Coverage for pygeodesy/rhumbaux.py: 96%

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1# -*- coding: utf-8 -*- 

2 

3u'''A pure Python version of I{Karney}'s C++ classes U{Rhumb 

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

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

6I{GeographicLib version 2.2+}. 

7 

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

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

10geodesic or perpendicular rhumb line. 

11 

12For more details, see the I{2.2} U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>} 

13documentation, especially the U{Class List<https://GeographicLib.SourceForge.io/C++/doc/annotated.html>}, 

14the background information on U{Rhumb lines<https://GeographicLib.SourceForge.io/C++/doc/rhumb.html>}, 

15utility U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/RhumbSolve.1.html>} and U{Online rhumb 

16line calculations<https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolve>}. 

17 

18Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2022-2023) and licensed under the MIT/X11 

19License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. 

20 

21@note: Class L{AuxDST} requires package U{numpy<https://PyPI.org/project/numpy>} to be installed, 

22 version 1.16 or newer and needed for I{exact} area calculations. 

23''' 

24# make sure int/int division yields float quotient 

25from __future__ import division as _; del _ # PYCHOK semicolon 

26 

27from pygeodesy.auxilats.auxAngle import AuxMu, AuxPhi, atan2d, hypot 

28from pygeodesy.auxilats.auxDLat import AuxDLat, _DClenshaw 

29# from pygeodesy.auxilats.auxDST import AuxDST # _MODS 

30from pygeodesy.auxilats.auxily import _Dlam, _Dp0Dpsi, _Ufloats 

31from pygeodesy.basics import _reverange, unsigned0, _zip, _xkwds_get 

32from pygeodesy.constants import EPS_2, MANT_DIG, NAN, PI4, isinf, \ 

33 _0_0, _4_0, _720_0, _log2, _over 

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

35# from pygeodesy.errors import itemsorted, _xkwds_get # from .basics, ... 

36from pygeodesy.karney import Caps, _diff182, GDict, _norm180, _EWGS84 

37# from pygeodesy.fmath import hypot # from .auxilats.auxAngle 

38from pygeodesy.interns import NN, _COMMASPACE_ 

39from pygeodesy.lazily import _ALL_LAZY, _ALL_DOCS, _ALL_MODS as _MODS 

40# from pygeodesy.props import Property, Property_RO # from .rhumbBase 

41from pygeodesy.rhumbBase import RhumbBase, RhumbLineBase, itemsorted, \ 

42 pairs, Property, Property_RO 

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

44# from pygeodesy.utily import atan2d # from .auxilats.auxAngle 

45 

46from math import ceil as _ceil, fabs, radians 

47 

48__all__ = _ALL_LAZY.rhumbaux 

49__version__ = '23.08.20' 

50 

51# DIGITS = (sizeof(real) * 8) bits 

52# = (ctypes.sizeof(ctypes.c_double(1.0)) * 8) bits 

53# For |n| <= 0.99, actual max for doubles is 2163. This scales 

54# as DIGITS and for long doubles (GEOGRAPHICLIB_PRECISION = 3, 

55# DIGITS = 64), this becomes 2163 * 64 / 53 = 2612. Round this 

56# up to 2^12 = 4096 and scale this by DIGITS//64 if DIGITS > 64. 

57# 

58# 64 = DIGITS for long double, 6 = 12 - _log2(64) 

59_Lbits = 1 << (int(_ceil(_log2(max(MANT_DIG, 64)))) + 6) 

60 

61 

62class RhumbAux(RhumbBase): 

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

64 on I{Auxiliary Latitudes} for accuracy near the poles. 

65 

66 @note: Package U{numpy<https://PyPI.org/project/numpy>} must be 

67 installed, version 1.16 or later. 

68 ''' 

69 

70 def __init__(self, a_earth=_EWGS84, f=None, exact=True, name=NN, **TMorder): # PYCHOK signature 

71 '''New C{rhumbaux.RhumbAux}. 

72 

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

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

75 (equatorial) radius (C{scalar}). 

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

77 a C{scalar}, ignored otherwise. 

78 @kwarg exact: If C{True}, use the exact expressions for the I{Auxiliary 

79 Latitudes}, otherwise use the I{Fourier} series expansion 

80 (C{bool}), see also property C{exact}. 

