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>} (2008-2023) 

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.fmath import hypot as _hypot 

47from pygeodesy.fsums import fsumf_, fsum1f_ 

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

49 _sincos12, _sin1cos2, _xnC4 

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

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

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

53 _fix90, GDict, GeodesicError, _K_2_0, \ 

54 _norm2, _norm180, _polynomial, _signBit, \ 

55 _sincos2, _sincos2d, _sincos2de, _unsigned2 

56from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS 

57from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple 

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

59from pygeodesy.streprs import Fmt, pairs 

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

61 

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

63 

64__all__ = () 

65__version__ = '24.02.21' 

66 

67_MAXIT1 = 20 

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

69 

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

71# case 52.784459512564 0 -52.784459512563990912 179.634407464943777557 

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

73_TOL0 = EPS 

74_TOL1 = _TOL0 * -200 # negative 

75_TOL2 = _EPSqrt # == sqrt(_TOL0) 

76_TOL3 = _TOL2 * _0_1 

77_TOLb = _TOL2 * _TOL0 # Check on bisection interval 

78_THR1 = _TOL2 * _1000_0 + _1_0 

79 

80_TINY3 = _TINY * _3_0 

81_TOL08 = _TOL0 * _8_0 

82_TOL016 = _TOL0 * _16_0 

83 

84 

85def _atan12(*sincos12, **sineg0): 

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

87 ''' 

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

89 

90 

91def _eTOL2(f): 

92 # Using the auxiliary sphere solution with dnm computed at 

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

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

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

96 

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

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

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

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

101 

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

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

104 # nearly spherical case. 

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

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

107 

108 

109class _PDict(GDict): 

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

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

112 ''' 

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

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

115 ''' 

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

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

118 

119 def toGDict(self): # PYCHOK no cover 

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

121 ''' 

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

123 return GDict(rest) 

124 

125 return _rest(**self) 

126 

127 

128class GeodesicExact(_GeodesicBase): 

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

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

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

132 geographiclib/geographiclib-python>}. 

133 ''' 

134 _datum = _WGS84 

135 _nC4 = 30 # default C4order 

136 

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

138 C4Order=None): # for backward compatibility 

139 '''New L{GeodesicExact} instance. 

140 

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

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

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

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

145 is specified as C{scalar}. 

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

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

148 L{C4order}, default C{30}. 

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

150 

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

152 ''' 

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

154 if name: 

155 self.name = name 

156 if C4order: # XXX private copy, always? 

157 self.C4order = C4order 

158 elif C4Order: # for backward compatibility 

159 self.C4Order = C4Order 

160 

161 @Property_RO 

162 def a(self): 

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

164 ''' 

165 return self.ellipsoid.a 

166 

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

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

169 

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

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

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

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

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

175 the quantities to be returned. 

176 

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

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

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

180 

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

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

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

184 ''' 

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

186 

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

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

189 

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

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

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

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

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

195 the capabilities the L{GeodesicLineExact} instance 

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

197 returned by calls to L{GeodesicLineExact.Position} 

198 and L{GeodesicLineExact.ArcPosition}. 

199 

200 @return: A L{GeodesicLineExact} instance. 

201 

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

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

204 

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

206 

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

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

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

210 ''' 

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

212 

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

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

215 perimeter of a polygon. 

216 

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

218 area and perimeter (C{bool}). 

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

220 

221 @return: A L{GeodesicAreaExact} instance. 

222 

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

224 to the returned L{GeodesicAreaExact} instance. 

225 ''' 

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

227 name=name or self.name) 

228 if self.debug: 

229 gaX.verbose = True 

230 return gaX 

231 

232 @Property_RO 

233 def b(self): 

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

235 ''' 

236 return self.ellipsoid.b 

237 

238 @Property_RO 

239 def c2x(self): 

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

241 ''' 

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

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

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

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

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

247 return self.ellipsoid.c2x 

248 

249 c2 = c2x # in this particular case 

250 

251 def C4f(self, eps): 

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

253 

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

255 

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

257 ''' 

258 def _c4(nC4, C4x): 

259 i, x, e = 0, _1_0, eps 

260 _p = _polynomial 

261 for r in range(nC4, 0, -1): 

262 j = i + r 

263 yield _p(e, C4x, i, j) * x 

264 x *= e 

265 i = j 

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

267 

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

269 

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

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

272 ''' 

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

274 

275 @Property 

276 def C4order(self): 

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

278 ''' 

279 return self._nC4 

280 

281 @C4order.setter # PYCHOK .setter! 

