Coverage for pygeodesy/geodesicx/gx.py: 93%

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1 

2# -*- coding: utf-8 -*- 

3 

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

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

6 

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

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

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

10 

11Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2024) 

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

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

14''' 

15# make sure int/int division yields float quotient 

16from __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 

39from pygeodesy.basics import _copysign, _xinstanceof, _xor, unsigned0 

40from 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 

44from pygeodesy.datums import _earth_datum, _WGS84, _EWGS84 

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

46from pygeodesy.errors import GeodesicError, _xkwds_pop2 

47from pygeodesy.fmath import hypot as _hypot, Fmt 

48from pygeodesy.fsums import fsumf_, fsum1f_ 

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

50 _sincos12, _sin1cos2, _sinf1cos2d, \ 

51 _TINY, _xnC4 

52from pygeodesy.geodesicx.gxline import _GeodesicLineExact, _update_glXs 

53from pygeodesy.interns import NN, _DOT_, _UNDER_ 

54from pygeodesy.karney import GDict, _around, _atan2d, Caps, _cbrt, _diff182, \ 

55 _fix90, _K_2_0, _norm2, _norm180, _polynomial, \ 

56 _signBit, _sincos2, _sincos2d, _sincos2de, _unsigned2 

57from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS 

58from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple 

59from pygeodesy.props import deprecated_Property, Property, Property_RO, property_RO 

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

61from pygeodesy.utily import atan2, atan2d as _atan2d_reverse, _unrollon, \ 

62 _Wrap, wrap360 

63 

64from math import copysign, cos, degrees, fabs, radians, sqrt 

65 

66__all__ = () 

67__version__ = '24.11.24' 

68 

69_MAXIT1 = 20 

70_MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits 

71 

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

73# case 52.784459512564 0 -52.784459512563990912 179.634407464943777557 

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

75_TOL0 = EPS 

76_TOL1 = _TOL0 * -200 # negative 

77_TOL2 = _EPSqrt # == sqrt(_TOL0) 

78_TOL3 = _TOL2 * _0_1 

79_TOLb = _TOL2 * _TOL0 # Check on bisection interval 

80_THR1 = _TOL2 * _1000_0 + _1_0 

81 

82_TINY3 = _TINY * _3_0 

83_TOL08 = _TOL0 * _8_0 

84_TOL016 = _TOL0 * _16_0 

85 

86 

87def _atan12(*sincos12, **sineg0): 

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

89 ''' 

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

91 

92 

93def _eTOL2(f): 

94 # Using the auxiliary sphere solution with dnm computed at 

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

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

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

98 

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

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

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

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

103 

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

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

106 # nearly spherical case. 

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

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

109 

110 

111class _PDict(GDict): 

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

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

114 ''' 

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

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

117 ''' 

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

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

120 

121 def toGDict(self): # PYCHOK no cover 

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

123 ''' 

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

125 return GDict(rest) 

126 

127 return _rest(**self) 

128 

129 

130class GeodesicExact(_GeodesicBase): 

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

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

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

134 geographiclib/geographiclib-python>}. 

135 ''' 

136 _datum = _WGS84 

137 _nC4 = 30 # default C4order 

138 

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

140 '''New L{GeodesicExact} instance. 

141 

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

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

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

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

146 is specified as C{scalar}. 

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

148 L{C4order}, default C{30}. 

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

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

151 

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

153 ''' 

154 if name_C4Order: 

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

156 if name: 

157 self.name = name 

158 else: 

159 name = {} # name_C4Order 

160 

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

162 if C4order: # XXX private copy, always? 

163 self.C4order = C4order 

164 

165 @Property_RO 

166 def a(self): 

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

168 ''' 

169 return self.ellipsoid.a 

170 

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

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

173 

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

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

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

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

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

179 the quantities to be returned. 

