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>} from 

7I{GeographicLib version 2.0}. 

8 

9Class L{RhumbLine} has been enhanced with methods C{intersection2} and C{nearestOn4} to iteratively 

10find the intersection of two rhumb lines, respectively the nearest point on a rumb line along a 

11geodesic or perpendicular rhumb line. 

12 

13For more details, see the C++ U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>} 

14documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>}, 

15the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>}, 

16the utily U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online 

17rhumb line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}. 

18 

19Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2014-2022) and licensed under the MIT/X11 

20License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. 

21''' 

22# make sure int/int division yields float quotient 

23from __future__ import division as _; del _ # PYCHOK semicolon 

24 

25from pygeodesy.basics import copysign0, neg, unsigned0, _zip 

26from pygeodesy.constants import NAN, PI_2, _0_0s, _0_0, _0_5, \ 

27 _1_0, _2_0, _4_0, _720_0, _over 

28# from pygeodesy.ellipsoids import _EWGS84 # from .karney 

29from pygeodesy.errors import itemsorted, RhumbError, _Xorder 

30from pygeodesy.fmath import hypot, hypot1 

31# from pygeodesy.fsums import fsum1f_ # _MODS 

32from pygeodesy.interns import NN, _COMMASPACE_ 

33from pygeodesy.karney import _atan2d, Caps, _diff182, GDict, _GTuple, \ 

34 _norm180, _EWGS84 

35from pygeodesy.ktm import KTransverseMercator, _Xs, \ 

36 _AlpCoeffs, _BetCoeffs # PYCHOK used! 

37from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

38from pygeodesy.props import deprecated_method, Property, Property_RO, property_RO 

39from pygeodesy.rhumbBase import RhumbBase, RhumbLineBase, pairs, _update_all_rls 

40# from pygeodesy.streprs import pairs # from .rhumbBase 

41from pygeodesy.units import Int 

42from pygeodesy.utily import sincos2_ 

43 

44from math import asinh, atan, cos, cosh, fabs, radians, sin, sinh, sqrt, tan 

45 

46__all__ = _ALL_LAZY.rhumbx 

47__version__ = '23.08.20' 

48 

49 

50class Rhumb(RhumbBase): 

51 '''Class to solve the I{direct} and I{inverse rhumb} problems, based on 

52 I{elliptic functions} or I{Krüger} series expansion. 

53 

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

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

56 ''' 

57 _mRA = 6 # see .RAorder 

58 

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

60 '''New C{rhumbx.Rhumb}. 

61 

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

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

64 (equatorial) radius (C{scalar}). 

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

66 a C{scalar}, ignored otherwise. 

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

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

69 series expansion (C{bool} or C{None}), see also properties 

70 C{exact} and C{TMorder}. 

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

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

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

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

75 

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

77 ''' 

78 RhumbBase.__init__(self, a_earth, f, exact, name) 

79 if RA_TMorder: 

80 self.orders(**RA_TMorder) 

81 

82 @Property_RO 

83 def _A2(self): # Conformal2RectifyingCoeffs 

84 m = self.TMorder 

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

86 

87 @Property_RO 

88 def _B2(self): # Rectifying2ConformalCoeffs 

89 m = self.TMorder 

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

91 

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

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

94 

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

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

97 

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

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

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

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

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

103 

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

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

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

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

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

109 

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

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

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

113 

114 @note: If the given point is a pole, the cosine of its latitude is 

115 taken to be C{sqrt(L{EPS})}. This position is extremely 

116 close to the actual pole and allows the calculation to be 

117 carried out in finite terms. 

118 ''' 

119 rl = RhumbLine(self, lat1, lon1, azi12, caps=Caps.LINE_OFF, 

120 name=self.name) 

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

122 

123 @deprecated_method 

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

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

126 

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

128 ''' 

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

130 

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

132 E = self.ellipsoid 

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

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

135 

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

137 if self.exact: 

138 E = self.ellipsoid 

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

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

141 r = _over(self._DRectifyingt( tphix, tphiy, t), 

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

143 else: 

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

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

146 return r 

147 

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

149 E = self.ellipsoid 

150 tbetx = E.f1 * tphix 

151 tbety = E.f1 * tphiy 

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

153 * _DfEt( tbetx, tbety, self._eF) 

154 * _Datan(tbetx, tbety)) / E.L 

155 

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

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

158 

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

160 E = self.ellipsoid 

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

162 if self.exact: 

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

164 r = _over(self._DIsometrict(phix, phiy, tphix, tphiy, t), 

165 self._DRectifyingt( tphix, tphiy, t)) 

166 else: 

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

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

169 return r 

170 

171 @Property_RO 

172 def _eF(self): 

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

174 ''' 

175 # .k2 = 0.006739496742276434 

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

177 

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

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

180 

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

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

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

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

185 

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

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

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

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

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

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

192 

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

194 on opposite meridians, there are two shortest rhumb lines 

195 and the East-going one is chosen. 