282 def C4order(self, order): 

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

284 

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

286 ''' 

287 _xnC4(C4order=order) 

288 if self._nC4 != order: 

289 GeodesicExact._C4x._update(self) 

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

291 self._nC4 = order 

292 

293 @deprecated_Property 

294 def C4Order(self): 

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

296 ''' 

297 return self.C4order 

298 

299 @C4Order.setter # PYCHOK .setter! 

300 def C4Order(self, order): 

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

302 ''' 

303 _xnC4(C4Order=order) 

304 self.C4order = order 

305 

306 @Property_RO 

307 def _C4x(self): 

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

309 

310 @see: Property L{C4order}. 

311 ''' 

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

313 def _C4(nC4): 

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

315 _p = _polynomial 

316 for r in range(nC4 + 1, 1, -1): 

317 for j in range(1, r): 

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

319 yield _p(n, cs, i, j) / cs[j] 

320 i = j + 1 

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

322 

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

324 

325 @property_RO 

326 def datum(self): 

327 '''Get the datum (C{Datum}). 

328 ''' 

329 return self._datum 

330 

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

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

333 

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

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

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

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

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

339 the quantities to be returned. 

340 

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

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

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

344 

345 @see: C++ U{GeodesicExact.Direct 

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

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

348 ''' 

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

350 

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

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

353 (final bearing) in C{degrees}. 

354 

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

356 ''' 

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

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

359 

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

361 '''(INTERNAL) Short-cut version. 

362 ''' 

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

364 

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

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

367 

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

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

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

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

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

373 the capabilities the L{GeodesicLineExact} instance 

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

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

376 

377 @return: A L{GeodesicLineExact} instance. 

378 

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

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

381 

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

383 

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

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

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

387 ''' 

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

389 

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

391 '''(INTERNAL) Helper. 

392 ''' 

393 if self.f < 0: # PYCHOK no cover 

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

395 else: 

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

397 return dn 

398 

399 @Property_RO 

400 def e2(self): 

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

402 ''' 

403 return self.ellipsoid.e2 

404 

405 @Property_RO 

406 def _e2a2(self): 

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

408 ''' 

409 return self.e2 * self.ellipsoid.a2 

410 

411 @Property_RO 

412 def _e2_f1(self): 

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

414 ''' 

415 return self.e2 / self.f1 

416 

417 @Property_RO 

418 def _eF(self): 

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

420 ''' 

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

422 

423 def _eF_reset_cHe2_f1(self, x, y): 

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

425 ''' 

426 self._eF_reset_k2(x) 

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

428 

429 def _eF_reset_k2(self, x): 

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

431 ''' 

432 ep2 = self.ep2 

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

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

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

436 return k2 

437 

438 @Property_RO 

439 def ellipsoid(self): 

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

441 ''' 

442 return self.datum.ellipsoid 

443 

444 @Property_RO 

445 def ep2(self): 

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

447 ''' 

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

449 

450 e22 = ep2 # for ellipsoid compatibility 

451 

452 @Property_RO 

453 def _eTOL2(self): 

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

455 ''' 

456 return _eTOL2(self.f) 

457 

458 @Property_RO 

459 def flattening(self): 

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

461 ''' 

462 return self.ellipsoid.f 

463 

464 f = flattening 

465 

466 @Property_RO 

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

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

469 ''' 

470 return self.ellipsoid.f1 

471 

472 @Property_RO 

473 def _f180(self): 

474 '''(INTERNAL) Cached/memoized. 

