Coverage for pygeodesy/geodesicx/gx.py: 94%

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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:Charles@Karney.com>} (2008-2022) 

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# - beta = latitude on auxiliary sphere 

25# - omega = longitude on auxiliary sphere 

26# - lambda = longitude 

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 _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 

44# from pygeodesy.datums import _a_ellipsoid # from .karney 

45from pygeodesy.fsums import fsum_, fsum1_ 

46from pygeodesy.geodesicx.gxbases import _cosSeries, _GeodesicBase, \ 

47 _sincos12, _sin1cos2, _xnC4 

48from pygeodesy.geodesicx.gxline import _GeodesicLineExact, _TINY, _update_glXs 

49from pygeodesy.interns import NN, _COMMASPACE_, _DOT_, _UNDER_ 

50from pygeodesy.karney import _around, _atan2d, Caps, _cbrt, _copysign, _diff182, \ 

51 _a_ellipsoid, _EWGS84, GDict, GeodesicError, _fix90, \ 

52 _hypot, _K_2_0, _norm2, _norm180, _polynomial, \ 

53 _signBit, _sincos2, _sincos2d, _sincos2de, _unsigned2 

54from pygeodesy.lazily import _ALL_DOCS, _ALL_MODS as _MODS 

55from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple 

56from pygeodesy.props import deprecated_Property, Property, Property_RO 

57from pygeodesy.streprs import Fmt, pairs 

58from pygeodesy.utily import atan2d as _atan2d_reverse, unroll180, wrap360 

59 

60from math import atan2, copysign, cos, degrees, fabs, radians, sqrt 

61 

62__all__ = () 

63__version__ = '23.04.11' 

64 

65_MAXIT1 = 20 

66_MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits 

67 

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

69# case 52.784459512564 0 -52.784459512563990912 179.634407464943777557 

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

71_TOL0 = EPS 

72_TOL1 = _TOL0 * -200 # negative 

73_TOL2 = _EPSqrt # == sqrt(_TOL0) 

74_TOL3 = _TOL2 * _0_1 

75_TOLb = _TOL2 * _TOL0 # Check on bisection interval 

76_THR1 = _TOL2 * _1000_0 + _1_0 

77 

78_TINY3 = _TINY * _3_0 

79_TOL08 = _TOL0 * _8_0 

80_TOL016 = _TOL0 * _16_0 

81 

82 

83def _atan12(*sincos12, **sineg0): 

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

85 ''' 

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

87 

88 

89def _eTOL2(f): 

90 # Using the auxiliary sphere solution with dnm computed at 

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

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

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

94 

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

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

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

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

99 

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

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

102 # nearly spherical case. 

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

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

105 

106 

107class _PDict(GDict): 

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

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

110 ''' 

111 def setsigs(self, ssig1, csig1, ssig2, csig2): 

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

113 ''' 

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

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

116 

117 def toGDict(self): # PYCHOK no cover 

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

119 ''' 

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

121 return GDict(rest) 

122 

123 return _rest(**self) 

124 

125 

126class GeodesicExact(_GeodesicBase): 

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

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

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

130 geographiclib/geographiclib-python>}. 

131 ''' 

132 _E = _EWGS84 

133 _nC4 = 30 # default C4order 

134 

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

136 C4Order=None): # for backward compatibility 

137 '''New L{GeodesicExact} instance. 

138 

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

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

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

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

143 is specified as C{scalar}. 

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

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

146 L{C4order}, default C{30}. 

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

148 

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

150 ''' 

151 if a_ellipsoid not in (GeodesicExact._E, None): 

152 self._E = _a_ellipsoid(a_ellipsoid, f, name=name) 

153 

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): 

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) 

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 / fsum_(_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 def _coeffs(self, nC4): 

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

308 ''' 

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

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

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

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

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

314 n = _xnC4(nC4=nC4) 

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

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

317 return _coeffs 

318 

319 @Property_RO 

320 def _C4x(self): 

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

322 

323 @see: Property L{C4order}. 

324 ''' 

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

326 def _C4(nC4): 

327 i, n, cs = 0, self.n, self._coeffs(nC4) 

328 _p = _polynomial 

329 for r in range(nC4 + 1, 1, -1): 

330 for j in range(1, r): 

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

332 yield _p(n, cs, i, j) / cs[j] 

333 i = j + 1 

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

335 

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

337 

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

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

340 

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

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

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

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

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

346 the quantities to be returned. 

347 

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

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

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

351 

352 @see: C++ U{GeodesicExact.Direct 

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

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

355 ''' 

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

357 

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

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

360 (final bearing) in C{degrees}. 

