124 '''

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

126 return GDict(rest)

127

128 return _rest(**self)

129

130

131 class GeodesicExact(_GeodesicBase):

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

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

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

135 geographiclib/geographiclib-python>}.

136 '''

137 _datum = _WGS84

138 _nC4 = 30 # default C4order

139

140 def __init__(self, a_ellipsoid=_EWGS84, f=None, C4order=None, **name_C4Order): # for backward compatibility

141 '''New L{GeodesicExact} instance.

142

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

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

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

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

147 is specified as C{scalar}.

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

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

150 @kwarg name_C4Order: Optional C{B{name}=NN} (C{str}) and the DEPRECATED

151 keyword argument C{C4Order}, use B{C{C4order}} instead.

152

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

154 '''

155 if name_C4Order:

156 C4order, name = _xkwds_pop2(name_C4Order, C4Order=C4order)

157 if name:

158 self.name = name

159 else:

160 name = {} # name_C4Order

161

162 _earth_datum(self, a_ellipsoid, f=f, **name)

163 if C4order: # XXX private copy, always?

164 self.C4order = C4order

165

166 @Property_RO

167 def a(self):

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

169 '''

170 return self.ellipsoid.a

171

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

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

174

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

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

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

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

179 @kwarg outmask: Bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>}

180 values specifying the quantities to be returned.

181

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

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

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

185

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

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

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

189 '''

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

191

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

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

194

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

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

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

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

199 @kwarg caps: Bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>} values

200 specifying the capabilities the L{GeodesicLineExact} instance

201 should possess, i.e., which quantities can be returned by methods

202 L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}.

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

204

205 @return: A L{GeodesicLineExact} instance.

206

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

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

209

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

211

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

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

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

215 '''

216 return GeodesicLineExact(self, lat1, lon1, azi1, caps=caps, **name)._GenSet(self._debug, a12=a12)

217

218 def Area(self, polyline=False, **name):

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

220 perimeter of a polygon.

221

222 @kwarg polyline: If C{True}, compute the perimeter only, otherwise

223 the perimeter and area (C{bool}).

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

225

226 @return: A L{GeodesicAreaExact} instance.

227

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

229 to the returned L{GeodesicAreaExact} instance.

230 '''

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

232 name=self._name__(name))

233 if self.debug:

234 gaX.verbose = True

235 return gaX

236

237 @Property_RO

238 def b(self):

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

240 '''

241 return self.ellipsoid.b

242

243 @Property_RO

244 def c2x(self):

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

246 '''

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

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

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

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

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

252 return self.ellipsoid.c2x

253

254 c2 = c2x # in this particular case

255

256 def C4f(self, eps):

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

258

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

260

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

262 '''

263 def _c4(nC4, C4x):

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

265 _p = _polynomial

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

267 j = i + r

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

269 x *= e

270 i = j

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

272

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

274

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

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

277 '''

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

279

280 @Property

281 def C4order(self):

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

283 '''

284 return self._nC4

285

286 @C4order.setter # PYCHOK .setter!

287 def C4order(self, order):

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

289

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

291 '''

292 _xnC4(C4order=order)

293 if self._nC4 != order:

294 GeodesicExact._C4x._update(self)

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

296 self._nC4 = order

297

298 @deprecated_Property

299 def C4Order(self):

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

301 '''

302 return self.C4order

303

304 @C4Order.setter # PYCHOK .setter!

305 def C4Order(self, order):

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

307 '''

308 _xnC4(C4Order=order)

309 self.C4order = order

310

311 @Property_RO

312 def _C4x(self):

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

314

315 @see: Property L{C4order}.

316 '''

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

318 def _C4(nC4):

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

320 _p = _polynomial

321 for r in range(nC4 + 1, 1, -1): # _reverange

322 for j in range(1, r):

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

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

325 i = j + 1

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

327

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

329

330 @property_RO

331 def datum(self):

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

333 '''

334 return self._datum

335

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

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

338

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

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

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

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

343 @kwarg outmask: Bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>}

344 values specifying the quantities to be returned.

345

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

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

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

349

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

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

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

353 '''

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

355

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

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

358 (final bearing) in C{degrees}.

359

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

361 '''

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

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

364

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

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

367 '''

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

369

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

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

372

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

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

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

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

377 @kwarg caps: Bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>} values

378 specifying the capabilities the L{GeodesicLineExact} instance

379 should possess, i.e., which quantities can be returned by methods

380 L{GeodesicLineExact.Position}.

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

382

383 @return: A L{GeodesicLineExact} instance.