180 

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

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

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

184 

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

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

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

188 ''' 

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

190 

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

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

193 

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

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

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

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

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

199 the capabilities the L{GeodesicLineExact} instance 

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

201 returned by calls to L{GeodesicLineExact.Position} 

202 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} values specifying 

344 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} values specifying 

378 the capabilities the L{GeodesicLineExact} instance 

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

380 returned by calls to 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} values specifying 

738 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} values specifying 

792 the capabilities the L{GeodesicLineExact} instance 

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

794 returned by calls to L{GeodesicLineExact.Position} 

795 and L{GeodesicLineExact.ArcPosition}. 

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

797 

798 @return: A L{GeodesicLineExact} instance. 

799 

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

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

802 

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

804 

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

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

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

808 ''' 

809 Cs = Caps 

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

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

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

813 

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

815 salp2, calp2, 

816 somg12, comg12, p): 

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

818 

819 @return: Area C{S12}. 

820 ''' 

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

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

823 A4 = calp0 * salp0 

824 if A4: 

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

826 k2 = calp0**2 * self.ep2 

827 C4a = self._C4f_k2(k2) 

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

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

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

831 A4 *= self._e2a2 

832 S12 = A4 * (B42 - B41) 

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

834 A4 = B41 = B42 = k2 = S12 = _0_0 

835 

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

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

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

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

840 # (tan(bet1/2) + tan(bet2/2)) / 

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

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

843 dbet1 = p.cbet1 + _1_0 

844 dbet2 = p.cbet2 + _1_0 

845 domg12 = comg12 + _1_0 

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

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

848 alp12 = atan2(salp12, calp12) * _2_0 

849 else: 

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

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

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

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

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

855 # Following ensures the correct behavior. 

856 if salp12 == 0 and calp12 < 0: 

857 alp12 = _copysign(PI, calp1) 

858 else: 

859 alp12 = atan2(salp12, calp12) 

860 

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

862 return S12 + self.c2x * alp12 

863 

864 def _InverseStart6(self, lam12, p): 

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

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

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

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

869 

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

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

872 ''' 

873 sig12 = None # use Newton 

874 salp1 = calp1 = salp2 = calp2 = dnm = NAN 

875 

876 # bet12 = bet2 - bet1 in [0, PI) 

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

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

879 if shortline: 

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

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

882 t = (p.sbet1 + p.sbet2)**2 

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

884 dnm = sqrt(_1_0 + self.ep2 * s) 

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

886 else: 

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

888 

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

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

891 

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

893 s = somg12**2 / c 

894 t = p.sbet1 * p.cbet2 * s 

895 salp1 = p.cbet2 * somg12 

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

897 

898 ssig12 = _hypot(salp1, calp1) 

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

900 

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

902 t = c if comg12 < 0 else s 

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

904 sbet12 - p.cbet1 * p.sbet2 * t) 

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

906 

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

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

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

910 

911 else: 

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

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

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

915 f = self.f 

916 if f < 0: # PYCHOK no cover 

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

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

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

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

921 Caps.REDUCEDLENGTH, p) 

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

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

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

925 y = lam12x / sca 

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

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

928 x = lam12x / sca # dlon 

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

930 

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

932 if f < 0: # PYCHOK no cover 

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

934 salp1 = sqrt(_1_0 - calp1**2) 

935 else: 

936 salp1 = min( _1_0, -x) 

937 calp1 = -sqrt(_1_0 - salp1**2) 

938 else: 

939 # Estimate alp1, by solving the astroid problem. 

940 # 

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

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

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

944 # 

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

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

947 # number of Newton iterations for astroid cases from 2.24 

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

949 # number of iterations are as follows: 

950 # 

951 # change percent 

952 # 1 5 

953 # 0 78 

954 # -1 16 

955 # -2 0.6 

956 # -3 0.04 

957 # -4 0.002 

958 # 

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

960 # estimating alp1 directly, n = number of iterations 

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

962 # 

963 # iter m n 

964 # 0 148 186 

965 # 1 13046 13845 

966 # 2 93315 102225 

967 # 3 36189 32341 

968 # 4 5396 7 

969 # 5 455 1 

970 # 6 56 0 

971 # 

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

973 k = _Astroid(x, y) 

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

975 (x * k / (k + _1_0)) 

976 s, c = _sincos2(-sca) # omg12a 

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

978 salp1 = p.cbet2 * s 

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

980 

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

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

983 

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

985 

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

987 '''(INTERNAL) Helper. 