361 

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

363 ''' 

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

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

366 

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

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

369 

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

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

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

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

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

375 the capabilities the L{GeodesicLineExact} instance 

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

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

378 

379 @return: A L{GeodesicLineExact} instance. 

380 

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

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

383 

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

385 

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

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

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

389 ''' 

390 return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps) 

391 

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

393 '''(INTERNAL) Helper. 

394 ''' 

395 if self.f < 0: # PYCHOK no cover 

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

397 else: 

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

399 return dn 

400 

401 @Property_RO 

402 def e2(self): 

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

404 ''' 

405 return self.ellipsoid.e2 

406 

407 @Property_RO 

408 def _e2a2(self): 

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

410 ''' 

411 return self.e2 * self.ellipsoid.a2 

412 

413 @Property_RO 

414 def _e2_f1(self): 

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

416 ''' 

417 return self.e2 / self.f1 

418 

419 @Property_RO 

420 def _eF(self): 

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

422 ''' 

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

424 

425 def _eF_reset_cHe2_f1(self, x, y): 

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

427 ''' 

428 self._eF_reset_k2(x) 

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

430 

431 def _eF_reset_k2(self, x): 

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

433 ''' 

434 ep2 = self.ep2 

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

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

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

438 return k2 

439 

440 @Property_RO 

441 def ellipsoid(self): 

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

443 ''' 

444 return self._E 

445 

446 @Property_RO 

447 def ep2(self): 

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

449 ''' 

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

451 

452 e22 = ep2 # for ellipsoid compatibility 

453 

454 @Property_RO 

455 def _eTOL2(self): 

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

457 ''' 

458 return _eTOL2(self.f) 

459 

460 @Property_RO 

461 def f(self): 

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

463 ''' 

464 return self.ellipsoid.f 

465 

466 flattening = f 

467 

468 @Property_RO 

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

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

471 ''' 

472 return self.ellipsoid.f1 

473 

474 @Property_RO 

475 def _f180(self): 

476 '''(INTERNAL) Cached/memoized. 

477 ''' 

478 return self.f * _180_0 

479 

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

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

482 

483 @return: A L{GDict} ... 

484 ''' 

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

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

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

488 

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

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

491 

492 @return: A L{GDict} ... 

493 ''' 

494 Cs = Caps 

495 if self._debug: # PYCHOK no cover 

496 outmask |= Cs._DEBUG_INVERSE & self._debug 

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

498 # compute longitude difference carefully (with _diff182): 

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

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

501 lon12, lon12s = _diff182(lon1, lon2) 

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

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

504 r = GDict(lon1=lon1, lon2=fsum_(lon1, lon12, lon12s)) 

505 else: 

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

507 if _K_2_0: # GeographicLib 2.0 

508 # make longitude difference positive 

509 lon12, lon_ = _unsigned2(lon12) 

510 if lon_: 

511 lon12s = -lon12s 

512 lam12 = radians(lon12) 

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

514 slam12, clam12 = _sincos2de(lon12, lon12s) 

515 # supplementary longitude difference 

516 lon12s = fsum_(_180_0, -lon12, -lon12s) 

517 else: # GeographicLib 1.52 

518 # make longitude difference positive and if very close 

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

520 if lon12 < 0: # _signBit(lon12) 

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

522 lon12s = _around(fsum_(_180_0, -lon12, lon12s)) 

523 else: 

524 lon_, lon12 = False, _around(lon12) 

525 lon12s = _around(fsum_(_180_0, -lon12, -lon12s)) 

526 lam12 = radians(lon12) 

527 if lon12 > _90_0: 

528 slam12, clam12 = _sincos2d(lon12s) 

529 clam12 = -clam12 

530 else: 

531 slam12, clam12 = _sincos2(lam12) 

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

533 lat1 = _around(_fix90(lat1)) 

534 lat2 = _around(_fix90(lat2)) 

535 r.set_(lat1=lat1, lat2=lat2) 

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

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

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

539 if swap_: 

540 lat1, lat2 = lat2, lat1 

541 lon_ = not lon_ 

542 if _signBit(lat1): 

543 lat_ = False # note, False 

544 else: # make lat1 <= -0 

545 lat_ = True # note, True 

546 lat1, lat2 = -lat1, -lat2 

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

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

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

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

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

552 # this enforces some symmetries in the results returned. 