384

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

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

387

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

389

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

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

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

393 '''

394 return GeodesicLineExact(self, lat1, lon1, azi1, caps=caps, **name)._GenSet(self._debug, s12=s12)

395

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

397 '''(INTERNAL) Helper.

398 '''

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

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

401 else:

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

403 return dn

404

405 @Property_RO

406 def e2(self):

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

408 '''

409 return self.ellipsoid.e2

410

411 @Property_RO

412 def _e2a2(self):

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

414 '''

415 return self.e2 * self.ellipsoid.a2

416

417 @Property_RO

418 def _e2_f1(self):

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

420 '''

421 return self.e2 / self.f1

422

423 @Property_RO

424 def _eF(self):

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

426 '''

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

428

429 def _eF_reset_cHe2_f1(self, x, y):

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

431 '''

432 self._eF_reset_k2(x)

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

434

435 def _eF_reset_k2(self, x):

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

437 '''

438 ep2 = self.ep2

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

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

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

442 return k2

443

444 @Property_RO

445 def ellipsoid(self):

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

447 '''

448 return self.datum.ellipsoid

449

450 @Property_RO

451 def ep2(self):

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

453 '''

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

455

456 e22 = ep2 # for ellipsoid compatibility

457

458 @Property_RO

459 def _eTOL2(self):

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

461 '''

462 return _eTOL2(self.f)

463

464 @Property_RO

465 def flattening(self):

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

467 '''

468 return self.ellipsoid.f

469

470 f = flattening

471

472 @Property_RO

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

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

475 '''

476 return self.ellipsoid.f1

477

478 @Property_RO

479 def _f180(self):

480 '''(INTERNAL) Cached/memoized.

481 '''

482 return self.f * _180_0

483

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

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

486

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

488 '''

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

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

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

492

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

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

495

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

497 '''

498 Cs = Caps

499 if self._debug: # PYCHOK no cover

500 outmask |= Cs._DEBUG_INVERSE & self._debug

501 outmask &= Cs._OUT_MASK # incl. _SALP_CALPs_ and _DEBUG_

502 # compute longitude difference carefully (with _diff182):

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

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

505 lon12, lon12s = _diff182(lon1, lon2)

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

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

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

509 elif (outmask & Cs.LONGITUDE):

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

511 else:

512 r = GDict()

513 if _K_2_0: # GeographicLib 2.0

514 # make longitude difference positive

515 lon12, lon_ = _unsigned2(lon12)

516 if lon_:

517 lon12s = -lon12s

518 lam12 = radians(lon12)

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

520 slam12, clam12 = _sincos2de(lon12, lon12s)

521 # supplementary longitude difference

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

523 else: # GeographicLib 1.52

524 # make longitude difference positive and if very close

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

526 if lon12 < 0: # _signBit(lon12)

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

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

529 else:

530 lon_, lon12 = False, _around(lon12)

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

532 lam12 = radians(lon12)

533 if lon12 > _90_0:

534 slam12, clam12 = _sincos2d(lon12s)

535 clam12 = -clam12

536 else:

537 slam12, clam12 = _sincos2(lam12)

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

539 lat1 = _around(_fix90(lat1))

540 lat2 = _around(_fix90(lat2))

541 if (outmask & Cs.LATITUDE):

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

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

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

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

546 if swap_:

547 lat1, lat2 = lat2, lat1

548 lon_ = not lon_

549 if _signBit(lat1):

550 lat_ = False # note, False

551 else: # make lat1 <= -0

552 lat_ = True # note, True

553 lat1, lat2 = -lat1, -lat2

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

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

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

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

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

559 # this enforces some symmetries in the results returned.

560

561 # Initialize for the meridian. No longitude calculation is done in

562 # this case to let the parameter default to 0.

563 sbet1, cbet1 = _sinf1cos2d(lat1, self.f1)

564 sbet2, cbet2 = _sinf1cos2d(lat2, self.f1)

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

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

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

568 # in assigning calp2 in _Lambda6. Sometimes these quantities

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

570 # example where is is necessary is the inverse problem

571 # 48.522876735459 0 -48.52287673545898293 179.599720456223079643

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

573 if cbet1 < -sbet1:

574 if cbet2 == cbet1:

575 sbet2 = copysign(sbet1, sbet2)

576 elif fabs(sbet2) == -sbet1:

577 cbet2 = cbet1

578

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

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

581

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

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

584 # Endpoints are on a single full meridian,

585 # so the geodesic might lie on a meridian.