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

82 @kwarg TMorder: Optional keyword argument B{C{TMorder}}, see property 

83 C{TMorder}. 

84 

85 @raise ImportError: Package C{numpy} not found or not installed, only 

86 required for area C{S12} when C{B{exact} is True}. 

87 

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

89 ''' 

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

91 if TMorder: 

92 self.Tmorder = _xkwds_get(TMorder, TMorder=RhumbBase._mTM) 

93 

94 def areaux(self, **exact): 

95 '''Get this ellipsoid's B{C{exact}} surface area (C{meter} I{squared}). 

96 

97 @kwarg exact: Optional C{exact} (C{bool}), overriding this rhumb's 

98 C{exact} setting, if C{True}, use the exact expression 

99 for the authalic radius otherwise the I{Taylor} series. 

100 

101 @return: The (signed?) surface area (C{meter} I{squared}). 

102 

103 @raise AuxError: If C{B{exact}=False} and C{abs(flattening)} exceeds 

104 property C{f_max}. 

105 

106 @note: The area of a polygon encircling a pole can be found by adding 

107 C{areaux / 2} to the sum of C{S12} for each side of the polygon. 

108 

109 @see: U{The area of rhumb polygons<https://ArXiv.org/pdf/2303.03219.pdf>} 

110 and method L{auxilats.AuxLat.AuthalicRadius2}. 

111 ''' 

112 x = _xkwds_get(exact, exact=self.exact) 

113 a = (self._c2 * _720_0) if bool(x) is self.exact else \ 

114 (self._auxD.AuthalicRadius2(exact=x, f_max=self.f_max) * PI4) 

115 return a 

116 

117 @Property_RO 

118 def _auxD(self): 

119 return AuxDLat(self.ellipsoid) 

120 

121 @Property_RO 

122 def _c2(self): # radians makes _c2 a factor per degree 

123 return radians(self._auxD.AuthalicRadius2(exact=self.exact, f_max=self.f_max)) 

124 

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

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

127 

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

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

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

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

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

133 

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

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

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

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

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

139 

140 @raise ImportError: Package C{numpy} not found or not installed, 

141 only required for area C{S12} when C{B{exact} 

142 is True}. 

143 

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

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

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

147 

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

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

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

151 carried out in finite terms. 

152 ''' 

153 rl = RhumbLineAux(self, lat1, lon1, azi12, caps=Caps.LINE_OFF, 

154 name=self.name) 

155 return rl.Position(s12, outmask) # lat2, lon2, S12 

156 

157 def _DMu_DPsi(self, Phi1, Phi2, Chi1, Chi2): 

158 xD = self._auxD 

159 return _over(xD.DRectifying(Phi1, Phi2), 

160 xD.DIsometric( Phi1, Phi2)) if self.exact else \ 

161 _over(xD.CRectifying(Chi1, Chi2), 

162 _Dlam(Chi1.tan, Chi2.tan)) # not Lambertian! 

163 

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

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

166 

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

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

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

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

171 

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

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

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

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

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

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

178 

179 @raise ImportError: Package C{numpy} not found or not installed, 

180 only required for area C{S12} when C{B{exact} 

181 is True}. 

182 

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

184 on opposite meridians, there are two shortest rhumb lines 

185 and the East-going one is chosen. 

186 

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

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

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

190 carried out in finite terms. 

191 ''' 

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

193 if (outmask & Cs.AZIMUTH_DISTANCE_AREA): 

194 psi1, Chi1, Phi1 = self._psiChiPhi3(lat1) 

195 psi2, Chi2, Phi2 = self._psiChiPhi3(lat2) 

196 

197 psi12 = psi2 - psi1 

198 lon12, _ = _diff182(lon1, lon2, K_2_0=True) 

199 lam12 = radians(lon12) 

200 if (outmask & Cs.AZIMUTH): 

201 r.set_(azi12=atan2d(lam12, psi12)) 

202 if (outmask & Cs.DISTANCE): 

203 if isinf(psi1) or isinf(psi2): # PYCHOK no cover 

204 d = Phi2.toMu(self).toRadians 

205 d -= Phi1.toMu(self).toRadians 

206 s = fabs(d) 