553 

554 # Initialize for the meridian. No longitude calculation is 

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

556 sbet1, cbet1 = self._sinf1cos2d(lat1) 

557 sbet2, cbet2 = self._sinf1cos2d(lat2) 

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

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

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

561 # in assigning calp2 in _Lambda6. Sometimes these quantities 

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

563 # example where is is necessary is the inverse problem 

564 # 48.522876735459 0 -48.52287673545898293 179.599720456223079643 

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

566 if cbet1 < -sbet1: 

567 if cbet2 == cbet1: 

568 sbet2 = copysign(sbet1, sbet2) 

569 elif fabs(sbet2) == -sbet1: 

570 cbet2 = cbet1 

571 

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

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

574 

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

576 if lat1 == -90 or slam12 == 0: 

577 # Endpoints are on a single full meridian, 

578 # so the geodesic might lie on a meridian. 

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

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

581 # tan(bet) = tan(sig) * cos(alp) 

582 p.setsigs(sbet1, calp1 * cbet1, sbet2, calp2 * cbet2) 

583 # sig12 = sig2 - sig1 

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

585 s12x, m12x, _, \ 

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

587 # Add the check for sig12 since zero length geodesics 

588 # might yield m12 < 0. Test case was 

589 # echo 20.001 0 20.001 0 | GeodSolve -i 

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

591 # geodesic which is not a shortest path. 

592 if m12x >= 0 or sig12 < _1_0: 

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

594 # Prevent negative s12 or m12 from geographiclib 1.52 

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

596 sig12 = m12x = s12x = _0_0 

597 else: 

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

599 _meridian = False 

600 C = 1 

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

602 # _meridian = True 

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

604 

605 if _meridian: 

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

607 if _b: # Geodesic runs along equator 

608 calp1 = calp2 = _0_0 

609 salp1 = salp2 = _1_0 

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

611 somg12, comg12 = _sincos2(sig12) 

612 m12x = self.b * somg12 

613 s12x = self.a * lam12 

614 M12 = M21 = comg12 

615 a12 = lon12 / self.f1 

616 C = 2 

617 else: 

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

619 # meridian and geodesic is neither meridional or equatorial. 

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

621 # Figure a starting point for Newton's method 

622 sig12, salp1, calp1, \ 

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

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

625 # pre-compute the constant _Lambda6 term, once 

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

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

628 ((sbet1 + sbet2) * (sbet1 - sbet2)))) 

629 sig12, salp1, calp1, \ 

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

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

632 if (outmask & Cs.AREA): 

633 # omg12 = lam12 - domg12 

634 s, c = _sincos2(domg12) 

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

636 C = 3 # Newton 

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

638 C = 4 # Short lines 

639 s, c = _sincos2(sig12 / dnm) 

640 m12x = dnm**2 * s 

641 s12x = dnm * sig12 

642 M12 = M21 = c 

643 if (outmask & Cs.AREA): 

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

645 

646 else: # _meridian is False 

647 somg12 = comg12 = NAN 

648 

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

650 

651 if (outmask & Cs.DISTANCE): 

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

653 

654 if (outmask & Cs.REDUCEDLENGTH): 

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

656 

657 if (outmask & Cs.GEODESICSCALE): 

658 if swap_: 

659 M12, M21 = M21, M12 

660 r.set_(M12=unsigned0(M12), 

661 M21=unsigned0(M21)) 

662 

663 if (outmask & Cs.AREA): 

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

665 salp2, calp2, 

666 somg12, comg12, p) 

667 if _xor(swap_, lat_, lon_): 

668 S12 = -S12 

669 r.set_(S12=unsigned0(S12)) 

670 

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

672 if swap_: 

673 salp1, salp2 = salp2, salp1 

674 calp1, calp2 = calp2, calp1 

675 if _xor(swap_, lon_): 

676 salp1, salp2 = -salp1, -salp2 

677 if _xor(swap_, lat_): 

678 calp1, calp2 = -calp1, -calp2 

679 

680 if (outmask & Cs.AZIMUTH): 

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

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

683 if (outmask & Cs._SALPs_CALPs): 

684 r.set_(salp1=salp1, calp1=calp1, 

685 salp2=salp2, calp2=calp2) 

686 

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

688 E, eF = self.ellipsoid, self._eF 

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

690 e=E.e, e2=self.e2, ep2=self.ep2, 

691 c2=E.c2, c2x=self.c2x, 

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

693 eFk2=eF.k2, eFa2=eF.alpha2) 

694 p.update(r) # r overrides p 

695 r = p.toGDict() 

696 return self._iter2tion(r, p) 

697 

698 def _GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask): 

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

700 

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

702 s12, m12, M12, M21, S12)}. 

703 ''' 

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

705 return r.toDirect9Tuple() 

706 

707 def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12, caps): 

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

709 

710 @return: A L{GeodesicLineExact} instance. 