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

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

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

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

590 # sig12 = sig2 - sig1

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

592 s12x, m12x, _, \

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

594 # Add the check for sig12 since zero length geodesics

595 # might yield m12 < 0. Test case was

596 # echo 20.001 0 20.001 0 | GeodSolve -i

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

598 # geodesic which is not a shortest path.

599 if m12x >= 0 or sig12 < _1_0:

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

601 # Prevent negative s12 or m12 from geographiclib 1.52

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

603 sig12 = m12x = s12x = _0_0

604 else:

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

606 _meridian = False

607 C = 1

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

609 # _meridian = True

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

611

612 if _meridian:

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

614 if _b: # Geodesic runs along equator

615 calp1 = calp2 = _0_0

616 salp1 = salp2 = _1_0

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

618 somg12, comg12 = _sincos2(sig12)

619 m12x = self.b * somg12

620 s12x = self.a * lam12

621 M12 = M21 = comg12

622 a12 = lon12 / self.f1

623 C = 2

624 else:

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

626 # meridian and geodesic is neither meridional or equatorial.

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

628 # Figure a starting point for Newton's method

629 sig12, salp1, calp1, \

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

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

632 # pre-compute the constant _Lambda6 term, once

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

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

635 ((sbet1 + sbet2) * (sbet1 - sbet2))))

636 sig12, salp1, calp1, \

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

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

639 if (outmask & Cs.AREA):

640 # omg12 = lam12 - domg12

641 s, c = _sincos2(domg12)

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

643 C = 3 # Newton

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

645 C = 4 # Short lines

646 s, c = _sincos2(sig12 / dnm)

647 m12x = dnm**2 * s

648 s12x = dnm * sig12

649 M12 = M21 = c

650 if (outmask & Cs.AREA):

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

652

653 else: # _meridian is False

654 somg12 = comg12 = NAN

655

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

657

658 if (outmask & Cs.DISTANCE):

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

660

661 if (outmask & Cs.REDUCEDLENGTH):

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

663

664 if (outmask & Cs.GEODESICSCALE):

665 if swap_:

666 M12, M21 = M21, M12

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

668 M21=unsigned0(M21))

669

670 if (outmask & Cs.AREA):

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

672 salp2, calp2,

673 somg12, comg12, p)

674 if _xor(swap_, lat_, lon_):

675 S12 = -S12

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

677

678 if (outmask & (Cs.AZIMUTH | Cs._SALP_CALPs_)):

679 if swap_:

680 salp1, salp2 = salp2, salp1

681 calp1, calp2 = calp2, calp1

682 if _xor(swap_, lon_):

683 salp1, salp2 = -salp1, -salp2

684 if _xor(swap_, lat_):

685 calp1, calp2 = -calp1, -calp2

686

687 if (outmask & Cs.AZIMUTH):

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

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

690 if (outmask & Cs._SALP_CALPs_):

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

692 salp2=salp2, calp2=calp2)

693

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

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

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

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

698 c2=E.c2, c2x=self.c2x,

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

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

701 p.update(r) # r overrides p

702 r = p.toGDict()

703 return self._iter2tion(r, **p)

704

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

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

707

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

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

710 '''

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

712 return r.toDirect9Tuple()

713

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

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

716

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

718 m12, M12, M21, S12)}.

719 '''

720 r = self._GDictInverse(lat1, lon1, lat2, lon2, outmask | Caps._SALP_CALPs_)

721 return r.toInverse10Tuple()

722

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

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

725 '''

726 if wrap:

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

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

729

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

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

732

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

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

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

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

737 @kwarg outmask: Bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>}

738 values specifying the quantities to be returned.

739

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

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

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

743

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

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

746

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

748

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

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

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

752 '''

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

754

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

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

757

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

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

760 '''

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

762 # and .HeightIDWkarney._distances

763 if wrap:

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

765 return fabs(self._GDictInverse(lat1, lon1, lat2, lon2, Caps.EMPTY).a12) # a12 always

766

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

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

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

770

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

772 '''

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

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

775 iteration=r.iteration)

776

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

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

779 '''

780 if wrap:

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

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

783

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

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

786

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

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

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

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

791 @kwarg caps: Bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>} values

792 specifying the capabilities the L{GeodesicLineExact} instance

793 should possess, i.e., which quantities can be returned by methods

794 L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}.

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

796

797 @return: A L{GeodesicLineExact} instance.

