Coverage for pygeodesy/rhumbx.py: 98%

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1 

2# -*- coding: utf-8 -*- 

3 

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

5<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Rhumb.html>} and U{RhumbLine 

6<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1RhumbLine.html>}. 

7 

8Class L{RhumbLine} has been enhanced with methods C{intersection2} and C{nearestOn4} to find 

9the intersection of two rhumb lines, respectively the nearest point on a rumb line. 

10 

11For more details, see the C++ U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>} 

12documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>}, 

13the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>}, 

14the utily U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online 

15rhumb line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}. 

16 

17Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2022) 

18and licensed under the MIT/X11 License. For more information, see the 

19U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. 

20''' 

21# make sure int/int division yields float quotient 

22from __future__ import division as _; del _ # PYCHOK semicolon 

23 

24from pygeodesy.basics import copysign0, neg, _xinstanceof, _zip 

25from pygeodesy.constants import INT0, _EPSqrt as _TOL, NAN, PI_2, isnan, _0_0s, \ 

26 _0_0, _0_5, _1_0, _2_0, _4_0, _90_0, _180_0, _720_0 

27# from pygeodesy.datums import _spherical_datum # in Rhumb.ellipsoid.setter 

28from pygeodesy.errors import IntersectionError, itemsorted, _ValueError, \ 

29 _xdatum, _xkwds 

30# from pygeodesy.etm import ExactTransverseMercator # in ._RhumbLine.xTM 

31from pygeodesy.fmath import euclid, favg, fsum1_, hypot, hypot1 

32# from pygeodesy.fsums import fsum1_ # from .fmath 

33from pygeodesy.interns import NN, _azi12_, _coincident_, _COMMASPACE_, \ 

34 _intersection_, _lat1_, _lat2_, _lon1_, _lon2_, \ 

35 _no_, _s12_, _S12_, _UNDER 

36from pygeodesy.karney import _a12_, _atan2d, Caps, _CapsBase as _RhumbBase, \ 

37 _diff182, Direct9Tuple, _EWGS84, _fix90, GDict, \ 

38 _GTuple, Inverse10Tuple, _norm180 

39from pygeodesy.ktm import KTransverseMercator, _Xorder, _Xs, \ 

40 _AlpCoeffs, _BetCoeffs # PYCHOK used! 

41from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

42from pygeodesy.namedTuples import Distance2Tuple, LatLon2Tuple, NearestOn4Tuple 

43from pygeodesy.props import deprecated_method, Property, Property_RO, property_RO, \ 

44 _update_all 

45from pygeodesy.streprs import Fmt, pairs, unstr 

46from pygeodesy.units import Bearing as _Azi, Degrees as _Deg, Int, Lat, Lon, \ 

47 Meter as _M, Meter2 as _M2 

48from pygeodesy.utily import sincos2_, sincos2d 

49from pygeodesy.vector3d import _intersect3d3, Vector3d # in .intersection2 below 

50 

51from math import asinh, atan, cos, cosh, fabs, radians, sin, sinh, sqrt, tan 

52 

53__all__ = _ALL_LAZY.rhumbx 

54__version__ = '23.04.10' 

55 

56_rls = [] # instances of C{RbumbLine} to be updated 

57_TRIPS = 65 # .intersection2, 18+ 

58 

59 

60class _Lat(Lat): 

61 '''(INTERNAL) Latitude B{C{lat}}. 

62 ''' 

63 def __init__(self, *lat, **Error_name): 

64 kwds = _xkwds(Error_name, clip=0, Error=RhumbError) 

65 Lat.__new__(_Lat, *lat, **kwds) 

66 

67 

68class _Lon(Lon): 

69 '''(INTERNAL) Longitude B{C{lon}}. 

70 ''' 

71 def __init__(self, *lon, **Error_name): 

72 kwds = _xkwds(Error_name, clip=0, Error=RhumbError) 

73 Lon.__new__(_Lon, *lon, **kwds) 

74 

75 

76def _update_all_rls(r): 

77 '''(INTERNAL) Zap cached/memoized C{Property[_RO]}s 

78 of any L{RhumbLine} instances tied to the given 

79 L{Rhumb} instance B{C{r}}. 

80 ''' 

81 _xinstanceof(r, Rhumb) 

82 _update_all(r) 

83 for rl in _rls: # PYCHOK use weakref? 

84 if rl._rhumb is r: 

85 _update_all(rl) 

86 

87 

88class Rhumb(_RhumbBase): 

89 '''Class to solve of the I{direct} and I{inverse rhumb} problems, accurately. 

90 

91 @see: The U{Detailed Description<https://GeographicLib.SourceForge.io/C++/doc/ 

92 classGeographicLib_1_1Rhumb.html>} of I{Karney}'s C++ C{Rhumb Class}. 

93 ''' 

94 _E = _EWGS84 

95 _exact = True 

96 _mRA = 6 

97 _mTM = 6 

98 

99 def __init__(self, a_earth=_EWGS84, f=None, exact=True, name=NN, **RA_TMorder): 

100 '''New L{Rhumb}. 

101 

102 @kwarg a_earth: This rhumb's earth (L{Ellipsoid}, L{Ellipsoid2}, 

103 L{a_f2Tuple}, L{Datum}, 2-tuple C{(a, f)}) or the 

104 (equatorial) radius (C{scalar}). 

105 @kwarg f: The ellipsoid's flattening (C{scalar}), iff B{C{a_earth}} is 

106 a C{scalar}, ignored otherwise. 

107 @kwarg exact: If C{True}, use an addition theorem for elliptic integrals 

108 to compute I{Divided differences}, otherwise use the Krüger 

109 series expansion (C{bool}), see also property C{exact}. 

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

111 @kwarg RA_TMorder: Optional keyword arguments B{C{RAorder}} and B{C{TMorder}} 

112 to set the respective C{order}, see properties C{RAorder} 

113 and C{TMorder} and method C{orders}. 

114 

115 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{RA_TMorder}}. 

116 ''' 

117 if f is not None: 

118 self.ellipsoid = a_earth, f 

119 elif a_earth not in (_EWGS84, None): 

120 self.ellipsoid = a_earth 

121 if not exact: 

122 self._exact = False 

123 if name: 

124 self.name = name 

125 if RA_TMorder: 

126 self.orders(**RA_TMorder) 

127 

128 @Property_RO 

129 def _A2(self): # Conformal2RectifyingCoeffs 

130 m = self.TMorder 

131 return _Xs(_AlpCoeffs, m, self.ellipsoid), m 

132 

133 @Property_RO 

134 def _B2(self): # Rectifying2ConformalCoeffs 

135 m = self.TMorder 

136 return _Xs(_BetCoeffs, m, self.ellipsoid), m 

137 

138 def _DConformal2Rectifying(self, x, y): # radians 

139 return _1_0 + (_sincosSeries(True, x, y, *self._A2) if self.f else _0_0) 

140 

141 def Direct(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE): 

142 '''Solve the I{direct rhumb} problem, optionally with the area. 