988 

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

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

991 ''' 

992 eF = self._eF 

993 f1 = self.f1 

994 

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

996 # Break degeneracy of equatorial line 

997 calp1 = -_TINY 

998 

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

1000 salp0, calp0 = _sin1cos2(salp1, calp1, p.sbet1, p.cbet1) 

1001 # tan(bet1) = tan(sig1) * cos(alp1) 

1002 # tan(omg1) = sin(alp0) * tan(sig1) 

1003 # = sin(bet1) * tan(alp1) 

1004 somg1 = salp0 * p.sbet1 

1005 comg1 = calp1 * p.cbet1 

1006 ssig1, csig1 = _norm2(p.sbet1, comg1) 

1007 # Without normalization we have schi1 = somg1 

1008 cchi1 = f1 * p.dn1 * comg1 

1009 

1010 # Enforce symmetries in the case abs(bet2) = -bet1. 

1011 # Need to be careful about this case, since this can 

1012 # yield singularities in the Newton iteration. 

1013 # sin(alp2) * cos(bet2) = sin(alp0) 

1014 salp2 = (salp0 / p.cbet2) if p.cbet2 != p.cbet1 else salp1 

1015 # calp2 = sqrt(1 - sq(salp2)) 

1016 # = sqrt(sq(calp0) - sq(sbet2)) / cbet2 

1017 # and subst for calp0 and rearrange to give (choose 

1018 # positive sqrt to give alp2 in [0, PI/2]). 

1019 calp2 = fabs(calp1) if p.bet12 is None else ( 

1020 sqrt((calp1 * p.cbet1)**2 + p.bet12) / p.cbet2) 

1021 # tan(bet2) = tan(sig2) * cos(alp2) 

1022 # tan(omg2) = sin(alp0) * tan(sig2). 

1023 somg2 = salp0 * p.sbet2 

1024 comg2 = calp2 * p.cbet2 

1025 ssig2, csig2 = _norm2(p.sbet2, comg2) 

1026 # without normalization we have schi2 = somg2 

1027 cchi2 = f1 * p.dn2 * comg2 

1028 

1029 # omg12 = omg2 - omg1, limit to [0, PI] 

1030 somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, sineg0=True) 

1031 # chi12 = chi2 - chi1, limit to [0, PI] 

1032 schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, sineg0=True) 

1033 

1034 p.set_sigs(ssig1, csig1, ssig2, csig2) 

1035 # sig12 = sig2 - sig1, limit to [0, PI] 

1036 sig12 = _atan12(ssig1, csig1, ssig2, csig2, sineg0=True) 

1037 

1038 eta12 = self._eF_reset_cHe2_f1(calp0, salp0) * _2__PI # then ... 

1039 eta12 *= fsum1f_(eF.deltaH(*p.sncndn2), 

1040 -eF.deltaH(*p.sncndn1), sig12) 

1041 # eta = chi12 - lam12 

1042 lam12 = _atan12(p.slam12, p.clam12, schi12, cchi12) - eta12 

1043 # domg12 = chi12 - omg12 - deta12 

1044 domg12 = _atan12( somg12, comg12, schi12, cchi12) - eta12 

1045 

1046 dlam12 = NAN # dv > 0 in ._Newton6 

1047 if diffp: 

1048 d = calp2 * p.cbet2 

1049 if d: 

1050 _, dlam12, _, _, _ = self._Length5(sig12, Caps.REDUCEDLENGTH, p) 