475 ''' 

476 return self.f * _180_0 

477 

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

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

480 

481 @return: A L{GDict} ... 

482 ''' 

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

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

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

486 

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

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

489 

490 @return: A L{GDict} ... 

491 ''' 

492 Cs = Caps 

493 if self._debug: # PYCHOK no cover 

494 outmask |= Cs._DEBUG_INVERSE & self._debug 

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

496 # compute longitude difference carefully (with _diff182): 

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

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

499 lon12, lon12s = _diff182(lon1, lon2) 

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

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

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

503 else: 

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

505 if _K_2_0: # GeographicLib 2.0 

506 # make longitude difference positive 

507 lon12, lon_ = _unsigned2(lon12) 

508 if lon_: 

509 lon12s = -lon12s 

510 lam12 = radians(lon12) 

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

512 slam12, clam12 = _sincos2de(lon12, lon12s) 

513 # supplementary longitude difference 

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

515 else: # GeographicLib 1.52 

516 # make longitude difference positive and if very close 

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

518 if lon12 < 0: # _signBit(lon12) 

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

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

521 else: 

522 lon_, lon12 = False, _around(lon12) 

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

524 lam12 = radians(lon12) 

525 if lon12 > _90_0: 

526 slam12, clam12 = _sincos2d(lon12s) 

527 clam12 = -clam12 

528 else: 

529 slam12, clam12 = _sincos2(lam12) 

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

531 lat1 = _around(_fix90(lat1)) 

532 lat2 = _around(_fix90(lat2)) 

533 r.set_(lat1=lat1, lat2=lat2) 

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

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

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

537 if swap_: 

538 lat1, lat2 = lat2, lat1 

539 lon_ = not lon_ 

540 if _signBit(lat1): 

541 lat_ = False # note, False 

542 else: # make lat1 <= -0 

543 lat_ = True # note, True 

544 lat1, lat2 = -lat1, -lat2 

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

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

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

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

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

550 # this enforces some symmetries in the results returned. 

551 

552 # Initialize for the meridian. No longitude calculation is 

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

554 sbet1, cbet1 = self._sinf1cos2d(lat1) 

555 sbet2, cbet2 = self._sinf1cos2d(lat2) 

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

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

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

559 # in assigning calp2 in _Lambda6. Sometimes these quantities 

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

561 # example where is is necessary is the inverse problem 

562 # 48.522876735459 0 -48.52287673545898293 179.599720456223079643 

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

564 if cbet1 < -sbet1: 

565 if cbet2 == cbet1: 

566 sbet2 = copysign(sbet1, sbet2) 

567 elif fabs(sbet2) == -sbet1: 

568 cbet2 = cbet1 

569 

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

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

572 

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

574 if lat1 == -90 or slam12 == 0: 

575 # Endpoints are on a single full meridian, 

576 # so the geodesic might lie on a meridian. 

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

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

579 # tan(bet) = tan(sig) * cos(alp) 

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

581 # sig12 = sig2 - sig1 

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

583 s12x, m12x, _, \ 

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

585 # Add the check for sig12 since zero length geodesics 

586 # might yield m12 < 0. Test case was 

587 # echo 20.001 0 20.001 0 | GeodSolve -i 

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

589 # geodesic which is not a shortest path. 

590 if m12x >= 0 or sig12 < _1_0: 

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

592 # Prevent negative s12 or m12 from geographiclib 1.52 

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

594 sig12 = m12x = s12x = _0_0 

595 else: 

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

597 _meridian = False 

598 C = 1 

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

600 # _meridian = True 

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

602 

603 if _meridian: 

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

605 if _b: # Geodesic runs along equator 

606 calp1 = calp2 = _0_0 

607 salp1 = salp2 = _1_0 

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

609 somg12, comg12 = _sincos2(sig12) 

610 m12x = self.b * somg12 

611 s12x = self.a * lam12 

612 M12 = M21 = comg12 

613 a12 = lon12 / self.f1 

614 C = 2 

615 else: 

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

617 # meridian and geodesic is neither meridional or equatorial. 