798

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

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

801

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

803

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

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

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

807 '''

808 Cs = Caps

809 r = self._GDictInverse(lat1, lon1, lat2, lon2, caps | Cs._SALP_CALPs_)

810 return GeodesicLineExact(self, lat1, lon1, None, caps=caps, _s_calp1=(r.salp1, r.calp1),

811 **name)._GenSet(self._debug, **r)

812

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

814 salp2, calp2,

815 somg12, comg12, p):

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

817

818 @return: Area C{S12}.

819 '''

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

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

822 A4 = calp0 * salp0

823 if A4:

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

825 k2 = calp0**2 * self.ep2

826 C4a = self._C4f_k2(k2)

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

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

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

830 A4 *= self._e2a2

831 S12 = A4 * (B42 - B41)

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

833 A4 = B41 = B42 = k2 = S12 = _0_0

834

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

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

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

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

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

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

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

842 dbet1 = p.cbet1 + _1_0

843 dbet2 = p.cbet2 + _1_0

844 domg12 = comg12 + _1_0

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

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

847 alp12 = atan2(salp12, calp12) * _2_0

848 else:

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

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

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

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

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

854 # Following ensures the correct behavior.

855 if salp12 == 0 and calp12 < 0:

856 alp12 = _copysign(PI, calp1)

857 else:

858 alp12 = atan2(salp12, calp12)

859

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

861 return S12 + self.c2x * alp12

862

863 def _InverseStart6(self, lam12, p):

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

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

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

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

868

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

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

871 '''

872 sig12 = None # use Newton

873 salp1 = calp1 = salp2 = calp2 = dnm = NAN

874

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

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

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

878 if shortline:

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

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

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

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

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

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

885 else:

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

887

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

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

890

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

892 s = somg12**2 / c

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

894 salp1 = p.cbet2 * somg12

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

896

897 ssig12 = _hypot(salp1, calp1)

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

899

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

901 t = c if comg12 < 0 else s

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

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

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

905

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

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

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

909

910 else:

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

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

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

914 f = self.f

915 if f < 0: # PYCHOK no cover

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

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

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

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

920 Caps.REDUCEDLENGTH, p)

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

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

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

924 y = lam12x / sca

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

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

927 x = lam12x / sca # dlon

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

929

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

931 if f < 0: # PYCHOK no cover

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

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

934 else:

935 salp1 = min( _1_0, -x)

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

937 else:

938 # Estimate alp1, by solving the astroid problem.

939 #

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

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

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

943 #

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

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

946 # number of Newton iterations for astroid cases from 2.24

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

948 # number of iterations are as follows:

949 #

950 # change percent

951 # 1 5

952 # 0 78

953 # -1 16

954 # -2 0.6

955 # -3 0.04

956 # -4 0.002

957 #

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

959 # estimating alp1 directly, n = number of iterations

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

961 #

962 # iter m n

963 # 0 148 186

964 # 1 13046 13845

965 # 2 93315 102225

966 # 3 36189 32341

967 # 4 5396 7

968 # 5 455 1

969 # 6 56 0

970 #

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

972 k = _Astroid(x, y)

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

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

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

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

977 salp1 = p.cbet2 * s

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

979

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

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

982

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

984

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

986 '''(INTERNAL) Helper.

987

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

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

990 '''

991 eF = self._eF

992 f1 = self.f1

993

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

995 # Break degeneracy of equatorial line

996 calp1 = -_TINY

997

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

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

1000 # tan(bet1) = tan(sig1) * cos(alp1)

1001 # tan(omg1) = sin(alp0) * tan(sig1)

1002 # = sin(bet1) * tan(alp1)

1003 somg1 = salp0 * p.sbet1

1004 comg1 = calp1 * p.cbet1

1005 ssig1, csig1 = _norm2(p.sbet1, comg1)

1006 # Without normalization we have schi1 = somg1

1007 cchi1 = f1 * p.dn1 * comg1

1008

1009 # Enforce symmetries in the case abs(bet2) = -bet1.