143 

144 @arg lat1: Latitude of the first point (C{degrees90}). 

145 @arg lon1: Longitude of the first point (C{degrees180}). 

146 @arg azi12: Azimuth of the rhumb line (compass C{degrees}). 

147 @arg s12: Distance along the rhumb line from the given to 

148 the destination point (C{meter}), can be negative. 

149 

150 @return: L{GDict} with 2 up to 8 items C{lat2, lon2, a12, S12, 

151 lat1, lon1, azi12, s12} with the destination point's 

152 latitude C{lat2} and longitude C{lon2} in C{degrees}, 

153 the rhumb angle C{a12} in C{degrees} and area C{S12} 

154 under the rhumb line in C{meter} I{squared}. 

155 

156 @note: If B{C{s12}} is large enough that the rhumb line crosses 

157 a pole, the longitude of the second point is indeterminate 

158 and C{NAN} is returned for C{lon2} and area C{S12}. 

159 

160 If the given point is a pole, the cosine of its latitude is 

161 taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52. 

162 This position is extremely close to the actual pole and 

163 allows the calculation to be carried out in finite terms. 

164 ''' 

165 rl = _RhumbLine(self, lat1, lon1, azi12, caps=Caps.LINE_OFF, 

166 name=self.name) 

167 return rl.Position(s12, outmask | self._debug) # lat2, lon2, S12 

168 

169 @deprecated_method 

170 def Direct7(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA): 

171 '''DEPRECATED, use method L{Rhumb.Direct8}. 

172 

173 @return: A I{DEPRECATED} L{Rhumb7Tuple}. 

174 ''' 

175 return self.Direct8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple() 

176 

177 def Direct8(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA): 

178 '''Like method L{Rhumb.Direct} but returning a L{Rhumb8Tuple} with area C{S12}. 

179 ''' 

180 return self.Direct(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple() 

181 

182 def DirectLine(self, lat1, lon1, azi12, name=NN, **caps): # caps=Caps.STANDARD 

183 '''Define a L{RhumbLine} in terms of the I{direct} rhumb problem. 

184 

185 @arg lat1: Latitude of the first point (C{degrees90}). 

186 @arg lon1: Longitude of the first point (C{degrees180}). 

187 @arg azi12: Azimuth of the rhumb line (compass C{degrees}). 

188 @kwarg caps: Optional C{caps}, see L{RhumbLine} C{B{caps}}. 

189 

190 @return: A L{RhumbLine} instance and invoke its method 

191 L{RhumbLine.Position} to compute each point. 

192 

193 @note: Updates to this rhumb are reflected in the returned 

194 rhumb line. 

195 ''' 

196 return RhumbLine(self, lat1=lat1, lon1=lon1, azi12=azi12, 

197 name=name or self.name, **caps) 

198 

199 def _DIsometrict(self, phix, phiy, tphix, tphiy, _Dtan_phix_phiy): 

200 E = self.ellipsoid 

201 return _Dtan_phix_phiy * _Dasinh(tphix, tphiy) - \ 

202 _Dsin(phix, phiy) * _DeatanhE(sin(phix), sin(phiy), E) 

203 

204 def _DIsometric2Rectifyingd(self, psix, psiy): # degrees 

205 if self.exact: 

206 E = self.ellipsoid 

207 phix, phiy, tphix, tphiy = _Eaux4(E.auxIsometric, psix, psiy) 

208 t = _Dtant(phix - phiy, tphix, tphiy) 

209 r = self._DRectifyingt( tphix, tphiy, t) / \ 

210 self._DIsometrict(phix, phiy, tphix, tphiy, t) 

211 else: 

212 x, y = radians(psix), radians(psiy) 

213 r = self._DConformal2Rectifying(_gd(x), _gd(y)) * _Dgd(x, y) 

214 return r 

215 

216 def _DRectifyingt(self, tphix, tphiy, _Dtan_phix_phiy): 

217 E = self.ellipsoid 

218 tbetx = E.f1 * tphix 

219 tbety = E.f1 * tphiy 

220 return (E.f1 * _Dtan_phix_phiy * E.b * PI_2 

221 * _DfEt( tbetx, tbety, self._eF) 

222 * _Datan(tbetx, tbety)) / E.L 

223 

224 def _DRectifying2Conformal(self, x, y): # radians 

225 return _1_0 - (_sincosSeries(True, x, y, *self._B2) if self.f else _0_0) 

226 

227 def _DRectifying2Isometricd(self, mux, muy): # degrees 

228 E = self.ellipsoid 

229 phix, phiy, tphix, tphiy = _Eaux4(E.auxRectifying, mux, muy) 

230 if self.exact: 

231 t = _Dtant(phix - phiy, tphix, tphiy) 

232 r = self._DIsometrict(phix, phiy, tphix, tphiy, t) / \ 

233 self._DRectifyingt( tphix, tphiy, t) 

234 else: 

235 r = self._DRectifying2Conformal(radians(mux), radians(muy)) * \ 

236 _Dgdinv(E.es_taupf(tphix), E.es_taupf(tphiy)) 

237 return r 

238 

239 @Property_RO 

240 def _eF(self): 

241 '''(INTERNAL) Get the ellipsoid's elliptic function. 

242 ''' 

243 # .k2 = 0.006739496742276434 

244 return self._E._elliptic_e12 # _MODS.elliptic.Elliptic(-self._E._e12) 

245 

246 @Property 

247 def ellipsoid(self): 

248 '''Get this rhumb's ellipsoid (L{Ellipsoid}). 

249 ''' 

250 return self._E 

251 

252 @ellipsoid.setter # PYCHOK setter! 

253 def ellipsoid(self, a_earth_f): 

254 '''Set this rhumb's ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum}, 

255 L{a_f2Tuple}, 2-tuple C{(a, f)}) or the (equatorial) radius (C{scalar}). 

256 ''' 

257 E = _MODS.datums._spherical_datum(a_earth_f, Error=RhumbError).ellipsoid 

258 if self._E != E: 

259 _update_all_rls(self) 

260 self._E = E 

261 

262 @property_RO 

263 def equatoradius(self): 

264 '''Get the C{ellipsoid}'s equatorial radius, semi-axis (C{meter}). 

265 ''' 

266 return self.ellipsoid.a 

267 

268 a = equatoradius 

269 

270 @Property 

271 def exact(self): 

272 '''Get the I{exact} option (C{bool}). 

273 ''' 

274 return self._exact 

275 

276 @exact.setter # PYCHOK setter! 