1051 dlam12 *= f1 / d 

1052 elif p.sbet1: 

1053 dlam12 = -f1 * p.dn1 * _2_0 / p.sbet1 

1054 

1055 # p.set_(deta12=-eta12, lam12=lam12) 

1056 return lam12, sig12, salp2, calp2, domg12, dlam12 

1057 

1058 def _Length5(self, sig12, outmask, p): 

1059 '''(INTERNAL) Return M{m12b = (reduced length) / self.b} and 

1060 calculate M{s12b = distance / self.b} and M{m0}, the 

1061 coefficient of secular term in expression for reduced 

1062 length and the geodesic scales C{M12} and C{M21}. 

1063 

1064 @return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}. 

1065 ''' 

1066 s12b = m12b = m0 = M12 = M21 = NAN 

1067 

1068 Cs = Caps 

1069 eF = self._eF 

1070 

1071 # outmask &= Cs._OUT_MASK 

1072 if (outmask & Cs.DISTANCE): 

1073 # Missing a factor of self.b 

1074 s12b = eF.cE * _2__PI * fsum1f_(eF.deltaE(*p.sncndn2), 

1075 -eF.deltaE(*p.sncndn1), sig12) 

1076 

1077 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE): 

1078 m0x = -eF.k2 * eF.cD * _2__PI 

1079 J12 = -m0x * fsum1f_(eF.deltaD(*p.sncndn2), 

1080 -eF.deltaD(*p.sncndn1), sig12) 

1081 if (outmask & Cs.REDUCEDLENGTH): 

1082 m0 = m0x 

1083 # Missing a factor of self.b. Add parens around 

1084 # (csig1 * ssig2) and (ssig1 * csig2) to ensure 

1085 # accurate cancellation for coincident points. 

1086 m12b = fsum1f_(p.dn2 * (p.csig1 * p.ssig2), 

1087 -p.dn1 * (p.ssig1 * p.csig2), 

1088 J12 * (p.csig1 * p.csig2)) 

1089 if (outmask & Cs.GEODESICSCALE): 

1090 M12 = M21 = p.ssig1 * p.ssig2 + \ 

1091 p.csig1 * p.csig2 

1092 t = (p.cbet1 - p.cbet2) * self.ep2 * \ 

1093 (p.cbet1 + p.cbet2) / (p.dn1 + p.dn2) 

1094 M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1 

1095 M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2 

1096 

1097 return s12b, m12b, m0, M12, M21 

1098 

1099 def Line(self, lat1, lon1, azi1, caps=Caps.ALL, **name): 

1100 '''Set up a L{GeodesicLineExact} to compute several points 

1101 on a single geodesic. 

1102 

1103 @arg lat1: Latitude of the first point (C{degrees}). 

1104 @arg lon1: Longitude of the first point (C{degrees}). 

1105 @arg azi1: Azimuth at the first point (compass C{degrees}). 

1106 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

1107 the capabilities the L{GeodesicLineExact} instance 

1108 should possess, i.e., which quantities can be 

1109 returnedby calls to L{GeodesicLineExact.Position} 

1110 and L{GeodesicLineExact.ArcPosition}. 

1111 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1112 

1113 @return: A L{GeodesicLineExact} instance. 

1114 

1115 @note: If the point is at a pole, the azimuth is defined by keeping 

1116 B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking 

1117 the limit C{ε → 0+}. 

1118 

1119 @see: C++ U{GeodesicExact.Line 

1120 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} 

1121 and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/Python/doc/code.html>}. 

1122 ''' 

1123 return GeodesicLineExact(self, lat1, lon1, azi1, caps=caps, **name)._GenSet(self._debug) 

1124 

1125 @Property_RO 

1126 def n(self): 

1127 '''Get the C{ellipsoid}'s I{3rd flattening} (C{scalar}), M{f / (2 - f) == (a - b) / (a + b)}. 