196 

197 @note: If either point is a pole, the cosine of its latitude is 

198 taken to be C{sqrt(L{EPS})}. This position is extremely 

199 close to the actual pole and allows the calculation to be 

200 carried out in finite terms. 

201 ''' 

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

203 if (outmask & Cs.AZIMUTH_DISTANCE_AREA): 

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

205 E = self.ellipsoid 

206 psi1 = E.auxIsometric(lat1) 

207 psi2 = E.auxIsometric(lat2) 

208 psi12 = psi2 - psi1 

209 lon12, _ = _diff182(lon1, lon2) 

210 if (outmask & Cs.AZIMUTH): 

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

212 if (outmask & Cs.DISTANCE): 

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

214 s12 = a12 * E._L_90 

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

216 if (outmask & Cs.AREA): 

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

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

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

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

221 lon12=lon12, psi1=psi1, exact=self.exact, 

222 psi12=psi12, psi2=psi2) 

223 return r 

224 

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

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

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

228# 

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

230# ''' 

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

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

233 

234 @deprecated_method 

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

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

237 

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

239 ''' 

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

241 

242 def _meanSinXi(self, x, y): # radians 

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

244 if self.f: 

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

246 return s 

247 

248 @deprecated_method 

249 def orders(self, RAorder=None, TMorder=None): # PYCHOK expected 

250 '''DEPRECATED, use properties C{RAorder} and/or C{TMorder}. 

251 

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

253 

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

255 or 8). 

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

257 5, 6, 7 or 8). 

258 

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

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

261 ''' 

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

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

264 self.RAorder = RAorder 

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

266 self.TMorder = TMorder 

267 return t 

268 

269 @Property_RO 

270 def _RA2(self): 

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

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

273 m = self.RAorder 

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

275 

276 @Property 

277 def RAorder(self): 

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

279 ''' 

280 return self._mRA 

281 

282 @RAorder.setter # PYCHOK setter! 

283 def RAorder(self, order): 

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

285 ''' 

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

287 if self._mRA != n: 

288 _update_all_rls(self) 

289 self._mRA = n 

290 

291 @Property_RO 

292 def _RhumbLine(self): 

293 '''(INTERNAL) Get this module's C{RhumbLine} class. 

294 ''' 

295 return RhumbLine 

296 

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

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

299 ''' 

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

301 self.ellipsoid.area) * lon12 / _720_0 

302 r *= self._meanSinXi(radians(psi2), radians(psi1)) 

303 return r 

304 

305 @Property 

306 def TMorder(self): 

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

308 ''' 

309 return self._mTM 

310 

311 @TMorder.setter # PYCHOK setter! 

312 def TMorder(self, order): 

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

314 

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

316 ''' 

317 self.exact = self._TMorder(order) 

318 

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

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

321 

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

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

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

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

326 

327 @return: Tuple items (C{str}). 

328 ''' 

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

330 exact=self.exact, TMorder=self.TMorder) 

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

332 

333 

334class RhumbLine(RhumbLineBase): 

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

336 

337 Class C{RhumbLine} facilitates the determination of points on 

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

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

340 

341 Method C{RhumbLine.Position} returns the location of an other 

342 point at distance C{s12} along and the area C{S12} under the 

343 rhumb line. 

344 

345 Method C{RhumbLine.intersection2} finds the intersection between 

346 two rhumb lines. 

347 

348 Method C{RhumbLine.nearestOn4} computes the nearest point on and 

349 the distance to a rhumb line in different ways. 

350 ''' 

351 _Rhumb = Rhumb # rhumbx.Rhumb 

352 

353 def __init__(self, rhumb, lat1=0, lon1=0, azi12=None, **caps_name): # PYCHOK signature 

354 '''New C{rhumbx.RhumbLine}. 

355 

356 @arg rhumb: The rhumb reference (C{rhumbx.Rhumb}). 

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

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

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

360 @kwarg caps_name: Optional keyword arguments C{B{name}=NN} and 

361 C{B{caps}=0}, a bit-or'ed combination of L{Caps} 

362 values specifying the required capabilities. Include 

363 C{Caps.LINE_OFF} if updates to the B{C{rhumb}} should 

364 I{not} be reflected in this rhumb line. 

365 ''' 

366 RhumbLineBase.__init__(self, rhumb, lat1, lon1, azi12, **caps_name) 

367 

368 @Property_RO 

369 def _mu1(self): 

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

371 ''' 

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

373 

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

375 '''Compute a point at a distance on this rhumb line. 

376 

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

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

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

380 the quantities to be returned. 