277 def exact(self, exact): 

278 '''Set the I{exact} option (C{bool}). If C{True}, use I{exact} rhumb 

279 calculations, if C{False} results are less precise for more oblate 

280 or more prolate ellipsoids with M{abs(flattening) > 0.01} (C{bool}). 

281 

282 @see: Option U{B{-s}<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} 

283 and U{ACCURACY<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html#ACCURACY>}. 

284 ''' 

285 x = bool(exact) 

286 if self._exact != x: 

287 _update_all_rls(self) 

288 self._exact = x 

289 

290 def flattening(self): 

291 '''Get the C{ellipsoid}'s flattening (C{float}). 

292 ''' 

293 return self.ellipsoid.f 

294 

295 f = flattening 

296 

297 def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH_DISTANCE): 

298 '''Solve the I{inverse rhumb} problem. 

299 

300 @arg lat1: Latitude of the first point (C{degrees90}). 

301 @arg lon1: Longitude of the first point (C{degrees180}). 

302 @arg lat2: Latitude of the second point (C{degrees90}). 

303 @arg lon2: Longitude of the second point (C{degrees180}). 

304 

305 @return: L{GDict} with 5 to 8 items C{azi12, s12, a12, S12, 

306 lat1, lon1, lat2, lon2}, the rhumb line's azimuth C{azi12} 

307 in compass C{degrees} between C{-180} and C{+180}, the 

308 distance C{s12} and rhumb angle C{a12} between both points 

309 in C{meter} respectively C{degrees} and the area C{S12} 

310 under the rhumb line in C{meter} I{squared}. 

311 

312 @note: The shortest rhumb line is found. If the end points are 

313 on opposite meridians, there are two shortest rhumb lines 

314 and the East-going one is chosen. 

315 

316 If either point is a pole, the cosine of its latitude is 

317 taken to be C{epsilon}**-2 (where C{epsilon} is 2.0**-52). 

318 This position is extremely close to the actual pole and 

319 allows the calculation to be carried out in finite terms. 

320 ''' 

321 r, Cs = GDict(name=self.name), Caps 

322 if (outmask & Cs.AZIMUTH_DISTANCE_AREA): 

323 r.set_(lat1=lat1, lon1=lon1, lat2=lat2, lon2=lon2) 

324 E = self.ellipsoid 

325 psi1 = E.auxIsometric(lat1) 

326 psi2 = E.auxIsometric(lat2) 

327 psi12 = psi2 - psi1 

328 lon12, _ = _diff182(lon1, lon2) 

329 if (outmask & Cs.AZIMUTH): 

330 r.set_(azi12=_atan2d(lon12, psi12)) 

331 if (outmask & Cs.DISTANCE): 

332 a12 = hypot(lon12, psi12) * self._DIsometric2Rectifyingd(psi2, psi1) 

333 s12 = a12 * E._L_90 

334 r.set_(s12=s12, a12=copysign0(a12, s12)) 

335 if (outmask & Cs.AREA): 

336 r.set_(S12=self._S12d(lon12, psi2, psi1)) 

337 if ((outmask | self._debug) & Cs._DEBUG_INVERSE): # PYCHOK no cover 

338 r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L, 

339 b=E.b, e=E.e, e2=E.e2, k2=self._eF.k2, 

340 lon12=lon12, psi1=psi1, exact=self.exact, 

341 psi12=psi12, psi2=psi2) 

342 return r 

343 

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

345# '''Return the distance in C{meter} and the forward and 

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

347# 

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

349# ''' 

350# r = self.Inverse(lat1, lon1, lat2, lon2) 

351# return Distance3Tuple(r.s12, r.azi12, r.azi12) 

352 

353 @deprecated_method 

354 def Inverse7(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA): 

355 '''DEPRECATED, use method L{Rhumb.Inverse8}. 

356 

357 @return: A I{DEPRECATED} L{Rhumb7Tuple}. 

358 ''' 

359 return self.Inverse8(lat1, lon1, azi12, s12, outmask=outmask)._to7Tuple() 

360 

361 def Inverse8(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA): 

362 '''Like method L{Rhumb.Inverse} but returning a L{Rhumb8Tuple} with area C{S12}. 

363 ''' 

364 return self.Inverse(lat1, lon1, azi12, s12, outmask=outmask).toRhumb8Tuple() 

365 

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

367 '''Define a L{RhumbLine} in terms of the I{inverse} rhumb problem. 

368 

369 @arg lat1: Latitude of the first point (C{degrees90}). 

370 @arg lon1: Longitude of the first point (C{degrees180}). 

371 @arg lat2: Latitude of the second point (C{degrees90}). 

372 @arg lon2: Longitude of the second point (C{degrees180}). 

373 @kwarg caps: Optional C{caps}, see L{RhumbLine} C{B{caps}}. 

374 

375 @return: A L{RhumbLine} instance and invoke its method 

376 L{RhumbLine.Position} to compute each point. 

377 

378 @note: Updates to this rhumb are reflected in the returned 

379 rhumb line. 

380 ''' 

381 r = self.Inverse(lat1, lon1, lat2, lon2, outmask=Caps.AZIMUTH) 

382 return RhumbLine(self, lat1=lat1, lon1=lon1, azi12=r.azi12, 

383 name=name or self.name, **caps) 

384 

385 Line = DirectLine # synonyms 

386 

387 def _MeanSinXi(self, x, y): # radians 

388 s = _Dlog(cosh(x), cosh(y)) * _Dcosh(x, y) 

389 if self.f: 

390 s += _sincosSeries(False, _gd(x), _gd(y), *self._RA2) * _Dgd(x, y) 

391 return s 

392 

393 def orders(self, RAorder=None, TMorder=None): 

394 '''Get and set the I{RAorder} and/or I{TMorder}. 

395 

396 @kwarg RAorder: I{Rhumb Area} order (C{int}, 4, 5, 6, 7 

397 or 8). 

398 @kwarg TMorder: I{Transverse Mercator} order (C{int}, 4, 

399 5, 6, 7 or 8). 

400 

401 @return: L{RhumbOrder2Tuple}C{(RAorder, TMorder)} with 

402 the previous C{RAorder} and C{TMorder} setting. 

403 ''' 

404 t = RhumbOrder2Tuple(self.RAorder, self.TMorder) 

405 if RAorder not in (None, t.RAorder): # PYCHOK attr 

406 self.RAorder = RAorder 

407 if TMorder not in (None, t.TMorder): # PYCHOK attr 

408 self.TMorder = TMorder 

409 return t 

410 

411 @Property_RO 

412 def _RA2(self): 

413 # for WGS84: (0, -0.0005583633519275459, -3.743803759172812e-07, -4.633682270824446e-10, 

414 # RAorder 6: -7.709197397676237e-13, -1.5323287106694307e-15, -3.462875359099873e-18) 

415 m = self.RAorder 

416 return _Xs(_RACoeffs, m, self.ellipsoid, RA=True), m 

417 

418 @Property 

419 def RAorder(self): 

420 '''Get the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). 