1010 # Need to be careful about this case, since this can

1011 # yield singularities in the Newton iteration.

1012 # sin(alp2) * cos(bet2) = sin(alp0)

1013 salp2 = (salp0 / p.cbet2) if p.cbet2 != p.cbet1 else salp1

1014 # calp2 = sqrt(1 - sq(salp2))

1015 # = sqrt(sq(calp0) - sq(sbet2)) / cbet2

1016 # and subst for calp0 and rearrange to give (choose

1017 # positive sqrt to give alp2 in [0, PI/2]).

1018 calp2 = fabs(calp1) if p.bet12 is None else (

1019 sqrt((calp1 * p.cbet1)**2 + p.bet12) / p.cbet2)

1020 # tan(bet2) = tan(sig2) * cos(alp2)

1021 # tan(omg2) = sin(alp0) * tan(sig2).

1022 somg2 = salp0 * p.sbet2

1023 comg2 = calp2 * p.cbet2

1024 ssig2, csig2 = _norm2(p.sbet2, comg2)

1025 # without normalization we have schi2 = somg2

1026 cchi2 = f1 * p.dn2 * comg2

1027

1028 # omg12 = omg2 - omg1, limit to [0, PI]

1029 somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, sineg0=True)

1030 # chi12 = chi2 - chi1, limit to [0, PI]

1031 schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, sineg0=True)

1032

1033 p.set_sigs(ssig1, csig1, ssig2, csig2)

1034 # sig12 = sig2 - sig1, limit to [0, PI]

1035 sig12 = _atan12(ssig1, csig1, ssig2, csig2, sineg0=True)

1036

1037 eta12 = self._eF_reset_cHe2_f1(calp0, salp0) * _2__PI # then ...

1038 eta12 *= fsum1f_(eF.deltaH(*p.sncndn2),

1039 -eF.deltaH(*p.sncndn1), sig12)

1040 # eta = chi12 - lam12

1041 lam12 = _atan12(p.slam12, p.clam12, schi12, cchi12) - eta12

1042 # domg12 = chi12 - omg12 - deta12

1043 domg12 = _atan12( somg12, comg12, schi12, cchi12) - eta12

1044

1045 dlam12 = NAN # dv > 0 in ._Newton6

1046 if diffp:

1047 d = calp2 * p.cbet2

1048 if d:

1049 _, dlam12, _, _, _ = self._Length5(sig12, Caps.REDUCEDLENGTH, p)

1050 dlam12 *= f1 / d

1051 elif p.sbet1:

1052 dlam12 = -f1 * p.dn1 * _2_0 / p.sbet1

1053

1054 # p.set_(deta12=-eta12, lam12=lam12)

1055 return lam12, sig12, salp2, calp2, domg12, dlam12

1056

1057 def _Length5(self, sig12, outmask, p):

1058 '''(INTERNAL) Return M{m12b = (reduced length) / self.b} and

1059 calculate M{s12b = distance / self.b} and M{m0}, the

1060 coefficient of secular term in expression for reduced

1061 length and the geodesic scales C{M12} and C{M21}.

1062

1063 @return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}.

1064 '''

1065 s12b = m12b = m0 = M12 = M21 = NAN

1066

1067 Cs = Caps

1068 eF = self._eF

1069

1070 # outmask &= Cs._OUT_MASK

1071 if (outmask & Cs.DISTANCE):

1072 # Missing a factor of self.b

1073 s12b = eF.cE * _2__PI * fsum1f_(eF.deltaE(*p.sncndn2),

1074 -eF.deltaE(*p.sncndn1), sig12)

1075

1076 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE):

1077 m0x = -eF.k2 * eF.cD * _2__PI

1078 J12 = -m0x * fsum1f_(eF.deltaD(*p.sncndn2),

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

1080 if (outmask & Cs.REDUCEDLENGTH):

1081 m0 = m0x

1082 # Missing a factor of self.b. Add parens around

1083 # (csig1 * ssig2) and (ssig1 * csig2) to ensure

1084 # accurate cancellation for coincident points.