711 ''' 

712 azi1 = _norm180(azi1) 

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

714 s, c = _sincos2d(_around(azi1)) 

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

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

717 self._debug, s, c)._GenSet(arcmode, s12_a12) 

718 

719 def _GenInverse(self, lat1, lon1, lat2, lon2, outmask): 

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

721 

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

723 m12, M12, M21, S12)}. 

724 ''' 

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

726 return r.toInverse10Tuple() 

727 

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

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

730 

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

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

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

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

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

736 the quantities to be returned. 

737 

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

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

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

741 

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

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

744 

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

746 

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

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

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

750 ''' 

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

752 

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

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

755 ''' 

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

757 # and .HeightIDWkarney._distances 

758 _, lon2 = unroll180(lon1, lon2, wrap=wrap) # self.LONG_UNROLL 

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

760 

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

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

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

764 

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

766 ''' 

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

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

769 iteration=r.iteration) 

770 

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

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

773 

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

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

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

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

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

779 the capabilities the L{GeodesicLineExact} instance 

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

781 returned by calls to L{GeodesicLineExact.Position} 

782 and L{GeodesicLineExact.ArcPosition}. 

783 

784 @return: A L{GeodesicLineExact} instance. 

785 

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

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

788 

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

790 

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

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

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

794 ''' 

795 Cs = Caps 

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

797 C = (caps | Cs.DISTANCE) if (caps & Cs._DISTANCE_IN_OUT) else caps 

798 azi1 = _atan2d(r.salp1, r.calp1) 

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

800 self._debug, r.salp1, r.calp1)._GenSet(True, r.a12) 

801 

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

803 salp2, calp2, 

804 somg12, comg12, p): 

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

806 

807 @return: Area C{S12}. 

808 ''' 

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

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

811 A4 = calp0 * salp0 

812 if A4: 

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

814 k2 = calp0**2 * self.ep2 

815 C4a = self._C4f_k2(k2) 

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

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

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

819 A4 *= self._e2a2 

820 S12 = A4 * (B42 - B41) 

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

822 A4 = B41 = B42 = k2 = S12 = _0_0 

823 

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

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

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

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

828 # (tan(bet1/2) + tan(bet2/2)) / 

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

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

831 dbet1 = p.cbet1 + _1_0 

832 dbet2 = p.cbet2 + _1_0 

833 domg12 = comg12 + _1_0 

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

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

836 alp12 = _2_0 * atan2(salp12, calp12) 

837 else: 

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

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

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

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

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

843 # Following ensures the correct behavior. 

844 if salp12 == 0 and calp12 < 0: 

845 alp12 = _copysign(PI, calp1) 

846 else: 

847 alp12 = atan2(salp12, calp12) 

848 

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

850 return S12 + self.c2x * alp12 

851 

852 def _InverseStart6(self, lam12, p): 

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

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

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

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

857 

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

859 and C{p.setsigs} updated for Newton, C{sig12=None}. 

860 ''' 

861 sig12 = None # use Newton 

862 salp1 = calp1 = salp2 = calp2 = dnm = NAN 

863 

864 # bet12 = bet2 - bet1 in [0, PI) 

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

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

867 if shortline: 

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

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

870 t = (p.sbet1 + p.sbet2)**2 

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

872 dnm = sqrt(_1_0 + self.ep2 * s) 

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

874 else: 

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

876 

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

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

879 

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

881 s = somg12**2 / c 

882 t = p.sbet1 * p.cbet2 * s 

883 salp1 = p.cbet2 * somg12 

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

885 

886 ssig12 = _hypot(salp1, calp1) 

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

888 

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

890 t = c if comg12 < 0 else s 

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

892 sbet12 - p.cbet1 * p.sbet2 * t) 

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

894 

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

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

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

898 

899 else: 

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

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

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

903 f = self.f 

904 if f < 0: # PYCHOK no cover 

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

906 p.setsigs(p.sbet1, -p.cbet1, p.sbet2, p.cbet2) 

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

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

909 Caps.REDUCEDLENGTH, p) 

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

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

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

913 y = lam12x / sca 

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

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

916 x = lam12x / sca # dlon 

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

918 

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

920 if f < 0: # PYCHOK no cover 

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

922 salp1 = sqrt(_1_0 - calp1**2) 

923 else: 

924 salp1 = min( _1_0, -x) 

925 calp1 = -sqrt(_1_0 - salp1**2) 

926 else: 

927 # Estimate alp1, by solving the astroid problem. 