1128 ''' 

1129 return self.ellipsoid.n 

1130 

1131 @Property_RO 

1132 def _n_0_1(self): 

1133 '''(INTERNAL) Cached once. 

1134 ''' 

1135 return fabs(self.n) > _0_1 

1136 

1137 @Property_RO 

1138 def _n6PI(self): 

1139 '''(INTERNAL) Cached once. 

1140 ''' 

1141 return fabs(self.n) * _6_0 * PI 

1142 

1143 def _Newton6(self, salp1, calp1, p): 

1144 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length. 

1145 

1146 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12)} 

1147 and C{p.iter} and C{p.trip} updated. 

1148 ''' 

1149 _abs = fabs 

1150 # This is a straightforward solution of f(alp1) = lambda12(alp1) - 

1151 # lam12 = 0 with one wrinkle. f(alp) has exactly one root in the 

1152 # interval (0, PI) and its derivative is positive at the root. 

1153 # Thus f(alp) is positive for alp > alp1 and negative for alp < alp1. 

1154 # During the course of the iteration, a range (alp1a, alp1b) is 

1155 # maintained which brackets the root and with each evaluation of 

1156 # f(alp) the range is shrunk, if possible. Newton's method is 

1157 # restarted whenever the derivative of f is negative (because the 

1158 # new value of alp1 is then further from the solution) or if the 

1159 # new estimate of alp1 lies outside (0,PI); in this case, the new 

1160 # starting guess is taken to be (alp1a + alp1b) / 2. 

1161 salp1a = salp1b = _TINY 

1162 calp1a, calp1b = _1_0, _N_1_0 

1163 MAXIT1, TOL0 = _MAXIT1, _TOL0 

1164 HALF, TOLb = _0_5, _TOLb 

1165 tripb, TOLv = False, TOL0 

1166 for i in range(_MAXIT2): 

1167 # 1/4 meridian = 10e6 meter and random input, 

1168 # estimated max error in nm (nano meter, by 

1169 # checking Inverse problem by Direct). 

1170 # 

1171 # max iterations 

1172 # log2(b/a) error mean sd 

1173 # -7 387 5.33 3.68 

1174 # -6 345 5.19 3.43 

1175 # -5 269 5.00 3.05 

1176 # -4 210 4.76 2.44 

1177 # -3 115 4.55 1.87 

1178 # -2 69 4.35 1.38 

1179 # -1 36 4.05 1.03 

1180 # 0 15 0.01 0.13 

1181 # 1 25 5.10 1.53 

1182 # 2 96 5.61 2.09 

1183 # 3 318 6.02 2.74 

1184 # 4 985 6.24 3.22 

1185 # 5 2352 6.32 3.44 

1186 # 6 6008 6.30 3.45 

1187 # 7 19024 6.19 3.30 

1188 v, sig12, salp2, calp2, \ 

1189 domg12, dv = self._Lambda6(salp1, calp1, i < MAXIT1, p) 

1190 

1191 # 2 * _TOL0 is approximately 1 ulp [0, PI] 

1192 # reversed test to allow escape with NaNs 

1193 if tripb or _abs(v) < TOLv: 

1194 break 

1195 # update bracketing values 

1196 if v > 0 and (i > MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)): 

1197 salp1b, calp1b = salp1, calp1 

1198 elif v < 0 and (i > MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)): 

1199 salp1a, calp1a = salp1, calp1 

1200 

1201 if i < MAXIT1 and dv > 0: 

1202 dalp1 = -v / dv 

1203 if _abs(dalp1) < PI: 

1204 s, c = _sincos2(dalp1) 

1205 # nalp1 = alp1 + dalp1 

1206 s, c = _sincos12(-s, c, salp1, calp1) 

1207 if s > 0: 

1208 salp1, calp1 = _norm2(s, c) 

1209 # in some regimes we don't get quadratic convergence 

1210 # because slope -> 0. So use convergence conditions 

1211 # based on epsilon instead of sqrt(epsilon) 