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

619 # Figure a starting point for Newton's method 

620 sig12, salp1, calp1, \ 

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

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

623 # pre-compute the constant _Lambda6 term, once 

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

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

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

627 sig12, salp1, calp1, \ 

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

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

630 if (outmask & Cs.AREA): 

631 # omg12 = lam12 - domg12 

632 s, c = _sincos2(domg12) 

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

634 C = 3 # Newton 

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

636 C = 4 # Short lines 

637 s, c = _sincos2(sig12 / dnm) 

638 m12x = dnm**2 * s 

639 s12x = dnm * sig12 

640 M12 = M21 = c 

641 if (outmask & Cs.AREA): 

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

643 

644 else: # _meridian is False 

645 somg12 = comg12 = NAN 

646 

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

648 

649 if (outmask & Cs.DISTANCE): 

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

651 

652 if (outmask & Cs.REDUCEDLENGTH): 

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

654 

655 if (outmask & Cs.GEODESICSCALE): 

656 if swap_: 

657 M12, M21 = M21, M12 

658 r.set_(M12=unsigned0(M12), 

659 M21=unsigned0(M21)) 

660 

661 if (outmask & Cs.AREA): 

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

663 salp2, calp2, 

664 somg12, comg12, p) 

665 if _xor(swap_, lat_, lon_): 

666 S12 = -S12 

667 r.set_(S12=unsigned0(S12)) 

668 

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

670 if swap_: 

671 salp1, salp2 = salp2, salp1 

672 calp1, calp2 = calp2, calp1 

673 if _xor(swap_, lon_): 

674 salp1, salp2 = -salp1, -salp2 

675 if _xor(swap_, lat_): 

676 calp1, calp2 = -calp1, -calp2 

677 

678 if (outmask & Cs.AZIMUTH): 

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

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

681 if (outmask & Cs._SALPs_CALPs): 

682 r.set_(salp1=salp1, calp1=calp1, 

683 salp2=salp2, calp2=calp2) 

684 

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

686 E, eF = self.ellipsoid, self._eF 

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

688 e=E.e, e2=self.e2, ep2=self.ep2, 

689 c2=E.c2, c2x=self.c2x, 

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

691 eFk2=eF.k2, eFa2=eF.alpha2) 

692 p.update(r) # r overrides p 

693 r = p.toGDict() 

694 return self._iter2tion(r, p) 

695 

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

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

698 

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

700 s12, m12, M12, M21, S12)}. 

701 ''' 

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

703 return r.toDirect9Tuple() 

704 

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

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

707 

708 @return: A L{GeodesicLineExact} instance. 

709 ''' 

710 azi1 = _norm180(azi1) 

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

712 s, c = _sincos2d(_around(azi1)) 

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

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

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

716 

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

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

719 

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

721 m12, M12, M21, S12)}. 

722 ''' 

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

724 return r.toInverse10Tuple() 

725 

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

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

728 ''' 

729 if wrap: 

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

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

732 

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

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

735 

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

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

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

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

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

741 the quantities to be returned. 

742 

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

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

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

746 

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

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

749 

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

751 

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

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

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

755 ''' 

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

757 

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

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

760 

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

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

763 ''' 

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

765 # and .HeightIDWkarney._distances 

766 if wrap: 

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

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

769 

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

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

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

773 

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

775 ''' 

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

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

778 iteration=r.iteration) 

779 

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

781 '''(INTERNAL) Short-cut version. 

782 ''' 

783 if wrap: 

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

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

786 

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

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

789 

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

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

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

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

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

795 the capabilities the L{GeodesicLineExact} instance 

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

797 returned by calls to L{GeodesicLineExact.Position} 

798 and L{GeodesicLineExact.ArcPosition}. 

799 

800 @return: A L{GeodesicLineExact} instance. 

801 

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

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

804 

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

806 

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

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

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

810 ''' 

811 Cs = Caps 

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

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

814 azi1 = _atan2d(r.salp1, r.calp1) 

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

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

817 

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

819 salp2, calp2, 

820 somg12, comg12, p): 

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

822 

823 @return: Area C{S12}. 