928 # 

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

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

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

932 # 

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

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

935 # number of Newton iterations for astroid cases from 2.24 

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

937 # number of iterations are as follows: 

938 # 

939 # change percent 

940 # 1 5 

941 # 0 78 

942 # -1 16 

943 # -2 0.6 

944 # -3 0.04 

945 # -4 0.002 

946 # 

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

948 # estimating alp1 directly, n = number of iterations 

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

950 # 

951 # iter m n 

952 # 0 148 186 

953 # 1 13046 13845 

954 # 2 93315 102225 

955 # 3 36189 32341 

956 # 4 5396 7 

957 # 5 455 1 

958 # 6 56 0 

959 # 

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

961 k = _Astroid(x, y) 

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

963 (x * k / (k + _1_0)) 

964 s, c = _sincos2(-sca) # omg12a 

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

966 salp1 = p.cbet2 * s 

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

968 

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

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

971 

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

973 

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

975 '''(INTERNAL) Helper. 

976 

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

978 dlam12} and C{p.setsigs} updated. 

979 ''' 

980 eF = self._eF 

981 f1 = self.f1 

982 

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

984 # Break degeneracy of equatorial line 

985 calp1 = -_TINY 

986 

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

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

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

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

991 # = sin(bet1) * tan(alp1) 

992 somg1 = salp0 * p.sbet1 

993 comg1 = calp1 * p.cbet1 

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

995 # Without normalization we have schi1 = somg1 

996 cchi1 = f1 * p.dn1 * comg1 

997 

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

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

1000 # yield singularities in the Newton iteration. 

1001 # sin(alp2) * cos(bet2) = sin(alp0) 

1002 salp2 = (salp0 / p.cbet2) if p.cbet2 != p.cbet1 else salp1 

1003 # calp2 = sqrt(1 - sq(salp2)) 

1004 # = sqrt(sq(calp0) - sq(sbet2)) / cbet2 

1005 # and subst for calp0 and rearrange to give (choose 

1006 # positive sqrt to give alp2 in [0, PI/2]). 

1007 calp2 = fabs(calp1) if p.bet12 is None else ( 

1008 sqrt((calp1 * p.cbet1)**2 + p.bet12) / p.cbet2) 

1009 # tan(bet2) = tan(sig2) * cos(alp2) 

1010 # tan(omg2) = sin(alp0) * tan(sig2). 

1011 somg2 = salp0 * p.sbet2 

1012 comg2 = calp2 * p.cbet2 

1013 ssig2, csig2 = _norm2(p.sbet2, comg2) 

1014 # without normalization we have schi2 = somg2 

1015 cchi2 = f1 * p.dn2 * comg2 

1016 

1017 # omg12 = omg2 - omg1, limit to [0, PI] 

1018 somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, sineg0=True) 

1019 # chi12 = chi2 - chi1, limit to [0, PI] 

1020 schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, sineg0=True) 

1021 

1022 p.setsigs(ssig1, csig1, ssig2, csig2) 

1023 # sig12 = sig2 - sig1, limit to [0, PI] 

1024 sig12 = _atan12(ssig1, csig1, ssig2, csig2, sineg0=True) 

1025 

1026 eta12 = self._eF_reset_cHe2_f1(calp0, salp0) * _2__PI # then ... 

1027 eta12 *= fsum1_(eF.deltaH(*p.sncndn2), 

1028 -eF.deltaH(*p.sncndn1), sig12) 

1029 # eta = chi12 - lam12 

1030 lam12 = _atan12(p.slam12, p.clam12, schi12, cchi12) - eta12 

1031 # domg12 = chi12 - omg12 - deta12 

1032 domg12 = _atan12( somg12, comg12, schi12, cchi12) - eta12 

1033 

1034 dlam12 = NAN # dv > 0 in ._Newton6 

1035 if diffp: 

1036 d = calp2 * p.cbet2 

1037 if d: 

1038 _, dlam12, _, _, _ = self._Length5(sig12, Caps.REDUCEDLENGTH, p) 

1039 dlam12 *= f1 / d 

1040 elif p.sbet1: 

1041 dlam12 = -f1 * p.dn1 * _2_0 / p.sbet1 

1042 

1043 # p.set_(deta12=-eta12, lam12=lam12) 

1044 return lam12, sig12, salp2, calp2, domg12, dlam12 

1045 

1046 def _Length5(self, sig12, outmask, p): 

1047 '''(INTERNAL) Return M{m12b = (reduced length) / self.b} and 

1048 calculate M{s12b = distance / self.b} and M{m0}, the 

1049 coefficient of secular term in expression for reduced 

1050 length and the geodesic scales C{M12} and C{M21}. 

1051 

1052 @return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}. 