421 ''' 

422 return self._mRA 

423 

424 @RAorder.setter # PYCHOK setter! 

425 def RAorder(self, order): 

426 '''Set the I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). 

427 ''' 

428 n = _Xorder(_RACoeffs, RhumbError, RAorder=order) 

429 if self._mRA != n: 

430 _update_all_rls(self) 

431 self._mRA = n 

432 

433 def _S12d(self, lon12, psi2, psi1): # degrees 

434 '''(INTERNAL) Compute the area C{S12}. 

435 ''' 

436 r = (self.ellipsoid.areax if self.exact else 

437 self.ellipsoid.area) * lon12 / _720_0 

438 r *= self._MeanSinXi(radians(psi2), radians(psi1)) 

439 return r 

440 

441 @Property 

442 def TMorder(self): 

443 '''Get the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

444 ''' 

445 return self._mTM 

446 

447 @TMorder.setter # PYCHOK setter! 

448 def TMorder(self, order): 

449 '''Set the I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

450 

451 @note: Setting C{TMorder} turns property C{exact} off. 

452 ''' 

453 n = _Xorder(_AlpCoeffs, RhumbError, TMorder=order) 

454 if self._mTM != n: 

455 _update_all_rls(self) 

456 self._mTM = n 

457 self.exact = False 

458 

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

460 '''Return this C{Rhumb} as string. 

461 

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

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

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

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

466 

467 @return: Tuple items (C{str}). 

468 ''' 

469 d = dict(ellipsoid=self.ellipsoid, RAorder=self.RAorder, 

470 exact=self.exact, TMorder=self.TMorder) 

471 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec)) 

472 

473 

474class RhumbError(_ValueError): 

475 '''Raised for a L{Rhumb} or L{RhumbLine} issue. 

476 ''' 

477 pass 

478 

479 

480class _RhumbLine(_RhumbBase): 

481 '''(INTERNAL) Class L{RhumbLine} 

482 ''' 

483 _azi12 = _0_0 

484# _lat1 = _0_0 

485# _lon1 = _0_0 

486 _salp = _0_0 

487 _calp = _1_0 

488 _rhumb = None # L{Rhumb} instance 

489 

490 def __init__(self, rhumb, lat1, lon1, azi12, caps=0, name=NN): # case=Caps.? 

491 '''New C{RhumbLine}. 

492 ''' 

493 _xinstanceof(Rhumb, rhumb=rhumb) 

494 self._lat1 = _Lat(lat1=_fix90(lat1)) 

495 self._lon1 = _Lon(lon1= lon1) 

496 self._debug |= (caps | rhumb._debug) & Caps._DEBUG_DIRECT_LINE 

497 if azi12: # non-zero 

498 self.azi12 = azi12 

499 self._caps = caps 

500 if not (caps & Caps.LINE_OFF): 

501 _rls.append(self) 

502 n = name or rhumb.name 

503 if n: 

504 self.name=n 

505 self._rhumb = rhumb # last 

506 

507 def __del__(self): # XXX use weakref? 

508 if _rls: # may be empty or None 

509 try: # PYCHOK no cover 

510 _rls.remove(self) 

511 except (TypeError, ValueError): 

512 pass 

513 self._rhumb = None 

514 # _update_all(self) # throws TypeError during Python 2 cleanup 

515 

516 @Property 

517 def azi12(self): 

518 '''Get this rhumb line's I{azimuth} (compass C{degrees}). 

519 ''' 

520 return self._azi12 

521 

522 @azi12.setter # PYCHOK setter! 

523 def azi12(self, azi12): 

524 '''Set this rhumb line's I{azimuth} (compass C{degrees}). 

525 ''' 

526 z = _norm180(azi12) 

527 if self._azi12 != z: 

528 if self._rhumb: 

529 _update_all(self) 

530 self._azi12 = z 

531 self._salp, self._calp = sincos2d(z) # no NEG0 

532 

533 def distance2(self, lat, lon): 

534 '''Return the distance from and (initial) bearing at the given 

535 point to this rhumb line's start point. 

536 

537 @arg lat: Latitude of the point (C{degrees}). 

538 @arg lon: Longitude of the points (C{degrees}). 

539 

540 @return: A L{Distance2Tuple}C{(distance, initial)} with the C{distance} 

541 in C{meter} and C{initial} bearing in C{degrees}. 

542 

543 @see: Methods L{RhumbLine.intersection2} and L{RhumbLine.nearestOn4}. 

544 ''' 

545 r = self.rhumb.Inverse(self.lat1, self.lon1, lat, lon) 

546# outmask=Caps.AZIMUTH_DISTANCE) 

547 return Distance2Tuple(r.s12, r.azi12) 

548 

549 @Property_RO 

550 def ellipsoid(self): 

551 '''Get this rhumb line's ellipsoid (L{Ellipsoid}). 

552 ''' 

553 return self.rhumb.ellipsoid 

554 

555 @property_RO 

556 def exact(self): 

557 '''Get this rhumb line's I{exact} option (C{bool}). 

558 ''' 

559 return self.rhumb.exact 

560 

561 def intersection2(self, other, tol=_TOL, **eps): 

562 '''Iteratively find the intersection of this and an other rhumb line. 

563 

564 @arg other: The other rhumb line (L{RhumbLine}). 

565 @kwarg tol: Tolerance for longitudinal convergence (C{degrees}). 

566 @kwarg eps: Tolerance for L{intersection3d3} (C{EPS}). 

567 

568 @return: A L{LatLon2Tuple}{(lat, lon)} with the C{lat}- and 

569 C{lon}gitude of the intersection point. 

570 

571 @raise IntersectionError: No convergence for this B{C{tol}} or 

572 no intersection for an other reason. 

573 

574 @see: Methods L{RhumbLine.distance2} and L{RhumbLine.nearestOn4} 

575 and function L{pygeodesy.intersection3d3}. 

576 

577 @note: Each iteration involves a round trip to this rhumb line's 

578 L{ExactTransverseMercator} or L{KTransverseMercator} 

579 projection and invoking function L{intersection3d3} in 

580 that domain. 