1085 m12b = fsum1f_(p.dn2 * (p.csig1 * p.ssig2),

1086 -p.dn1 * (p.ssig1 * p.csig2),

1087 J12 * (p.csig1 * p.csig2))

1088 if (outmask & Cs.GEODESICSCALE):

1089 M12 = M21 = p.ssig1 * p.ssig2 + \

1090 p.csig1 * p.csig2

1091 t = (p.cbet1 - p.cbet2) * self.ep2 * \

1092 (p.cbet1 + p.cbet2) / (p.dn1 + p.dn2)

1093 M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1

1094 M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2

1095

1096 return s12b, m12b, m0, M12, M21

1097

1098 def Line(self, lat1, lon1, azi1, caps=Caps.ALL, **name):

1099 '''Set up a L{GeodesicLineExact} to compute several points

1100 on a single geodesic.

1101

1102 @arg lat1: Latitude of the first point (C{degrees}).

1103 @arg lon1: Longitude of the first point (C{degrees}).

1104 @arg azi1: Azimuth at the first point (compass C{degrees}).

1105 @kwarg caps: Bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>} values

1106 specifying the capabilities the L{GeodesicLineExact} instance

1107 should possess, i.e., which quantities can be returned by methods

1108 L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}.

1109 @kwarg name: Optional C{B{name}=NN} (C{str}).

1110

1111 @return: A L{GeodesicLineExact} instance.

1112

1113 @note: If the point is at a pole, the azimuth is defined by keeping

1114 B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking

1115 the limit C{ε → 0+}.

1116

1117 @see: C++ U{GeodesicExact.Line

1118 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>}

1119 and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/Python/doc/code.html>}.

1120 '''

1121 return GeodesicLineExact(self, lat1, lon1, azi1, caps=caps, **name)._GenSet(self._debug)

1122

1123 @Property_RO

1124 def n(self):

1125 '''Get the C{ellipsoid}'s I{3rd flattening} (C{scalar}), M{f / (2 - f) == (a - b) / (a + b)}.

1126 '''

1127 return self.ellipsoid.n

1128

1129 @Property_RO

1130 def _n_0_1(self):

1131 '''(INTERNAL) Cached once.

1132 '''

1133 return fabs(self.n) > _0_1

1134

1135 @Property_RO

1136 def _n6PI(self):

1137 '''(INTERNAL) Cached once.

1138 '''

1139 return fabs(self.n) * _6_0 * PI

1140

1141 def _Newton6(self, salp1, calp1, p):

1142 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length.

1143

1144 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12)}

1145 and C{p.iter} and C{p.trip} updated.

1146 '''

1147 _abs = fabs

1148 # This is a straightforward solution of f(alp1) = lambda12(alp1) -

1149 # lam12 = 0 with one wrinkle. f(alp) has exactly one root in the

1150 # interval (0, PI) and its derivative is positive at the root.

1151 # Thus f(alp) is positive for alp > alp1 and negative for alp < alp1.

1152 # During the course of the iteration, a range (alp1a, alp1b) is

1153 # maintained which brackets the root and with each evaluation of

1154 # f(alp) the range is shrunk, if possible. Newton's method is

1155 # restarted whenever the derivative of f is negative (because the

1156 # new value of alp1 is then further from the solution) or if the

1157 # new estimate of alp1 lies outside (0,PI); in this case, the new

1158 # starting guess is taken to be (alp1a + alp1b) / 2.

1159 salp1a = salp1b = _TINY

1160 calp1a, calp1b = _1_0, _N_1_0

1161 MAXIT1, TOL0 = _MAXIT1, _TOL0

1162 HALF, TOLb = _0_5, _TOLb

1163 tripb, TOLv = False, TOL0

1164 for i in range(_MAXIT2):

1165 # 1/4 meridian = 10e6 meter and random input,

1166 # estimated max error in nm (nano meter, by

1167 # checking Inverse problem by Direct).

1168 #

1169 # max iterations

1170 # log2(b/a) error mean sd

1171 # -7 387 5.33 3.68

1172 # -6 345 5.19 3.43

1173 # -5 269 5.00 3.05

1174 # -4 210 4.76 2.44

1175 # -3 115 4.55 1.87

1176 # -2 69 4.35 1.38

1177 # -1 36 4.05 1.03

1178 # 0 15 0.01 0.13

1179 # 1 25 5.10 1.53

1180 # 2 96 5.61 2.09

1181 # 3 318 6.02 2.74

1182 # 4 985 6.24 3.22

1183 # 5 2352 6.32 3.44

1184 # 6 6008 6.30 3.45

1185 # 7 19024 6.19 3.30

1186 v, sig12, salp2, calp2, \

1187 domg12, dv = self._Lambda6(salp1, calp1, i < MAXIT1, p)