1053 ''' 

1054 s12b = m12b = m0 = M12 = M21 = NAN 

1055 

1056 Cs = Caps 

1057 eF = self._eF 

1058 

1059 # outmask &= Cs._OUT_MASK 

1060 if (outmask & Cs.DISTANCE): 

1061 # Missing a factor of self.b 

1062 s12b = eF.cE * _2__PI * fsum1_(eF.deltaE(*p.sncndn2), 

1063 -eF.deltaE(*p.sncndn1), sig12) 

1064 

1065 if (outmask & Cs._REDUCEDLENGTH_GEODESICSCALE): 

1066 m0x = -eF.k2 * eF.cD * _2__PI 

1067 J12 = -m0x * fsum1_(eF.deltaD(*p.sncndn2), 

1068 -eF.deltaD(*p.sncndn1), sig12) 

1069 if (outmask & Cs.REDUCEDLENGTH): 

1070 m0 = m0x 

1071 # Missing a factor of self.b. Add parens around 

1072 # (csig1 * ssig2) and (ssig1 * csig2) to ensure 

1073 # accurate cancellation for coincident points. 

1074 m12b = fsum1_(p.dn2 * (p.csig1 * p.ssig2), 

1075 -p.dn1 * (p.ssig1 * p.csig2), 

1076 J12 * (p.csig1 * p.csig2)) 

1077 if (outmask & Cs.GEODESICSCALE): 

1078 M12 = M21 = p.ssig1 * p.ssig2 + \ 

1079 p.csig1 * p.csig2 

1080 t = (p.cbet1 - p.cbet2) * self.ep2 * \ 

1081 (p.cbet1 + p.cbet2) / (p.dn1 + p.dn2) 

1082 M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1 

1083 M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2 

1084 

1085 return s12b, m12b, m0, M12, M21 

1086 

1087 def Line(self, lat1, lon1, azi1, caps=Caps.ALL): 

1088 '''Set up a L{GeodesicLineExact} to compute several points 

1089 on a single geodesic. 

1090 

1091 @arg lat1: Latitude of the first point (C{degrees}). 

1092 @arg lon1: Longitude of the first point (C{degrees}). 

1093 @arg azi1: Azimuth at the first point (compass C{degrees}). 

1094 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

1095 the capabilities the L{GeodesicLineExact} instance 

1096 should possess, i.e., which quantities can be 

1097 returnedby calls to L{GeodesicLineExact.Position} 

1098 and L{GeodesicLineExact.ArcPosition}. 

1099 

1100 @return: A L{GeodesicLineExact} instance. 

1101 

1102 @note: If the point is at a pole, the azimuth is defined by keeping 

1103 B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking 

1104 the limit C{ε → 0+}. 

1105 

1106 @see: C++ U{GeodesicExact.Line 

1107 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicExact.html>} 

1108 and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/Python/doc/code.html>}. 

1109 ''' 

1110 return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug) 

1111 

1112 @Property_RO 

1113 def n(self): 

1114 '''Get the ellipsoid's I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}. 

1115 ''' 

1116 return self.ellipsoid.n 

1117 

1118 @Property_RO 

1119 def _n_0_1(self): 

1120 '''(INTERNAL) Cached once. 

1121 ''' 

1122 return fabs(self.n) > _0_1 

1123 

1124 @Property_RO 

1125 def _n6PI(self): 

1126 '''(INTERNAL) Cached once. 

1127 ''' 

1128 return fabs(self.n) * _6_0 * PI 

1129 

1130 def _Newton6(self, salp1, calp1, p): 

1131 '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length. 

1132 

1133 @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12)} 

1134 and C{p.iter} and C{p.trip} updated. 

1135 ''' 

1136 # This is a straightforward solution of f(alp1) = lambda12(alp1) - 

1137 # lam12 = 0 with one wrinkle. f(alp) has exactly one root in the 

1138 # interval (0, PI) and its derivative is positive at the root. 

1139 # Thus f(alp) is positive for alp > alp1 and negative for alp < alp1. 

1140 # During the course of the iteration, a range (alp1a, alp1b) is 

1141 # maintained which brackets the root and with each evaluation of 

1142 # f(alp) the range is shrunk, if possible. Newton's method is 

1143 # restarted whenever the derivative of f is negative (because the 

1144 # new value of alp1 is then further from the solution) or if the 

1145 # new estimate of alp1 lies outside (0,PI); in this case, the new 

1146 # starting guess is taken to be (alp1a + alp1b) / 2. 

1147 salp1a = salp1b = _TINY 

1148 calp1a, calp1b = _1_0, _N_1_0 

1149 MAXIT1, TOL0 = _MAXIT1, _TOL0 

1150 HALF, TOLb = _0_5, _TOLb 

1151 tripb, TOLv = False, TOL0 

1152 for i in range(_MAXIT2): 

1153 # 1/4 meridian = 10e6 meter and random input, 

1154 # estimated max error in nm (nano meter, by 

1155 # checking Inverse problem by Direct). 