581 ''' 

582 _xinstanceof(other, _RhumbLine) 

583 _xdatum(self.rhumb, other.rhumb, Error=RhumbError) 

584 try: 

585 if other is self: 

586 raise ValueError(_coincident_) 

587 # make globals and invariants locals 

588 _diff = euclid # approximate length 

589 _i3d3 = _intersect3d3 # NOT .vector3d.intersection3d3 

590 _LL2T = LatLon2Tuple 

591 _xTMr = self.xTM.reverse # ellipsoidal or spherical 

592 _s_3d, s_az = self._xTM3d, self.azi12 

593 _o_3d, o_az = other._xTM3d, other.azi12 

594 # use halfway point as initial estimate 

595 p = _LL2T(favg(self.lat1, other.lat1), 

596 favg(self.lon1, other.lon1)) 

597 for i in range(1, _TRIPS): 

598 v = _i3d3(_s_3d(p), s_az, # point + bearing 

599 _o_3d(p), o_az, useZ=False, **eps)[0] 

600 t = _xTMr(v.x, v.y, lon0=p.lon) # PYCHOK Reverse4Tuple 

601 d = _diff(t.lon - p.lon, t.lat) # PYCHOK t.lat + p.lat - p.lat 

602 p = _LL2T(t.lat + p.lat, t.lon) # PYCHOK t.lon + p.lon = lon0 

603 if d < tol: 

604 return _LL2T(p.lat, p.lon, iteration=i, # PYCHOK p... 

605 name=self.intersection2.__name__) 

606 except Exception as x: 

607 raise IntersectionError(_no_(_intersection_), cause=x) 

608 t = unstr(self.intersection2, tol=tol, **eps) 

609 raise IntersectionError(Fmt.no_convergence(d), txt=t) 

610 

611 @property_RO 

612 def lat1(self): 

613 '''Get this rhumb line's latitude (C{degrees90}). 

614 ''' 

615 return self._lat1 

616 

617 @property_RO 

618 def lon1(self): 

619 '''Get this rhumb line's longitude (C{degrees180}). 

620 ''' 

621 return self._lon1 

622 

623 @Property_RO 

624 def latlon1(self): 

625 '''Get this rhumb line's lat- and longitude (L{LatLon2Tuple}C{(lat, lon)}). 

626 ''' 

627 return LatLon2Tuple(self.lat1, self.lon1) 

628 

629 @Property_RO 

630 def _mu1(self): 

631 '''(INTERNAL) Get the I{rectifying auxiliary} latitude C{mu} (C{degrees}). 

632 ''' 

633 return self.ellipsoid.auxRectifying(self.lat1) 

634 

635 def nearestOn4(self, lat, lon, tol=_TOL, **eps): 

636 '''Iteratively locate the point on this rhumb line nearest to 

637 the given point. 

638 

639 @arg lat: Latitude of the point (C{degrees}). 

640 @arg lon: Longitude of the point (C{degrees}). 

641 @kwarg tol: Longitudinal convergence tolerance (C{degrees}). 

642 @kwarg eps: Tolerance for L{intersection3d3} (C{EPS}). 

643 

644 @return: A L{NearestOn4Tuple}C{(lat, lon, distance, normal)} with 

645 the C{lat}- and C{lon}gitude of the nearest point on and 

646 the C{distance} in C{meter} to this rhumb line and with the 

647 azimuth of the C{normal}, perpendicular to this rhumb line. 

648 

649 @raise IntersectionError: No convergence for this B{C{eps}} or 

650 no intersection for an other reason. 

651 

652 @see: Methods L{RhumbLine.distance2} and L{RhumbLine.intersection2} 

653 and function L{intersection3d3}. 

654 ''' 

655 z = _norm180(self.azi12 + _90_0) # perpendicular 

656 r = _RhumbLine(self.rhumb, lat, lon, z, caps=Caps.LINE_OFF) 

657 p = self.intersection2(r, tol=tol, **eps) 

658 t = r.distance2(p.lat, p.lon) 

659 return NearestOn4Tuple(p.lat, p.lon, t.distance, z, 

660 iteration=p.iteration) 

661 

662 @Property_RO 

663 def _psi1(self): 

664 '''(INTERNAL) Get the I{isometric auxiliary} latitude C{psi} (C{degrees}). 

665 ''' 

666 return self.ellipsoid.auxIsometric(self.lat1) 

667 

668 @property_RO 

669 def RAorder(self): 

670 '''Get this rhumb line's I{Rhumb Area} order (C{int}, 4, 5, 6, 7 or 8). 

671 ''' 

672 return self.rhumb.RAorder 

673 

674 @Property_RO 

675 def _r1rad(self): # PYCHOK no cover 

676 '''(INTERNAL) Get this rhumb line's parallel I{circle radius} (C{meter}). 

677 ''' 

678 return radians(self.ellipsoid.circle4(self.lat1).radius) 

679 

680 @Property_RO 

681 def rhumb(self): 

682 '''Get this rhumb line's rhumb (L{Rhumb}). 

683 ''' 

684 return self._rhumb 

685 

686 def Position(self, s12, outmask=Caps.LATITUDE_LONGITUDE): 

687 '''Compute a position at a distance on this rhumb line. 

688 

689 @arg s12: The distance along this rhumb between its point and 

690 the other point (C{meters}), can be negative. 

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

692 the quantities to be returned. 

693 

694 @return: L{GDict} with 4 to 8 items C{azi12, a12, s12, S12, lat2, 

695 lon2, lat1, lon1} with latitude C{lat2} and longitude 

696 C{lon2} of the other point in C{degrees}, the rhumb angle 

697 C{a12} between both points in C{degrees} and the area C{S12} 

698 under the rhumb line in C{meter} I{squared}. 

699 

700 @note: If B{C{s12}} is large enough that the rhumb line crosses a 

701 pole, the longitude of the second point is indeterminate and 

702 C{NAN} is returned for C{lon2} and area C{S12}. 

703 

704 If the first point is a pole, the cosine of its latitude is 

705 taken to be C{epsilon}**-2 (where C{epsilon} is 2**-52). 

706 This position is extremely close to the actual pole and 

707 allows the calculation to be carried out in finite terms. 