1188

1189 # 2 * _TOL0 is approximately 1 ulp [0, PI]

1190 # reversed test to allow escape with NaNs

1191 if tripb or _abs(v) < TOLv:

1192 break

1193 # update bracketing values

1194 if v > 0 and (i > MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)):

1195 salp1b, calp1b = salp1, calp1

1196 elif v < 0 and (i > MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)):

1197 salp1a, calp1a = salp1, calp1

1198

1199 if i < MAXIT1 and dv > 0:

1200 dalp1 = -v / dv

1201 if _abs(dalp1) < PI:

1202 s, c = _sincos2(dalp1)

1203 # nalp1 = alp1 + dalp1

1204 s, c = _sincos12(-s, c, salp1, calp1)

1205 if s > 0:

1206 salp1, calp1 = _norm2(s, c)

1207 # in some regimes we don't get quadratic convergence

1208 # because slope -> 0. So use convergence conditions

1209 # based on epsilon instead of sqrt(epsilon)

1210 TOLv = TOL0 if _abs(v) > _TOL016 else _TOL08

1211 continue

1212 TOLv = TOL0

1213 # Either dv was not positive or updated value was outside

1214 # legal range. Use the midpoint of the bracket as the next

1215 # estimate. This mechanism is not needed for the WGS84

1216 # ellipsoid, but it does catch problems with more eccentric

1217 # ellipsoids. Its efficacy is such for the WGS84 test set

1218 # with the starting guess set to alp1 = 90 deg: the WGS84

1219 # test set: mean = 5.21, stdev = 3.93, max = 24 and WGS84

1220 # with random input: mean = 4.74, stdev = 0.99

1221 salp1, calp1 = _norm2((salp1a + salp1b) * HALF,

1222 (calp1a + calp1b) * HALF)

1223 tripb = fsum1f_(calp1a, -calp1, _abs(salp1a - salp1)) < TOLb or \

1224 fsum1f_(calp1b, -calp1, _abs(salp1b - salp1)) < TOLb

1225 else:

1226 raise GeodesicError(Fmt.no_convergence(v, TOLv), txt=repr(self)) # self.toRepr()

1227

1228 p.set_(iter=i, trip=tripb) # like .geodsolve._GDictInvoke: iter NOT iteration!

1229 return sig12, salp1, calp1, salp2, calp2, domg12

1230

1231 Polygon = Area # for C{geographiclib} compatibility

1232

1233 def toStr(self, **prec_sep_name): # PYCHOK signature

1234 '''Return this C{GeodesicExact} as string.

1235

1236 @see: L{Ellipsoid.toStr<pygeodesy.ellipsoids.Ellipsoid.toStr>}

1237 for further details.

1238

1239 @return: C{GeodesicExact} (C{str}).

1240 '''

1241 t = GeodesicExact.caps, GeodesicExact.ellipsoid

1242 return self._instr(props=t, C4order=self.C4order, **prec_sep_name)

1243

1244

1245 class GeodesicLineExact(_GeodesicLineExact):

1246 '''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact

1247 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicLineExact.html>},

1248 modeled after I{Karney}'s Python class U{geodesicline.GeodesicLine<https://GitHub.com/

1249 geographiclib/geographiclib-python>}.

1250 '''

1251

1252 def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, **name):

1253 '''New L{GeodesicLineExact} instance, allowing points to be found along

1254 a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}.

1255

1256 @arg geodesic: The geodesic to use (L{GeodesicExact}).

1257 @arg lat1: Latitude of the first point (C{degrees}).

1258 @arg lon1: Longitude of the first point (C{degrees}).

1259 @arg azi1: Azimuth at the first points (compass C{degrees}).

1260 @kwarg caps: Bit-or'ed combination of L{Caps<pygeodesy.karney.Caps>} values

1261 specifying the capabilities the L{GeodesicLineExact} instance

1262 should possess, i.e., which quantities can bereturned by methods

1263 L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}.

1264 @kwarg name: Optional C{B{name}=NN} (C{str}).

1265

1266 @raise TypeError: Invalid B{C{geodesic}}.