1156 # 

1157 # max iterations 

1158 # log2(b/a) error mean sd 

1159 # -7 387 5.33 3.68 

1160 # -6 345 5.19 3.43 

1161 # -5 269 5.00 3.05 

1162 # -4 210 4.76 2.44 

1163 # -3 115 4.55 1.87 

1164 # -2 69 4.35 1.38 

1165 # -1 36 4.05 1.03 

1166 # 0 15 0.01 0.13 

1167 # 1 25 5.10 1.53 

1168 # 2 96 5.61 2.09 

1169 # 3 318 6.02 2.74 

1170 # 4 985 6.24 3.22 

1171 # 5 2352 6.32 3.44 

1172 # 6 6008 6.30 3.45 

1173 # 7 19024 6.19 3.30 

1174 v, sig12, salp2, calp2, \ 

1175 domg12, dv = self._Lambda6(salp1, calp1, i < MAXIT1, p) 

1176 

1177 # 2 * _TOL0 is approximately 1 ulp [0, PI] 

1178 # reversed test to allow escape with NaNs 

1179 if tripb or fabs(v) < TOLv: 

1180 break 

1181 # update bracketing values 

1182 if v > 0 and (i > MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)): 

1183 salp1b, calp1b = salp1, calp1 

1184 elif v < 0 and (i > MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)): 

1185 salp1a, calp1a = salp1, calp1 

1186 

1187 if i < MAXIT1 and dv > 0: 

1188 dalp1 = -v / dv 

1189 if fabs(dalp1) < PI: 

1190 s, c = _sincos2(dalp1) 

1191 # nalp1 = alp1 + dalp1 

1192 s, c = _sincos12(-s, c, salp1, calp1) 

1193 if s > 0: 

1194 salp1, calp1 = _norm2(s, c) 

1195 # in some regimes we don't get quadratic convergence 

1196 # because slope -> 0. So use convergence conditions 

1197 # based on epsilon instead of sqrt(epsilon) 

1198 TOLv = TOL0 if fabs(v) > _TOL016 else _TOL08 

1199 continue 

1200 

1201 # Either dv was not positive or updated value was outside 

1202 # legal range. Use the midpoint of the bracket as the next 

1203 # estimate. This mechanism is not needed for the WGS84 

1204 # ellipsoid, but it does catch problems with more eccentric 

1205 # ellipsoids. Its efficacy is such for the WGS84 test set 

1206 # with the starting guess set to alp1 = 90 deg: the WGS84 

1207 # test set: mean = 5.21, stdev = 3.93, max = 24 and WGS84 

1208 # with random input: mean = 4.74, stdev = 0.99 

1209 salp1, calp1 = _norm2((salp1a + salp1b) * HALF, 

1210 (calp1a + calp1b) * HALF) 

1211 tripb = fsum1_(calp1a, -calp1, fabs(salp1a - salp1)) < TOLb or \ 

1212 fsum1_(calp1b, -calp1, fabs(salp1b - salp1)) < TOLb 

1213 TOLv = TOL0 

1214 

1215 else: 

1216 raise GeodesicError(Fmt.no_convergence(v, TOLv), txt=repr(self)) # self.toRepr() 

1217 

1218 p.set_(iter=i, trip=tripb) # like .geodsolve._GDictInvoke: iter NOT iteration! 

1219 return sig12, salp1, calp1, salp2, calp2, domg12 

1220 

1221 Polygon = Area # for C{geographiclib} compatibility 

1222 

1223 def _sinf1cos2d(self, lat): 

1224 '''(INTERNAL) Helper, also for C{_G_GeodesicLineExact}. 

1225 ''' 

1226 sbet, cbet = _sincos2d(lat) 

1227 # ensure cbet1 = +epsilon at poles; doing the fix on beta means 

1228 # that sig12 will be <= 2*tiny for two points at the same pole 

1229 sbet, cbet = _norm2(sbet * self.f1, cbet) 

1230 return sbet, max(_TINY, cbet) 

1231 

1232 def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature 

1233 '''Return this C{GeodesicExact} as string. 

1234 

1235 @kwarg prec: The C{float} precision, number of decimal digits (0..9). 

1236 Trailing zero decimals are stripped for B{C{prec}} values 

1237 of 1 and above, but kept for negative B{C{prec}} values. 

1238 @kwarg sep: Separator to join (C{str}). 

1239 

1240 @return: Tuple items (C{str}). 