381 

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

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

384 C{lon2} of the point in C{degrees}, the rhumb angle C{a12} 

385 in C{degrees} from the start point of and the area C{S12} 

386 under this rhumb line in C{meter} I{squared}. 

387 

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

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

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

391 

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

393 taken to be C{sqrt(L{EPS})}. This position is extremely 

394 close to the actual pole and allows the calculation to be 

395 carried out in finite terms. 

396 ''' 

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

398 if (outmask & Cs.LATITUDE_LONGITUDE_AREA): 

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

400 a12 = s12 / E._L_90 

401 mu12 = self._calp * a12 

402 mu2, x90 = self._mu22(mu12, self._mu1) 

403 if x90: # PYCHOK no cover 

404 lat2 = E.auxRectifying(mu2, inverse=True) 

405 lon2 = NAN 

406 if (outmask & Cs.AREA): 

407 r.set_(S12=NAN) 

408 else: 

409 psi2 = self._psi1 

410 if self._calp: 

411 lat2 = E.auxRectifying(mu2, inverse=True) 

412 psi12 = R._DRectifying2Isometricd(mu2, 

413 self._mu1) * mu12 

414 lon2 = psi12 * self._salp / self._calp 

415 psi2 += psi12 

416 else: # PYCHOK no cover 

417 lat2 = self.lat1 

418 lon2 = self._salp * s12 / self._r1rad 

419 if (outmask & Cs.AREA): 

420 S12 = R._S12d(lon2, self._psi1, psi2) 

421 r.set_(S12=unsigned0(S12)) # like .gx 

422 if (outmask & Cs.LONGITUDE): 

423 if (outmask & Cs.LONG_UNROLL): 

424 lon2 += self.lon1 

425 else: 

426 lon2 = _norm180(self._lon12 + lon2) 

427 r.set_(azi12=self.azi12, s12=s12, a12=a12) 

428 if (outmask & Cs.LATITUDE): 

429 r.set_(lat2=lat2, lat1=self.lat1) 

430 if (outmask & Cs.LONGITUDE): 

431 r.set_(lon2=lon2, lon1=self.lon1) 

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

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

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

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

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

437 return r 

438 

439 @Property_RO 

440 def _psi1(self): 

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

442 ''' 

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

444 

445 @property_RO 

446 def RAorder(self): 

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

448 ''' 

449 return self.rhumb.RAorder 

450 

451 @Property_RO 

452 def _r1rad(self): # PYCHOK no cover 

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

454 ''' 

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

456 

457 

458class RhumbOrder2Tuple(_GTuple): 

459 '''2-Tuple C{(RAorder, TMorder)} with a I{Rhumb Area} and 

460 I{Transverse Mercator} order, both C{int}, DEPRECATED. 

461 ''' 

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

463 _Units_ = ( Int, Int) 

464 

465 

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

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

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

469# 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716> and @see 

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

471 

472def _Dasinh(x, y): 

473 hx = hypot1(x) 

474 d = x - y 

475 if d: 

476 hx *= y 

477 hy = x * hypot1(y) 

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

479 r = asinh(t) / d 

480 else: 

481 r = _1_0 / hx 

482 return r 

483 

484 

485def _Datan(x, y): 

486 xy = x * y 

487 r = xy + _1_0 

488 d = x - y 

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

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

491 else: 

492 r = _1_0 / r 

493 return r 

494 

495 

496def _Dcosh(x, y): 

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

498 

499 

500def _DeatanhE(x, y, E): # see .albers._Datanhee 

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

502 e = _1_0 - E.e2 * x * y 

503 if e: # assert not isnear0(e) 

504 d = x - y 

505 e = (E._es_atanh(d / e) / d) if d else (E.e2 / e) 

506 return e 

507 

508 

509def _DfEt(tx, ty, eF): # tangents 

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

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

512 d = x - y 

513 if (x * y) > 0: 

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

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

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

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

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

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

520 # = t = d * Dt 

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

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

523 # holds if x*y > 0): 

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

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

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

527 sx, cx, sy, cy = sincos2_(x, y) 

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

529 eF.fDelta(sx, cx) * sy) 

530 D = (sx + sy) * _Dsin(x, y) / D 

531 t = D * d 

532 t2 = _1_0 + t**2 

533 D *= _2_0 / t2 

534 s = D * d 

535 if s: 

536 c = (t + _1_0) * (_1_0 - t) / t2 

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

538 r = D * (r - eF.k2 * sx * sy) 

539 elif d: 

540 r = (eF.fE(x) - eF.fE(y)) / d 

541 return r 

542 

543 

544def _Dgd(x, y): 

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

546 

547 

548def _Dgdinv(x, y): # x, y are tangents 

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

550 

551 

552def _Dlog(x, y): 

553 d = (x - y) * _0_5 

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

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

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

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

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

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

560 

561 

562def _Dsin(x, y): 