708 ''' 

709 r, Cs = GDict(name=self.name), Caps 

710 if (outmask & Cs.LATITUDE_LONGITUDE_AREA): 

711 E, R = self.ellipsoid, self.rhumb 

712 mu12 = s12 * self._calp / E._L_90 

713 mu2 = mu12 + self._mu1 

714 if fabs(mu2) > 90: # PYCHOK no cover 

715 mu2 = _norm180(mu2) # reduce to [-180, 180) 

716 if fabs(mu2) > 90: # point on anti-meridian 

717 mu2 = _norm180(_180_0 - mu2) 

718 lat2x = E.auxRectifying(mu2, inverse=True) 

719 lon2x = NAN 

720 if (outmask & Cs.AREA): 

721 r.set_(S12=NAN) 

722 else: 

723 psi2 = self._psi1 

724 if self._calp: 

725 lat2x = E.auxRectifying(mu2, inverse=True) 

726 psi12 = R._DRectifying2Isometricd(mu2, 

727 self._mu1) * mu12 

728 lon2x = psi12 * self._salp / self._calp 

729 psi2 += psi12 

730 else: # PYCHOK no cover 

731 lat2x = self.lat1 

732 lon2x = self._salp * s12 / self._r1rad 

733 if (outmask & Cs.AREA): 

734 r.set_(S12=R._S12d(lon2x, self._psi1, psi2)) 

735 r.set_(s12=s12, azi12=self.azi12, a12=s12 / E._L_90) 

736 if (outmask & Cs.LATITUDE): 

737 r.set_(lat2=lat2x, lat1=self.lat1) 

738 if (outmask & Cs.LONGITUDE): 

739 if (outmask & Cs.LONG_UNROLL) and not isnan(lat2x): 

740 lon2x += self.lon1 

741 else: 

742 lon2x = _norm180(_norm180(self.lon1) + lon2x) 

743 r.set_(lon2=lon2x, lon1=self.lon1) 

744 if ((outmask | self._debug) & Cs._DEBUG_DIRECT_LINE): # PYCHOK no cover 

745 r.set_(a=E.a, f=E.f, f1=E.f1, L=E.L, exact=R.exact, 

746 b=E.b, e=E.e, e2=E.e2, k2=R._eF.k2, 

747 calp=self._calp, mu1 =self._mu1, mu12=mu12, 

748 salp=self._salp, psi1=self._psi1, mu2=mu2) 

749 return r 

750 

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

752 '''Return this C{RhumbLine} as string. 

753 

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

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

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

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

758 

759 @return: C{RhumbLine} (C{str}). 

760 ''' 

761 d = dict(rhumb=self.rhumb, lat1=self.lat1, lon1=self.lon1, 

762 azi12=self.azi12, exact=self.exact, 

763 TMorder=self.TMorder, xTM=self.xTM) 

764 return sep.join(pairs(itemsorted(d, asorted=False), prec=prec)) 

765 

766 @property_RO 

767 def TMorder(self): 

768 '''Get this rhumb line's I{Transverse Mercator} order (C{int}, 4, 5, 6, 7 or 8). 

769 ''' 

770 return self.rhumb.TMorder 

771 

772 @Property_RO 

773 def xTM(self): 

774 '''Get this rhumb line's I{Transverse Mercator} projection (L{ExactTransverseMercator} 

775 if I{exact} and I{ellipsoidal}, otherwise L{KTransverseMercator}). 

776 ''' 

777 E = self.ellipsoid 

778 # ExactTransverseMercator doesn't handle spherical earth models 

779 return _MODS.etm.ExactTransverseMercator(E) if self.exact and E.isEllipsoidal else \ 

780 KTransverseMercator(E, TMorder=self.TMorder) 

781 

782 def _xTM3d(self, latlon0, z=INT0, V3d=Vector3d): 

783 '''(INTERNAL) C{xTM.forward} this C{latlon1} to C{V3d} with B{C{latlon0}} 

784 as current intersection estimate and central meridian. 

785 ''' 

786 t = self.xTM.forward(self.lat1 - latlon0.lat, self.lon1, lon0=latlon0.lon) 

787 return V3d(t.easting, t.northing, z) 

788 

789 

790class RhumbLine(_RhumbLine): 

791 '''Compute one or several points on a single rhumb line. 

792 

793 Class L{RhumbLine} facilitates the determination of points on 

794 a single rhumb line. The starting point (C{lat1}, C{lon1}) 

795 and the azimuth C{azi12} are specified once. 

796 

797 Method L{RhumbLine.Position} returns the location of an other 

798 point and optionally the distance C{s12} along the corresponding 

799 area C{S12} under the rhumb line. 

800 

801 Method L{RhumbLine.intersection2} finds the intersection between 

802 two rhumb lines. 

803 

804 Method L{RhumbLine.nearestOn4} computes the nearest point on and 

805 its distance to a rhumb line. 

806 ''' 

807 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, caps=0, name=NN): # case=Caps.? 

808 '''New L{RhumbLine}. 

809 

810 @arg rhumb: The rhumb reference (L{Rhumb}). 

811 @kwarg lat1: Latitude of the start point (C{degrees90}). 

812 @kwarg lon1: Longitude of the start point (C{degrees180}). 

813 @kwarg azi12: Azimuth of this rhumb line (compass C{degrees}). 

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

815 the capabilities. Include C{Caps.LINE_OFF} if 

816 updates to B{C{rhumb}} should I{not} be reflected 

817 in this rhumb line. 

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

819 ''' 

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

821 rhumb = rhumb.copy(deep=False, name=_UNDER(rhumb.name)) 

822 _RhumbLine.__init__(self, rhumb, lat1, lon1, azi12, caps=caps, name=name) 

823 

824 

825class RhumbOrder2Tuple(_GTuple): 

826 '''2-Tuple C{(RAorder, TMorder)} with a I{Rhumb Area} and 

827 I{Transverse Mercator} order, both C{int}. 

828 ''' 

829 _Names_ = (Rhumb.RAorder.name, Rhumb.TMorder.name) 

830 _Units_ = ( Int, Int) 

831 

832 

833class Rhumb8Tuple(_GTuple): 

834 '''8-Tuple C{(lat1, lon1, lat2, lon2, azi12, s12, S12, a12)} with lat- C{lat1}, 

835 C{lat2} and longitudes C{lon1}, C{lon2} of both points, the azimuth of the 

836 rhumb line C{azi12}, the distance C{s12}, the area C{S12} under the rhumb 

837 line and the angular distance C{a12} between both points. 

838 ''' 

839 _Names_ = (_lat1_, _lon1_, _lat2_, _lon2_, _azi12_, _s12_, _S12_, _a12_) 

840 _Units_ = (_Lat, _Lon, _Lat, _Lon, _Azi, _M, _M2, _Deg) 

841 

842 def toDirect9Tuple(self, dflt=NAN, **a12_azi1_azi2_m12_M12_M21): 

843 '''Convert this L{Rhumb8Tuple} result to a 9-tuple, like I{Karney}'s 

844 method C{geographiclib.geodesic.Geodesic._GenDirect}. 

845 

846 @kwarg dflt: Default value for missing items (C{any}). 

847 @kwarg a12_azi1_azi2_m12_M12_M21: Optional keyword arguments 

848 to specify or override L{Inverse10Tuple} items. 