1267 '''

1268 _xinstanceof(GeodesicExact, geodesic=geodesic)

1269 if (caps & Caps.LINE_OFF): # copy to avoid updates

1270 geodesic = geodesic.copy(deep=False, name=_UNDER_(NN, geodesic.name))

1271# _update_all(geodesic)

1272 _GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, **name)

1273

1274

1275 def _Astroid(x, y):

1276 '''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1)

1277 * k^2 - (2 * k + 1) * y^2 = 0} for positive root k.

1278 '''

1279 p = x**2

1280 q = y**2

1281 r = fsumf_(_1_0, q, p, _N_2_0)

1282 if r > 0 or q:

1283 # avoid possible division by zero when r = 0

1284 # by multiplying s and t by r^3 and r, resp.

1285 S = p * q / _4_0 # S = r^3 * s

1286 if r:

1287 r = r / _6_0 # /= chokes PyChecker

1288 r3 = r**3

1289 T3 = r3 + S

1290 # discriminant of the quadratic equation for T3 is

1291 # zero on the evolute curve p^(1/3) + q^(1/3) = 1

1292 d = (r3 + T3) * S

1293 if d < 0:

1294 # T is complex, but u is defined for a real result

1295 a = atan2(sqrt(-d), -T3) / _3_0

1296 # There are 3 possible cube roots, choose the one which

1297 # avoids cancellation. Note d < 0 implies that r < 0.

1298 u = (cos(a) * _2_0 + _1_0) * r

1299 else:

1300 # pick the sign on the sqrt to maximize abs(T3) to

1301 # minimize loss of precision due to cancellation.

1302 if d:

1303 T3 += _copysign(sqrt(d), T3) # T3 = (r * t)^3

1304 # _cbrt always returns the real root, _cbrt(-8) = -2

1305 u = _cbrt(T3) # T = r * t

1306 if u: # T can be zero; but then r2 / T -> 0

1307 u += r**2 / u

1308 u += r

1309 elif S: # d == T3**2 == S**2: sqrt(d) == abs(S) == abs(T3)

1310 u = _cbrt(S * _2_0) # == T3 + _copysign(abs(S), T3)

1311 else:

1312 u = _0_0

1313 v = _hypot(u, y) # sqrt(u**2 + q)

1314 # avoid loss of accuracy when u < 0

1315 u = (q / (v - u)) if u < 0 else (v + u)

1316 w = (u - q) / (v + v) # positive?

1317 # rearrange expression for k to avoid loss of accuracy due to

1318 # subtraction, division by 0 impossible because u > 0, w >= 0

1319 k = u / (sqrt(w**2 + u) + w) # guaranteed positive

1320

1321 else: # q == 0 && r <= 0

1322 # y = 0 with |x| <= 1. Handle this case directly, for

1323 # y small, positive root is k = abs(y) / sqrt(1 - x^2)

1324 k = _0_0

1325

1326 return k

1327

1328

1329 def _C4coeffs(nC4): # in .geodesicx.__main__

1330 '''(INTERNAL) Get the C{C4_24}, C{_27} or C{_30} series coefficients.

1331 '''

1332 try: # from pygeodesy.geodesicx._C4_xx import _coeffs_xx as _coeffs

1333 _C4_xx = _DOT_(typename(_MODS.geodesicx), _UNDER_('_C4', nC4))

1334 _coeffs = _MODS.getattr(_C4_xx, _UNDER_('_coeffs', nC4))

1335 except (AttributeError, ImportError, TypeError) as x:

1336 raise GeodesicError(nC4=nC4, cause=x)

1337 n = _xnC4(nC4=nC4)

1338 if len(_coeffs) != n: # double check

1339 raise GeodesicError(_coeffs=len(_coeffs), _xnC4=n, nC4=nC4)

1340 return _coeffs

1341

1342

1343__all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact)

1344

1345# **) MIT License

1346#

1347# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.

1348#

1349# Permission is hereby granted, free of charge, to any person obtaining a

1350# copy of this software and associated documentation files (the "Software"),

1351# to deal in the Software without restriction, including without limitation

1352# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1353# and/or sell copies of the Software, and to permit persons to whom the

1354# Software is furnished to do so, subject to the following conditions:

1355#

1356# The above copyright notice and this permission notice shall be included

1357# in all copies or substantial portions of the Software.

1358#

1359# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1360# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1361# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1362# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1363# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1364# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1365# OTHER DEALINGS IN THE SOFTWARE.