207 else: # dmu/dpsi = dmu/dchi/dpsi/dchi 

208 s = self._DMu_DPsi(Phi1, Phi2, Chi1, Chi2) 

209 s *= hypot(lam12, psi12) 

210 r.set_(s12=self._rrm * s) 

211 if (outmask & Cs.AREA): 

212 S = self._c2SinXi(Chi1, Chi2) 

213 r.set_(S12=unsigned0(S * lon12)) # like .gx 

214 return r 

215 

216 def _c2SinXi(self, Chix, Chiy): 

217 pP, xD = self._RA, self._auxD 

218 

219 tx, Phix = Chix.tan, Chix.toPhi(self) 

220 ty, Phiy = Chiy.tan, Chiy.toPhi(self) 

221 dD = _DClenshaw(False, Phix.toBeta(self).normalized, 

222 Phiy.toBeta(self).normalized, 

223 pP, min(len(pP), 8)) # Fsum 

224 dD *= _over(xD.DParametric(Phix, Phiy), 

225 xD.DIsometric( Phix, Phiy)) if self.exact else \ 

226 _over(xD.CParametric(Chix, Chiy), _Dlam(tx, ty)) # not Lambertian! 

227 dD += _Dp0Dpsi(tx, ty) 

228 dD *= self._c2 

229 return float(dD) 

230 

231 def _psiChiPhi3(self, lat): 

232 Phi = AuxPhi.fromDegrees(lat) 

233 Chi = Phi.toChi(self) 

234 psi = Chi.toLambertianRadians 

235 return psi, Chi, Phi 

236 

237 @Property_RO 

238 def _RA(self): # get the coefficients for area calculation 

239 return tuple(_RAintegrate(self._auxD) if self.exact else 

240 _RAseries(self._auxD)) 

241 

242 @Property_RO 

243 def _RhumbLine(self): 

244 '''(INTERNAL) Get this module's C{RhumbLineAux} class. 

245 ''' 

246 return RhumbLineAux 

247 

248 @Property_RO 

249 def _rrm(self): 

250 return self._auxD.RectifyingRadius(exact=self.exact) 

251 

252 @Property 

253 def TMorder(self): 

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

255 ''' 

256 return self._mTM 

257 

258 @TMorder.setter # PYCHOK setter! 

259 def TMorder(self, order): 

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

261 ''' 

262 self._TMorder(order) 

263 

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

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

266 

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

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

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

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

271 

272 @return: Tuple items (C{str}). 

273 ''' 

274 d = dict(ellipsoid=self.ellipsoid, exact=self.exact, 

275 TMorder=self.TMorder) 

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

277 

278 

279class RhumbLineAux(RhumbLineBase): 

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

281 

282 Class C{RhumbLineAux} facilitates the determination of points 

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

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

285 

286 Method C{RhumbLineAux.Position} returns the location of an 

287 other point and optionally the distance C{s12} along and the 

288 area C{S12} under the rhumb line. 

289 

290 Method C{RhumbLineAux.intersection2} finds the intersection 

291 between two rhumb lines. 

292 

293 Method C{RhumbLineAux.nearestOn4} computes the nearest point 

294 on and the distance to a rhumb line in different ways. 

295 ''' 

296 _Rhumb = RhumbAux # rhumbaux.RhumbAux 

297 

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

299 '''New C{rhumbaux.RhumbLineAux}. 

300 

301 @arg rhumb: The rhumb reference (C{rhumbaux.RhumbAux}). 

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

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

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

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

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

307 values specifying the required capabilities. Include 

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

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

310 ''' 

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

312 

313 @Property_RO 

314 def _Chi1(self): 

315 return self._Phi1.toChi(self.rhumb) 

316 

317 @Property_RO 

318 def _mu1(self): 

319 return self._Phi1.toMu(self.rhumb).toDegrees 

320 

321 @Property_RO 

322 def _Phi1(self): 

323 return AuxPhi.fromDegrees(self.lat1) 

324 

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

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

327 

328 @arg s12: The distance along this rhumb line between its origin 

329 and the point (C{meters}), can be negative. 