849 

850 @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, s12, 

851 m12, M12, M21, S12)} 

852 ''' 

853 d = dict(azi1=self.azi12, M12=_1_0, m12=self.s12, # PYCHOK attr 

854 azi2=self.azi12, M21=_1_0) # PYCHOK attr 

855 if a12_azi1_azi2_m12_M12_M21: 

856 d.update(a12_azi1_azi2_m12_M12_M21) 

857 return self._toTuple(Direct9Tuple, dflt, d) 

858 

859 def toInverse10Tuple(self, dflt=NAN, **a12_m12_M12_M21_salp1_calp1_salp2_calp2): 

860 '''Convert this L{Rhumb8Tuple} to a 10-tuple, like I{Karney}'s 

861 method C{geographiclib.geodesic.Geodesic._GenInverse}. 

862 

863 @kwarg dflt: Default value for missing items (C{any}). 

864 @kwarg a12_m12_M12_M21_salp1_calp1_salp2_calp2: Optional keyword 

865 arguments to specify or override L{Inverse10Tuple} items. 

866 

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

868 m12, M12, M21, S12)}. 

869 ''' 

870 s, c = sincos2d(self.azi12) # PYCHOK attr 

871 d = dict(salp1=s, calp1=c, M12=_1_0, m12=self.s12, # PYCHOK attr 

872 salp2=s, calp2=c, M21=_1_0) 

873 if a12_m12_M12_M21_salp1_calp1_salp2_calp2: 

874 d.update(a12_m12_M12_M21_salp1_calp1_salp2_calp2) 

875 return self._toTuple(Inverse10Tuple, dflt, d) 

876 

877 def _toTuple(self, nTuple, dflt, updates={}): 

878 '''(INTERNAL) Convert this C{Rhumb8Tuple} to an B{C{nTuple}}. 

879 ''' 

880 _g = self.toGDict(**updates).get 

881 t = tuple(_g(n, dflt) for n in nTuple._Names_) 

882 return nTuple(t, name=self.name) 

883 

884 @deprecated_method 

885 def _to7Tuple(self): 

886 '''DEPRECATED, do not use! 

887 ''' 

888 return _MODS.deprecated.Rhumb7Tuple(self[:-1]) 

889 

890 

891# Use I{Divided Differences} to determine (mu2 - mu1) / (psi2 - psi1) accurately. 

892# Definition: _Df(x,y,d) = (f(x) - f(y)) / (x - y), @see W. M. Kahan & R. J. 

893# Fateman, "Symbolic computation of Divided Differences", SIGSAM Bull. 33(3), 

894# 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716>, @see 

895# U{UCB<https://www.CS.Berkeley.edu/~fateman/papers/divdiff.pdf>}, Dec 8, 1999. 

896 

897def _Dasinh(x, y): 

898 hx = hypot1(x) 

899 d = x - y 

900 if d: 

901 hx *= y 

902 hy = x * hypot1(y) 

903 t = (d * (x + y) / (hy + hx)) if (x * y) > 0 else (hy - hx) 

904 r = asinh(t) / d 

905 else: 

906 r = _1_0 / hx 

907 return r 

908 

909 

910def _Datan(x, y): 

911 xy = x * y 

912 r = xy + _1_0 

913 d = x - y 

914 if d: # 2 * xy > -1 == 2 * xy + 1 > 0 == xy + r > 0 == xy > -r 

915 r = (atan(d / r) if xy > -r else (atan(x) - atan(y))) / d 

916 else: 

917 r = _1_0 / r 

918 return r 

919 

920 

921def _Dcosh(x, y): 

922 return _Dsincos(x, y, sinh, sinh) 

923 

924 

925def _DeatanhE(x, y, E): 

926 # Deatanhe(x, y) = eatanhe((x - y) / (1 - e^2 * x * y)) / (x - y) 

927 e = _1_0 - E.e2 * x * y 

928 # assert not isnear0(e) 

929 d = x - y 

930 return (E._es_atanh(d / e) / d) if d else (E.e2 / e) 

931 

932 

933def _DfEt(tx, ty, eF): # tangents 

934 # eF = Elliptic(-E.e12) # -E.e2 / (1 - E.e2) 

935 r, x, y, = _1_0, atan(tx), atan(ty) 

936 d = x - y 

937 if (x * y) > 0: 

938 # See U{DLMF<https://DLMF.NIST.gov/19.11>}: 19.11.2 and 19.11.4 

939 # letting theta -> x, phi -> -y, psi -> z 

940 # (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d 

941 # tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y)) 

942 # = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) / 

943 # (sin(x)*Delta(y) + sin(y)*Delta(x)) 

944 # = t = d * Dt 

945 # sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2) 

946 # Alt (this only works for |z| <= pi/2 -- however, this conditions 

947 # holds if x*y > 0): 

948 # sin(z) = d * Dsin(x,y) * (sin(x) + sin(y)) / 

949 # (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x)) 

950 # cos(z) = sqrt((1-sin(z))*(1+sin(z))) 

951 sx, cx, sy, cy = sincos2_(x, y) 

952 D = (cx + cy) * (eF.fDelta(sy, cy) * sx + 

953 eF.fDelta(sx, cx) * sy) 

954 D = (sx + sy) * _Dsin(x, y) / D 

955 t = D * d 

956 t2 = t**2 + _1_0 

957 D *= _2_0 / t2 

958 s = D * d 

959 if s: 

960 c = (t + _1_0) * (_1_0 - t) / t2 

961 r = eF.fE(s, c, eF.fDelta(s, c)) / s 

962 r = D * (r - eF.k2 * sx * sy) 

963 elif d: 

964 r = (eF.fE(x) - eF.fE(y)) / d 

965 return r 

966 

967 

968def _Dgd(x, y): 

969 return _Datan(sinh(x), sinh(y)) * _Dsinh(x, y) 

970 

971 

972def _Dgdinv(x, y): # x, y are tangents 

973 return _Dasinh(x, y) / _Datan(x, y) 

974 

975 

976def _Dlog(x, y): 

977 d = (x - y) * _0_5 

978 # Changed atanh(t / (x + y)) to asinh(t / (2 * sqrt(x*y))) to 

979 # avoid taking atanh(1) when x is large and y is 1. This also 

980 # fixes bogus results being returned for the area when an endpoint 

981 # is at a pole. N.B. this routine is invoked with positive x 

982 # and y, so the sqrt is always taken of a positive quantity. 