1212 TOLv = TOL0 if _abs(v) > _TOL016 else _TOL08 

1213 continue 

1214 TOLv = TOL0 

1215 # Either dv was not positive or updated value was outside 

1216 # legal range. Use the midpoint of the bracket as the next 

1217 # estimate. This mechanism is not needed for the WGS84 

1218 # ellipsoid, but it does catch problems with more eccentric 

1219 # ellipsoids. Its efficacy is such for the WGS84 test set 

1220 # with the starting guess set to alp1 = 90 deg: the WGS84 

1221 # test set: mean = 5.21, stdev = 3.93, max = 24 and WGS84 

1222 # with random input: mean = 4.74, stdev = 0.99 

1223 salp1, calp1 = _norm2((salp1a + salp1b) * HALF, 

1224 (calp1a + calp1b) * HALF) 

1225 tripb = fsum1f_(calp1a, -calp1, _abs(salp1a - salp1)) < TOLb or \ 

1226 fsum1f_(calp1b, -calp1, _abs(salp1b - salp1)) < TOLb 

1227 else: 

1228 raise GeodesicError(Fmt.no_convergence(v, TOLv), txt=repr(self)) # self.toRepr() 

1229 

1230 p.set_(iter=i, trip=tripb) # like .geodsolve._GDictInvoke: iter NOT iteration! 

1231 return sig12, salp1, calp1, salp2, calp2, domg12 

1232 

1233 Polygon = Area # for C{geographiclib} compatibility 

1234 

1235 def toStr(self, **prec_sep_name): # PYCHOK signature 

1236 '''Return this C{GeodesicExact} as string. 

1237 

1238 @see: L{Ellipsoid.toStr<pygeodesy.ellipsoids.Ellipsoid.toStr>} 

1239 for further details. 

1240 

1241 @return: C{GeodesicExact} (C{str}). 

1242 ''' 

1243 t = GeodesicExact.caps, GeodesicExact.ellipsoid 

1244 return self._instr(props=t, C4order=self.C4order, **prec_sep_name) 

1245 

1246 

1247class GeodesicLineExact(_GeodesicLineExact): 

1248 '''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact 

1249 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicLineExact.html>}, 

1250 modeled after I{Karney}'s Python class U{geodesicline.GeodesicLine<https://GitHub.com/ 

1251 geographiclib/geographiclib-python>}. 

1252 ''' 

1253 

1254 def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, **name): 

1255 '''New L{GeodesicLineExact} instance, allowing points to be found along 

1256 a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}. 

1257 

1258 @arg geodesic: The geodesic to use (L{GeodesicExact}). 

1259 @arg lat1: Latitude of the first point (C{degrees}). 

1260 @arg lon1: Longitude of the first point (C{degrees}). 

1261 @arg azi1: Azimuth at the first points (compass C{degrees}). 

1262 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

1263 the capabilities the L{GeodesicLineExact} instance 

1264 should possess, i.e., which quantities can be 

1265 returned by calls to L{GeodesicLineExact.Position} 

1266 and L{GeodesicLineExact.ArcPosition}. 

1267 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1268 

1269 @raise TypeError: Invalid B{C{geodesic}}. 

1270 ''' 

1271 _xinstanceof(GeodesicExact, geodesic=geodesic) 

1272 if (caps & Caps.LINE_OFF): # copy to avoid updates 

1273 geodesic = geodesic.copy(deep=False, name=_UNDER_(NN, geodesic.name)) 

1274# _update_all(geodesic) 

1275 _GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, **name) 

1276 

1277 

1278def _Astroid(x, y): 

1279 '''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1) 

1280 * k^2 - (2 * k + 1) * y^2 = 0} for positive root k. 

1281 ''' 

1282 p = x**2 

1283 q = y**2 

1284 r = fsumf_(_1_0, q, p, _N_2_0) 

1285 if r > 0 or q: 

1286 # avoid possible division by zero when r = 0 

1287 # by multiplying s and t by r^3 and r, resp. 