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

331 the quantities to be returned. 

332 

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

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

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

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

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

338 

339 @raise ImportError: Package C{numpy} not found or not installed, 

340 only required for area C{S12} when C{B{exact} 

341 is True}. 

342 

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

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

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

346 

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

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

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

350 carried out in finite terms. 

351 ''' 

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

353 if (outmask & Cs.LATITUDE_LONGITUDE_AREA): 

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

355 r12 = _over(s12, radians(R._rrm)) 

356 mu2, x90 = self._mu22(self._calp * r12, self._mu1) 

357 Mu2 = AuxMu.fromDegrees(mu2) 

358 Phi2 = Mu2.toPhi(R) 

359 lat2 = Phi2.toDegrees 

360 if x90: # PYCHOK no cover 

361 lon2 = NAN 

362 if (outmask & Cs.AREA): 

363 r.set_(S12=NAN) 

364 else: 

365 Chi2 = Phi2.toChi(R) 

366 Chi1 = self._Chi1 

367 lon2 = R._DMu_DPsi(self._Phi1, Phi2, Chi1, Chi2) 

368 lon2 = _over(self._salp * r12, lon2) 

369 if (outmask & Cs.AREA): 

370 S = R._c2SinXi(Chi1, Chi2) 

371 r.set_(S12=unsigned0(S * lon2)) # like .gx 

372 if (outmask & Cs.LONGITUDE): 

373 if (outmask & Cs.LONG_UNROLL): 

374 lon2 += self.lon1 

375 else: 

376 lon2 = _norm180(self._lon12 + lon2) 

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

378 if (outmask & Cs.LATITUDE): 

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

380 if (outmask & Cs.LONGITUDE): 

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

382 return r 

383 

384# @Property_RO 

385# def _psi1(self): 

386# return self._Chi1.toLambertianRadians 

387 

388 

389def _RAintegrate(auxD): 

390 # Compute coefficients by Fourier transform of integrand 

391 L = 2 

392 fft = _MODS.auxilats.auxDST.AuxDST(L) 

393 f = auxD._qIntegrand 

394 c = fft.transform(f) 

395 # assert L < _Lbits 

396 while L < _Lbits: 

397 fft.reset(L) 

398 c = fft.refine(f, c) 

399 L *= 2 # == len(c) 

400 # assert len(c) == L 

401 pP, k = [], -1 

402 for j in range(1, L + 1): 

403 # Compute Fourier coefficients of integral 

404 p = -(c[j - 1] + (c[j] if j < L else _0_0)) / (_4_0 * j) 

405 if fabs(p) > EPS_2: 

406 k = -1 # run interrupted 

407 else: 

408 if k < 0: 

409 k = 1 # mark as first small value 

410 if (j - k) >= ((j + 7) // 8): 

411 # run of at least (j - 1) // 8 small values 

412 return pP[:j] 

413 pP.append(p) 

414 return pP # no convergence, use pP as-is 

415 

416 

417def _RAseries(auxD): 

418 # Series expansions in n for Fourier coeffients of the integral 

419 # @see: U{"Series expansions for computing rhumb areas" 

420 # <https:#DOI.org/10.5281/zenodo.7685484>}. 

421 d = n = auxD._n 

422 i = 0 

423 pP = [] 

424 aL = auxD.ALorder 

425 Cs = _RACoeffs[aL] 

426 # assert len(Cs) == (aL * (aL + 1)) // 2 

427 _p = _MODS.karney._polynomial 

428 for m in _reverange(aL): # order 

429 j = i + m + 1 

430 pP.append(_p(n, Cs, i, j) * d) 

431 d *= n 

432 i = j 

433 # assert i == len(pP) 

434 return pP 

435 

436 

437_f, _u = float, _Ufloats() 