563 return _Dsincos(x, y, sin, cos) 

564 

565 

566def _Dsincos(x, y, sin_, cos_): 

567 r = cos_((x + y) * _0_5) 

568 d = (x - y) * _0_5 

569 if d: 

570 r *= sin_(d) / d 

571 return r 

572 

573 

574def _Dsinh(x, y): 

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

576 

577 

578def _Dtan(x, y): # PYCHOK no cover 

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

580 

581 

582def _Dtant(dxy, tx, ty): 

583 txy = tx * ty 

584 r = txy + _1_0 

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

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

587 return r 

588 

589 

590def _Eaux4(E_aux, mu_psi_x, mu_psi_y): # degrees 

591 # get inverse auxiliary lats in radians and tangents 

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

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

594 return phix, phiy, tan(phix), tan(phiy) 

595 

596 

597def _gd(x): 

598 return atan(sinh(x)) 

599 

600 

601def _sincosSeries(sinp, x, y, C, n): 

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

603 # C[0] is ignored and n >= 0 

604 # Use Clenshaw summation to evaluate 

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

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

607 # where 

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

609 # SC = sinp ? sin : cos 

610 # CS = sinp ? cos : sin 

611 # ... 

612 d, _neg = (x - y), neg 

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

614 sd = (sd / d) if d else _1_0 

615 s = _neg(sp * sd) # negative 

616 # 2x2 matrices in row-major order 

617 a1 = s * d**2 

618 a2 = s * _4_0 

619 a0 = a3 = _2_0 * cp * cd # m 

620 b2 = b1 = _0_0s(4) 

621 if n > 0: 

622 b1 = C[n], _0_0, _0_0, C[n] 

623 

624 _fsum = _MODS.fsums.fsum1f_ 

625 for j in range(n - 1, 0, -1): # C[0] unused 

626 b1, b2, Cj = b2, b1, C[j] 

627 # b1 = a * b2 - b1 + C[j] * I 

628 m0, m1, m2, m3 = b2 

629 n0, n1, n2, n3 = map(_neg, b1) 

630 b1 = (_fsum(a0 * m0, a1 * m2, n0, Cj), 

631 _fsum(a0 * m1, a1 * m3, n1), 

632 _fsum(a2 * m0, a3 * m2, n2), 

633 _fsum(a2 * m1, a3 * m3, n3, Cj)) 

634 # Here are the full expressions for m and s 

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

636 # (1, 0, cd * cp, -2 * sd * sp) 

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

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

639 cd *= b1[2] 

640 sd *= b1[3] * _2_0 

641 s = _fsum(cd * sp, sd * cp) if sinp else \ 

642 _fsum(cd * cp, _neg(sd * sp), _neg(b2[2])) 

643 return s 

644 

645 

646_RACoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00 

647 4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4 

648 691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4 

649 1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3 

650 -1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2 

651 104, -31, 315, # R[3]/n^3, polynomial(n), order 1 

652 -41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20 

653 5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5 

654 -79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5 

655 41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4 

656 18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3 

657 -23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2 

658 5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1 

659 -668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27 

660 6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6 

661 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6 

662 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5 

663 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4 

664 1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3 

665 -4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2 

666 1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1 

667 -313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35 

668 7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7 

669 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7 

670 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6 

671 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5 

672 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4 

673 15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3 

674 -27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3 

675 4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3 

676 -3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44 

677 8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8 

678 71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8 

679 -13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7 

680 30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6 

681 -8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5 

682 50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4 

683 359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3 

684 -16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2 

685 930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1 

686 -673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54 

687} 

688 

689__all__ += _ALL_DOCS(Caps, Rhumb, RhumbLine) 

690 

691if __name__ == '__main__': 

692 

693 from pygeodesy.lazily import printf 

694 

695 def _re(fmt, r3, x3): 

696 e3 = [] 

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

698 e = fabs(r - x) / fabs(x) 

699 e3.append('%.g' % (e,)) 

700 printf((fmt % r3) + ' rel errors: ' + ', '.join(e3)) 

701 

702 # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve> version 2.0 

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

704 printf('# %r\n', rhumb) 

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

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

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

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

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

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

711 

712# % python3 -m pygeodesy.rhumbx 

713 

714# 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, e21=0.99330562, 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) 

715 

716# JFK NE lat2=71.68889988, lon2=0.25551982, S12=44095641862956.1 rel errors: 4e-11, 2e-08, 5e-16 

717# JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.4 rel errors: 3e-12, 5e-15, 0 

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

719 

720# **) MIT License 

721# 

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

723# 

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

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

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

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

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

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

730# 

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

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

733# 

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

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

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

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

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

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

740# OTHER DEALINGS IN THE SOFTWARE.