1288 S = p * q / _4_0 # S = r^3 * s 

1289 if r: 

1290 r = r / _6_0 # /= chokes PyChecker 

1291 r3 = r**3 

1292 T3 = r3 + S 

1293 # discriminant of the quadratic equation for T3 is 

1294 # zero on the evolute curve p^(1/3) + q^(1/3) = 1 

1295 d = (r3 + T3) * S 

1296 if d < 0: 

1297 # T is complex, but u is defined for a real result 

1298 a = atan2(sqrt(-d), -T3) / _3_0 

1299 # There are 3 possible cube roots, choose the one which 

1300 # avoids cancellation. Note d < 0 implies that r < 0. 

1301 u = (cos(a) * _2_0 + _1_0) * r 

1302 else: 

1303 # pick the sign on the sqrt to maximize abs(T3) to 

1304 # minimize loss of precision due to cancellation. 

1305 if d: 

1306 T3 += _copysign(sqrt(d), T3) # T3 = (r * t)^3 

1307 # _cbrt always returns the real root, _cbrt(-8) = -2 

1308 u = _cbrt(T3) # T = r * t 

1309 if u: # T can be zero; but then r2 / T -> 0 

1310 u += r**2 / u 

1311 u += r 

1312 elif S: # d == T3**2 == S**2: sqrt(d) == abs(S) == abs(T3) 

1313 u = _cbrt(S * _2_0) # == T3 + _copysign(abs(S), T3) 

1314 else: 

1315 u = _0_0 

1316 v = _hypot(u, y) # sqrt(u**2 + q) 

1317 # avoid loss of accuracy when u < 0 

1318 u = (q / (v - u)) if u < 0 else (v + u) 

1319 w = (u - q) / (v + v) # positive? 

1320 # rearrange expression for k to avoid loss of accuracy due to 

1321 # subtraction, division by 0 impossible because u > 0, w >= 0 

1322 k = u / (sqrt(w**2 + u) + w) # guaranteed positive 

1323 

1324 else: # q == 0 && r <= 0 

1325 # y = 0 with |x| <= 1. Handle this case directly, for 

1326 # y small, positive root is k = abs(y) / sqrt(1 - x^2) 

1327 k = _0_0 

1328 

1329 return k 

1330 

1331 

1332def _C4coeffs(nC4): # in .geodesicx.__main__ 

1333 '''(INTERNAL) Get the C{C4_24}, C{_27} or C{_30} series coefficients. 

1334 ''' 

1335 try: # from pygeodesy.geodesicx._C4_xx import _coeffs_xx as _coeffs 

1336 _C4_xx = _DOT_(_MODS.geodesicx.__name__, _UNDER_('_C4', nC4)) 

1337 _coeffs = _MODS.getattr(_C4_xx, _UNDER_('_coeffs', nC4)) 

1338 except (AttributeError, ImportError, TypeError) as x: 

1339 raise GeodesicError(nC4=nC4, cause=x) 

1340 n = _xnC4(nC4=nC4) 

1341 if len(_coeffs) != n: # double check 

1342 raise GeodesicError(_coeffs=len(_coeffs), _xnC4=n, nC4=nC4) 

1343 return _coeffs 

1344 

1345 

1346__all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact) 

1347 

1348# **) MIT License 

1349# 

1350# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

1351# 

1352# Permission is hereby granted, free of charge, to any person obtaining a 

1353# copy of this software and associated documentation files (the "Software"), 

1354# to deal in the Software without restriction, including without limitation 

1355# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1356# and/or sell copies of the Software, and to permit persons to whom the 

1357# Software is furnished to do so, subject to the following conditions: 

1358# 

1359# The above copyright notice and this permission notice shall be included 

1360# in all copies or substantial portions of the Software. 

1361# 

1362# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1363# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1364# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1365# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1366# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1367# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1368# OTHER DEALINGS IN THE SOFTWARE.