438_RACoeffs = { # Rhumb Area Coefficients in matrix Q 

439 4: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4 

440 596 / _f(2025), -398 / _f(945), 22 / _f(45), -1 / _f(3), 

441 1543 / _f(4725), -118 / _f(315), 1 / _f(5), 

442 152 / _f(945), -17 / _f(315), 

443 5 / _f(252)), 

444 5: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5 

445 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45), 

446 -1 / _f(3), 

447 -24562 / _f(155925), 1543 / _f(4725), -118 / _f(315), 1 / _f(5), 

448 -38068 / _f(155925), 152 / _f(945), -17 / _f(315), 

449 -752 / _f(10395), 5 / _f(252), 

450 -101 / _f(17325)), 

451 6: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6 

452 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025), 

453 -398 / _f(945), 22 / _f(45), -1 / _f(3), 

454 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725), 

455 -118 / _f(315), 1 / _f(5), 

456 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945), 

457 -17 / _f(315), 

458 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252), 

459 62464 / _f(2027025), -101 / _f(17325), 

460 11537 / _f(4054050)), 

461 7: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7 

462 -565017322 / _f(1915538625), 138734126 / _f(638512875), 

463 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45), 

464 -1 / _f(3), 

465 -1969276 / _f(58046625), 17749373 / _f(425675250), -24562 / _f(155925), 

466 1543 / _f(4725), -118 / _f(315), 1 / _f(5), 

467 -58573784 / _f(638512875), 1882432 / _f(8513505), -38068 / _f(155925), 

468 152 / _f(945), -17 / _f(315), 

469 -6975184 / _f(42567525), 268864 / _f(2027025), -752 / _f(10395), 

470 5 / _f(252), 

471 -112832 / _f(1447875), 62464 / _f(2027025), -101 / _f(17325), 

472 -4096 / _f(289575), 11537 / _f(4054050), 

473 -311 / _f(525525)), 

474 8: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8 

475 188270561816 / _f(488462349375), -565017322 / _f(1915538625), 

476 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025), 

477 -398 / _f(945), 22 / _f(45), -1 / _f(3), 

478 2332829602 / _f(23260111875), -1969276 / _f(58046625), 

479 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725), 

480 -118 / _f(315), 1 / _f(5), 

481 -41570288 / _f(930404475), -58573784 / _f(638512875), 

482 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945), 

483 -17 / _f(315), 

484 1538774036 / _f(10854718875), -6975184 / _f(42567525), 

485 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252), 

486 436821248 / _f(3618239625), -112832 / _f(1447875), 

487 62464 / _f(2027025), -101 / _f(17325), 

488 3059776 / _f(80405325), -4096 / _f(289575), 11537 / _f(4054050), 

489 4193792 / _f(723647925), -311 / _f(525525), 

490 1097653 / _f(1929727800)) 

491} 

492del _f, _u, _Ufloats 

493 

494 

495__all__ += _ALL_DOCS(Caps, RhumbAux, RhumbLineAux) 

496 

497if __name__ == '__main__': 

498 

499 from pygeodesy.lazily import printf 

500 from pygeodesy import RhumbAux # PYCHOK RhumbAux is __main__.RhumbAux 

501 

502 def _re(fmt, r3, x3): 

503 e3 = [] 

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

505 e = fabs(r - x) / fabs(x) 

506 e3.append('%.g' % (e,)) 

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

508 

509 # <https://GeographicLib.SourceForge.io/cgi-bin/RhumbSolveå -p 9> version 2.2 

510 rhumb = RhumbAux(exact=True) # WGS84 default 

511 printf('# %r\n', rhumb) 

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

513 _re('# JFK NE lat2=%.8f, lon2=%.8f, S12=%.1f', (r.lat2, r.lon2, r.S12), (71.688899882813, 0.2555198244234, 44095641862956.11)) 

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

515 _re('# JFK-LHR azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (77.7683897102557, 5771083.38332803, 37395209100030.39)) 

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

517 _re('# JFK-NRT azi12=%.8f, s12=%.3f S12=%.1f', (r.azi12, r.s12, r.S12), (-92.38888798169965, 12782581.067684170, -63760642939072.50)) 

518 

519# % python3 -m pygeodesy.rhumbaux 

520 

521# RhumbAux(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) 

522 

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

524# JFK-LHR azi12=77.76838971, s12=5771083.383 S12=37395209100030.3 rel errors: 3e-12, 5e-15, 6e-16 

525# JFK-NRT azi12=-92.38888798, s12=12782581.068 S12=-63760642939072.5 rel errors: 2e-16, 3e-16, 6e-16