1241 ''' 

1242 d = dict(ellipsoid=self.ellipsoid, C4order=self.C4order) 

1243 return sep.join(pairs(d, prec=prec)) 

1244 

1245 

1246class GeodesicLineExact(_GeodesicLineExact): 

1247 '''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact 

1248 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1GeodesicLineExact.html>}, 

1249 modeled after I{Karney}'s Python class U{geodesicline.GeodesicLine<https://GitHub.com/ 

1250 geographiclib/geographiclib-python>}. 

1251 ''' 

1252 

1253 def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, name=NN): 

1254 '''New L{GeodesicLineExact} instance, allowing points to be found along 

1255 a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}. 

1256 

1257 @arg geodesic: The geodesic to use (L{GeodesicExact}). 

1258 @arg lat1: Latitude of the first point (C{degrees}). 

1259 @arg lon1: Longitude of the first point (C{degrees}). 

1260 @arg azi1: Azimuth at the first points (compass C{degrees}). 

1261 @kwarg caps: Bit-or'ed combination of L{Caps} values specifying 

1262 the capabilities the L{GeodesicLineExact} instance 

1263 should possess, i.e., which quantities can be 

1264 returned by calls to L{GeodesicLineExact.Position} 

1265 and L{GeodesicLineExact.ArcPosition}. 

1266 @kwarg name: Optional name (C{str}). 

1267 

1268 @raise TypeError: Invalid B{C{geodesic}}. 

1269 ''' 

1270 _xinstanceof(GeodesicExact, geodesic=geodesic) 

1271 if (caps & Caps.LINE_OFF): # copy to avoid updates 

1272 geodesic = geodesic.copy(deep=False, name=NN(_UNDER_, geodesic.name)) 

1273# _update_all(geodesic) 

1274 _GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, 0, name=name) 

1275 

1276 

1277def _Astroid(x, y): 

1278 '''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1 ) * k^2 - 

1279 (2 * k + 1) * y^2 = 0} for positive root k. 

1280 ''' 

1281 p = x**2 

1282 q = y**2 

1283 r = fsum_(_1_0, q, p, _N_2_0, floats=True) 

1284 if q or r > 0: 

1285 r = r / _6_0 # /= chokes PyChecker 

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 r3 = r**3 

1290 T3 = r3 + S 

1291 # discriminant of the quadratic equation for T3 is 

1292 # zero on the evolute curve p^(1/3) + q^(1/3) = 1 

1293 d = S * (S + r3 * _2_0) 

1294 if d < 0: 

1295 # T is complex, but u is defined for a real result 

1296 a = atan2(sqrt(-d), -T3) / _3_0 

1297 # There are 3 possible cube roots, choose the one which 

1298 # avoids cancellation. Note d < 0 implies that r < 0. 

1299 u = (cos(a) * _2_0 + _1_0) * r 

1300 else: 

1301 # pick the sign on the sqrt to maximize abs(T3) to 

1302 # minimize loss of precision due to cancellation. 

1303 if d: 

1304 T3 += _copysign(sqrt(d), T3) # T3 = (r * t)^3 

1305 # _cbrt always returns the real root, _cbrt(-8) = -2 

1306 u = _cbrt(T3) # T = r * t 

1307 if u: # T can be zero; but then r2 / T -> 0 

1308 u += r**2 / u 

1309 u += r 

1310 v = _hypot(u, y) # sqrt(u**2 + q) 

1311 # avoid loss of accuracy when u < 0 

1312 u = (q / (v - u)) if u < 0 else (v + u) 

1313 w = (u - q) / (v + v) # positive? 

1314 # rearrange expression for k to avoid loss of accuracy due to 

1315 # subtraction, division by 0 impossible because u > 0, w >= 0 

1316 k = u / (sqrt(w**2 + u) + w) # guaranteed positive 

1317 

1318 else: # q == 0 && r <= 0 

1319 # y = 0 with |x| <= 1. Handle this case directly, for 

1320 # y small, positive root is k = abs(y) / sqrt(1 - x^2) 

1321 k = _0_0 

1322 

1323 return k 

1324 

1325 

1326__all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact) 

1327 

1328# **) MIT License 

1329# 

1330# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1331# 

1332# Permission is hereby granted, free of charge, to any person obtaining a 

1333# copy of this software and associated documentation files (the "Software"), 

1334# to deal in the Software without restriction, including without limitation 

1335# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1336# and/or sell copies of the Software, and to permit persons to whom the 

1337# Software is furnished to do so, subject to the following conditions: 

1338# 

1339# The above copyright notice and this permission notice shall be included 

1340# in all copies or substantial portions of the Software. 

1341# 

1342# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1343# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1344# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1345# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1346# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1347# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1348# OTHER DEALINGS IN THE SOFTWARE.