824 ''' 

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

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

827 A4 = calp0 * salp0 

828 if A4: 

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

830 k2 = calp0**2 * self.ep2 

831 C4a = self._C4f_k2(k2) 

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

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

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

835 A4 *= self._e2a2 

836 S12 = A4 * (B42 - B41) 

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

838 A4 = B41 = B42 = k2 = S12 = _0_0 

839 

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

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

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

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

844 # (tan(bet1/2) + tan(bet2/2)) / 

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

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

847 dbet1 = p.cbet1 + _1_0 

848 dbet2 = p.cbet2 + _1_0 

849 domg12 = comg12 + _1_0 

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

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

852 alp12 = atan2(salp12, calp12) * _2_0 

853 else: 

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

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

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

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

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

859 # Following ensures the correct behavior. 

860 if salp12 == 0 and calp12 < 0: 

861 alp12 = _copysign(PI, calp1) 

862 else: 

863 alp12 = atan2(salp12, calp12) 

864 

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

866 return S12 + self.c2x * alp12 

867 

868 def _InverseStart6(self, lam12, p): 

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

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

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

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

873 

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

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

876 ''' 

877 sig12 = None # use Newton 

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

879 

880 # bet12 = bet2 - bet1 in [0, PI) 

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

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

883 if shortline: 

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

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

886 t = (p.sbet1 + p.sbet2)**2 

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

888 dnm = sqrt(_1_0 + self.ep2 * s) 

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

890 else: 

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

892 

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

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

895 

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

897 s = somg12**2 / c 

898 t = p.sbet1 * p.cbet2 * s 

899 salp1 = p.cbet2 * somg12 

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

901 

902 ssig12 = _hypot(salp1, calp1) 

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

904 

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

906 t = c if comg12 < 0 else s 

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

908 sbet12 - p.cbet1 * p.sbet2 * t) 

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

910 

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

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

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

914 

915 else: 

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

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

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

919 f = self.f 

920 if f < 0: # PYCHOK no cover 

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

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

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

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

925 Caps.REDUCEDLENGTH, p) 

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

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

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

929 y = lam12x / sca 

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

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

932 x = lam12x / sca # dlon 

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

934 

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

936 if f < 0: # PYCHOK no cover 

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

938 salp1 = sqrt(_1_0 - calp1**2) 

939 else: 

940 salp1 = min( _1_0, -x) 

941 calp1 = -sqrt(_1_0 - salp1**2) 

942 else: 

943 # Estimate alp1, by solving the astroid problem. 

944 # 

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

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

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

948 # 

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

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

951 # number of Newton iterations for astroid cases from 2.24 

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

953 # number of iterations are as follows: 

954 # 

955 # change percent 

956 # 1 5 

957 # 0 78 

958 # -1 16 

959 # -2 0.6 

960 # -3 0.04 

961 # -4 0.002 

962 # 

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

964 # estimating alp1 directly, n = number of iterations 

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

966 # 

967 # iter m n 

968 # 0 148 186 

969 # 1 13046 13845 

970 # 2 93315 102225 

971 # 3 36189 32341 

972 # 4 5396 7 

973 # 5 455 1 

974 # 6 56 0 

975 # 

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

977 k = _Astroid(x, y) 

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

979 (x * k / (k + _1_0)) 

980 s, c = _sincos2(-sca) # omg12a 

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

982 salp1 = p.cbet2 * s 

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

984 

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

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

987 

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

989 

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

991 '''(INTERNAL) Helper. 

992 

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

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

995 ''' 

996 eF = self._eF 

997 f1 = self.f1 

998 

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

1000 # Break degeneracy of equatorial line 

1001 calp1 = -_TINY 

1002 

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

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

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

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

1007 # = sin(bet1) * tan(alp1) 

1008 somg1 = salp0 * p.sbet1 

1009 comg1 = calp1 * p.cbet1 

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

1011 # Without normalization we have schi1 = somg1 

1012 cchi1 = f1 * p.dn1 * comg1 

1013 

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

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

1016 # yield singularities in the Newton iteration. 