983 return (asinh(d / sqrt(x * y)) / d) if d else (_1_0 / x) 

984 

985 

986def _Dsin(x, y): 

987 return _Dsincos(x, y, sin, cos) 

988 

989 

990def _Dsincos(x, y, sin_, cos_): 

991 r = cos_((x + y) * _0_5) 

992 d = (x - y) * _0_5 

993 if d: 

994 r *= sin_(d) / d 

995 return r 

996 

997 

998def _Dsinh(x, y): 

999 return _Dsincos(x, y, sinh, cosh) 

1000 

1001 

1002def _Dtan(x, y): # PYCHOK no cover 

1003 return _Dtant(x - y, tan(x), tan(y)) 

1004 

1005 

1006def _Dtant(dxy, tx, ty): 

1007 txy = tx * ty 

1008 r = txy + _1_0 

1009 if dxy: # 2 * txy > -1 == 2 * txy + 1 > 0 == txy + r > 0 == txy > -r 

1010 r = ((tan(dxy) * r) if txy > -r else (tx - ty)) / dxy 

1011 return r 

1012 

1013 

1014def _Eaux4(E_aux, mu_psi_x, mu_psi_y): # degrees 

1015 # get inverse auxiliary lats in radians and tangents 

1016 phix = radians(E_aux(mu_psi_x, inverse=True)) 

1017 phiy = radians(E_aux(mu_psi_y, inverse=True)) 

1018 return phix, phiy, tan(phix), tan(phiy) 

1019 

1020 

1021def _gd(x): 

1022 return atan(sinh(x)) 

1023 

1024 

1025def _sincosSeries(sinp, x, y, C, n): 

1026 # N.B. C[] has n+1 elements of which 

1027 # C[0] is ignored and n >= 0 

1028 # Use Clenshaw summation to evaluate 

1029 # m = (g(x) + g(y)) / 2 -- mean value 

1030 # s = (g(x) - g(y)) / (x - y) -- average slope 

1031 # where 

1032 # g(x) = sum(C[j] * SC(2 * j * x), j = 1..n) 

1033 # SC = sinp ? sin : cos 

1034 # CS = sinp ? cos : sin 

1035 # ... 

1036 d = x - y 

1037 sp, cp, sd, cd = sincos2_(x + y, d) 

1038 sd = (sd / d) if d else _1_0 

1039 m = cp * cd * _2_0 

1040 s = neg(sp * sd) # negative 

1041 # 2x2 matrices in row-major order 

1042 a0, a1 = m, (s * d**2) 

1043 a2, a3 = (s * _4_0), m 

1044 b2 = b1 = _0_0s(4) 

1045 if n > 0: 

1046 b1 = C[n], _0_0, _0_0, C[n] 

1047 _fsum1_, _neg = fsum1_, neg 

1048 for j in range(n - 1, 0, -1): 

1049 b1, b2, Cj = b2, b1, C[j] # C[0] unused 

1050 # b1 = a * b2 - b1 + C[j] * I 

1051 m0, m1, m2, m3 = b2 

1052 n0, n1, n2, n3 = map(_neg, b1) 

1053 b1 = (_fsum1_(a0 * m0, a1 * m2, n0, Cj, floats=True), 

1054 _fsum1_(a0 * m1, a1 * m3, n1, floats=True), 

1055 _fsum1_(a2 * m0, a3 * m2, n2, floats=True), 

1056 _fsum1_(a2 * m1, a3 * m3, n3, Cj, floats=True)) 

1057 # Here are the full expressions for m and s 

1058 # f01, f02, f11, f12 = (0, 0, cd * sp, 2 * sd * cp) if sinp else \ 

1059 # (1, 0, cd * cp, -2 * sd * sp) 

1060 # m = -b2[1] * f02 + (C[0] - b2[0]) * f01 + b1[0] * f11 + b1[1] * f12 

1061 # s = -b2[2] * f01 + (C[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12 

1062 cd *= b1[2] 

1063 sd *= b1[3] * _2_0 

1064 s = _fsum1_(cd * sp, sd * cp, floats=True) if sinp else \ 

1065 _fsum1_(cd * cp, _neg(sd * sp), _neg(b2[2]), floats=True) 

1066 return s 

1067 

1068 

1069_RACoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00 

1070 4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4 

1071 691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4 

1072 1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3 

1073 -1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2 

1074 104, -31, 315, # R[3]/n^3, polynomial(n), order 1 

1075 -41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20 

1076 5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5 

1077 -79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5 

1078 41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4 

1079 18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3 

1080 -23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2 

1081 5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1 

1082 -668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27 

1083 6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6 

1084 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6 

1085 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5 

1086 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4 

1087 1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3 

1088 -4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2 

1089 1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1 

1090 -313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35 

1091 7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7 

1092 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7 

1093 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6 

1094 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5 

1095 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4 

1096 15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3 

1097 -27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3 

1098 4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3 

1099 -3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44 

1100 8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8 

1101 71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8 

1102 -13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7 

1103 30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6 

1104 -8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5 

1105 50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4 

1106 359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3 

1107 -16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2 

1108 930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1 

1109 -673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54 

1110} 

1111 

1112__all__ += _ALL_DOCS(Caps, _RhumbLine) 

1113 

1114if __name__ == '__main__': 

1115 

1116 def _re(fmt, r3, x3): 

1117 e3 = [] 

1118 for r, x in _zip(r3, x3): # strict=True 

1119 e = fabs(r - x) / fabs(x) 

1120 e3.append('%.g' % (e,)) 

1121 print((fmt % r3) + ' rel errors: ' + ', '.join(e3)) 

1122 

1123 # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve> 

1124 rhumb = Rhumb(exact=True) # WGS84 default 

1125 print('# %r\n' % rhumb) 

1126 r = rhumb.Direct8(40.6, -73.8, 51, 5.5e6) # from JFK about NE 

1127 _re('# JFK NE lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.68889988, 0.25551982, 44095641862956.148438)) 

1128 r = rhumb.Inverse8(40.6, -73.8, 51.6, -0.5) # JFK to LHR 

1129 _re('# JFK-LHR azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.76838971, 5771083.383328, 37395209100030.367188)) 

1130 r = rhumb.Inverse8(40.6, -73.8, 35.8, 140.3) # JFK to Tokyo Narita 

1131 _re('# JFK-NRT azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.388887981699639, 12782581.0676841792, -63760642939072.492)) 

1132 

1133# % python3 -m pygeodesy.rhumbx 

1134 

1135# Rhumb(RAorder=6, TMorder=6, ellipsoid=Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594), exact=True) 

1136 

1137# JFK NE lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 rel errors: 4e-11, 2e-08, 2e-15 

1138# JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.4 rel errors: 3e-12, 5e-15, 2e-16 

1139# JFK-NRT azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 rel errors: 2e-16, 3e-16, 0 

1140 

1141# **) MIT License 

1142# 

1143# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1144# 

1145# Permission is hereby granted, free of charge, to any person obtaining a 

1146# copy of this software and associated documentation files (the "Software"), 

1147# to deal in the Software without restriction, including without limitation 

1148# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1149# and/or sell copies of the Software, and to permit persons to whom the 

1150# Software is furnished to do so, subject to the following conditions: 

1151# 

1152# The above copyright notice and this permission notice shall be included 

1153# in all copies or substantial portions of the Software. 

1154# 

1155# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1156# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1157# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1158# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1159# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1160# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1161# OTHER DEALINGS IN THE SOFTWARE.