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

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

1019 # calp2 = sqrt(1 - sq(salp2)) 

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

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

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

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

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

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

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

1027 somg2 = salp0 * p.sbet2 

1028 comg2 = calp2 * p.cbet2 

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

1030 # without normalization we have schi2 = somg2 

1031 cchi2 = f1 * p.dn2 * comg2 

1032 

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

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

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

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

1037 

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

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

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

1041 

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

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

1044 -eF.deltaH(*p.sncndn1), sig12) 

1045 # eta = chi12 - lam12 

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

1047 # domg12 = chi12 - omg12 - deta12 

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

1049 

1050 dlam12 = NAN # dv > 0 in ._Newton6 

1051 if diffp: 

1052 d = calp2 * p.cbet2 

1053 if d: 

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

1055 dlam12 *= f1 / d 

1056 elif p.sbet1: 

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

1058 

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

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

1061 

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

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

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

1065 coefficient of secular term in expression for reduced 

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

1067 

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

1069 ''' 

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

1071 

1072 Cs = Caps 

1073 eF = self._eF 

1074 

1075 # outmask &= Cs._OUT_MASK 

1076 if (outmask & Cs.DISTANCE): 

1077 # Missing a factor of self.b 

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

1079 -eF.deltaE(*p.sncndn1), sig12) 

1080 

1081 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE): 

1082 m0x = -eF.k2 * eF.cD * _2__PI 

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

1084 -eF.deltaD(*p.sncndn1), sig12) 

1085 if (outmask & Cs.REDUCEDLENGTH): 

1086 m0 = m0x 

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

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

1089 # accurate cancellation for coincident points. 

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

1091 -p.dn1 * (p.ssig1 * p.csig2), 

1092 J12 * (p.csig1 * p.csig2)) 

1093 if (outmask & Cs.GEODESICSCALE): 

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

1095 p.csig1 * p.csig2 

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

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

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

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

1100 

1101 return s12b, m12b, m0, M12, M21 

1102 

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

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

1105 on a single geodesic. 

1106 

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

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

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

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

1111 the capabilities the L{GeodesicLineExact} instance 

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

1113 returnedby calls to L{GeodesicLineExact.Position} 

1114 and L{GeodesicLineExact.ArcPosition}. 

1115 

1116 @return: A L{GeodesicLineExact} instance. 

1117 

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

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

1120 the limit C{ε → 0+}. 

1121 

1122 @see: C++ U{GeodesicExact.Line 

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

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

1125 ''' 

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

1127 

1128 @Property_RO 

1129 def n(self): 

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

1131 ''' 

1132 return self.ellipsoid.n 

1133 

1134 @Property_RO 

1135 def _n_0_1(self): 

1136 '''(INTERNAL) Cached once. 

1137 ''' 

1138 return fabs(self.n) > _0_1 

1139 

1140 @Property_RO 

1141 def _n6PI(self): 

1142 '''(INTERNAL) Cached once. 

1143 ''' 

1144 return fabs(self.n) * _6_0 * PI 

1145 

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

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

1148 

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

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

1151 ''' 

1152 _abs = fabs 

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

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

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

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

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

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

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

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

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

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

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

1164 salp1a = salp1b = _TINY 

1165 calp1a, calp1b = _1_0, _N_1_0 

1166 MAXIT1, TOL0 = _MAXIT1, _TOL0 

1167 HALF, TOLb = _0_5, _TOLb 

1168 tripb, TOLv = False, TOL0 

1169 for i in range(_MAXIT2): 

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

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

1172 # checking Inverse problem by Direct). 

1173 # 

1174 # max iterations 

1175 # log2(b/a) error mean sd 

1176 # -7 387 5.33 3.68 

1177 # -6 345 5.19 3.43 

1178 # -5 269 5.00 3.05 

1179 # -4 210 4.76 2.44 

1180 # -3 115 4.55 1.87 

1181 # -2 69 4.35 1.38 

1182 # -1 36 4.05 1.03 

1183 # 0 15 0.01 0.13 

1184 # 1 25 5.10 1.53 

1185 # 2 96 5.61 2.09 

1186 # 3 318 6.02 2.74 

1187 # 4 985 6.24 3.22 

1188 # 5 2352 6.32 3.44 

1189 # 6 6008 6.30 3.45 

1190 # 7 19024 6.19 3.30 

1191 v, sig12, salp2, calp2, \ 

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

1193 

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

1195 # reversed test to allow escape with NaNs 

1196 if tripb or _abs(v) < TOLv: 

1197 break 

1198 # update bracketing values 

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

1200 salp1b, calp1b = salp1, calp1 

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

1202 salp1a, calp1a = salp1, calp1 

1203 

1204 if i < MAXIT1 and dv > 0: 

1205 dalp1 = -v / dv 

1206 if _abs(dalp1) < PI: 

1207 s, c = _sincos2(dalp1) 

1208 # nalp1 = alp1 + dalp1 

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

1210 if s > 0: 

1211 salp1, calp1 = _norm2(s, c) 

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

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

1214 # based on epsilon instead of sqrt(epsilon) 

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

1216 continue 

1217 TOLv = TOL0 

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

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

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

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

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

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

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

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

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

1227 (calp1a + calp1b) * HALF) 

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

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

1230 else: 

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

1232 

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

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

1235 

1236 Polygon = Area # for C{geographiclib} compatibility 

1237 

1238 def _sinf1cos2d(self, lat): 

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

1240 ''' 

1241 sbet, cbet = _sincos2d(lat) 

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

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

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

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

1246 

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

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

1249 

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

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

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

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

1254 

1255 @return: Tuple items (C{str}). 

1256 ''' 

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

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

1259 

1260 

1261class GeodesicLineExact(_GeodesicLineExact): 

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

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

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

1265 geographiclib/geographiclib-python>}. 

1266 ''' 

1267 

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

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

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

1271 

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

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

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

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

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

1277 the capabilities the L{GeodesicLineExact} instance 

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

1279 returned by calls to L{GeodesicLineExact.Position} 

1280 and L{GeodesicLineExact.ArcPosition}. 

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

1282 

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

1284 ''' 

1285 _xinstanceof(GeodesicExact, geodesic=geodesic) 

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

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

1288# _update_all(geodesic) 

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

1290 

1291 

1292def _Astroid(x, y): 

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

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

1295 ''' 

1296 p = x**2 

1297 q = y**2 

1298 r = fsumf_(_1_0, q, p, _N_2_0) 

1299 if r > 0 or q: 

1300 # avoid possible division by zero when r = 0 

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

1302 S = p * q / _4_0 # S = r^3 * s 

1303 if r: 

1304 r = r / _6_0 # /= chokes PyChecker 

1305 r3 = r**3 

1306 T3 = r3 + S 

1307 # discriminant of the quadratic equation for T3 is 

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

1309 d = (r3 + T3) * S 

1310 if d < 0: 

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

1312 a = atan2(sqrt(-d), -T3) / _3_0 

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

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

1315 u = (cos(a) * _2_0 + _1_0) * r 

1316 else: 

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

1318 # minimize loss of precision due to cancellation. 

1319 if d: 

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

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

1322 u = _cbrt(T3) # T = r * t 

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

1324 u += r**2 / u 

1325 u += r 

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

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

1328 else: 

1329 u = _0_0 

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

1331 # avoid loss of accuracy when u < 0 

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

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

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

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

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

1337 

1338 else: # q == 0 && r <= 0 

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

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

1341 k = _0_0 

1342 

1343 return k 

1344 

1345 

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

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

1348 ''' 

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

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

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

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

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

1354 n = _xnC4(nC4=nC4) 

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

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

1357 return _coeffs 

1358 

1359 

1360__all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact) 

1361 

1362# **) MIT License 

1363# 

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

1365# 

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

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

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

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

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

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

1372# 

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

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1375# 

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

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

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