1# -*- coding: utf-8 -*-

2

3 u'''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

6 I{GeographicLib version 2.2+}.

7

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

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

10 geodesic or perpendicular rhumb line.

11

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

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

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

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

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

17

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

19 License. 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

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

26

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

28 from pygeodesy.auxilats.auxDLat import AuxDLat, _DClenshaw

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

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

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

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

33 _0_0, _4_0, _720_0, _log2, _over

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

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

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

37 from pygeodesy.interns import NN, _COMMASPACE_

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

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

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

41 pairs, Property, Property_RO

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

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

44

45 from math import ceil as _ceil, fabs, radians

46

47__all__ = _ALL_LAZY.rhumbaux

48__version__ = '23.08.06'

49

50# DIGITS = (sizeof(real) * 8) bits # assert DIGITS > MANT_DIG

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

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

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

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

55#

56# 64 = DIGITS for long double, 6 = 12 - _log2(64)

57_Lbits = 1 << (int(_ceil(_log2(max(MANT_DIG, 64)))) + 6)

58_rls = [] # instances of C{RbumbLineAux} to be updated

59

60

61 class RhumbAux(RhumbBase):

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

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

64

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

66 installed, version 1.16 or later.

67 '''

68

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

70 '''New C{rhumbaux.RhumbAux}.

71

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

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

74 (equatorial) radius (C{scalar}).

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

76 a C{scalar}, ignored otherwise.

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

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

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

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

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

82 C{TMorder}.

83

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

85 required when C{B{exact} is True}.

86

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

88 '''

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

90 if TMorder:

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

92

93 def areaux(self, **exact):

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

95

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

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

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

99

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

101

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

103 property C{f_max}.

104

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

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

107

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

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

110 '''

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

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

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

114 return a

115

116 @Property_RO

117 def _auxD(self):

118 return AuxDLat(self.ellipsoid)

119

120 @Property_RO

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

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

123

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

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

126

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

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

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

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

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

132

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

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

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

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

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

138

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

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

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

142

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

144 taken to be C{sqrt(EPS)}. This position is extremely close

145 to the actual pole and allows the calculation to be carried

146 out in finite terms.

147 '''

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

149 name=self.name)

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

151

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

153 xD = self._auxD

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

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

156 _over(xD.DRectifyingZ(Chi1, Chi2),

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

158

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

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

161

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

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

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

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

166

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

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

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

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

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

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

173

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

175 on opposite meridians, there are two shortest rhumb lines

176 and the East-going one is chosen.

177

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

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

180 This position is extremely close to the actual pole and

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

182 '''

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

184 if (outmask & Cs.AZIMUTH_DISTANCE_AREA):

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

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

187

188 psi12 = psi2 - psi1

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

190 lam12 = radians(lon12)

191 if (outmask & Cs.AZIMUTH):

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

193 if (outmask & Cs.DISTANCE):

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

195 d = Phi2.toMu(self).toRadians

196 d -= Phi1.toMu(self).toRadians

197 s = fabs(d)

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

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

200 s *= hypot(lam12, psi12)

201 r.set_(s12=self._rrm * s)

202 if (outmask & Cs.AREA):

203 S = self._c2SinXi(Chi1, Chi2)

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

205 return r

206

207 def _c2SinXi(self, Chix, Chiy):

208 pP, xD = self._RA, self._auxD

209

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

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

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

213 Phiy.toBeta(self).normalized,

214 pP, min(len(pP), 8)) # Fsum

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

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

217 _over(xD.DParametricZ(Chix, Chiy), _Dlam(tx, ty)) # not Lambertian!

218 dD += _Dp0Dpsi(tx, ty)

219 dD *= self._c2

220 return float(dD)

221

222 def _psiChiPhi3(self, lat):

223 Phi = AuxPhi.fromDegrees(lat)

224 Chi = Phi.toChi(self)

225 psi = Chi.toLambertianRadians

226 return psi, Chi, Phi

227

228 @Property_RO

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

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

231 _RAseries(self._auxD))

232

233 @Property_RO

234 def _RhumbLine(self):

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

236 '''

237 return RhumbLineAux

238

239 @Property_RO

240 def _rrm(self):

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

242

243 @Property

244 def TMorder(self):

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

246 '''

247 return self._mTM

248

249 @TMorder.setter # PYCHOK setter!

250 def TMorder(self, order):

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

252 '''

253 self._TMorder(order)

254

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

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

257

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

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

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

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

262

263 @return: Tuple items (C{str}).

264 '''

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

266 TMorder=self.TMorder)

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

268

269

270 class RhumbLineAux(RhumbLineBase):

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

272

273 Class C{RhumbLineAux} facilitates the determination of points

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

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

276

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

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

279 area C{S12} under the rhumb line.

280

281 Method C{RhumbLineAux.intersection2} finds the intersection

282 between two rhumb lines.

283

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

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

286 '''

287 _Rhumb = RhumbAux # rhumbaux.RhumbAux

288

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

290 '''New C{rhumbaux.RhumbLineAux}.

291

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

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

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

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

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

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

298 values specifying the required capabilities. Include

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

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

301 '''

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

303

304 @Property_RO

305 def _Chi1(self):

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

307

308 @Property_RO

309 def _mu1(self):

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

311

312 @Property_RO

313 def _Phi1(self):

314 return AuxPhi.fromDegrees(self.lat1)

315

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

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

318

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

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

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

322 the quantities to be returned.

323

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

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

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

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

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

329

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

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

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

333

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

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

336 This position is extremely close to the actual pole and

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

338 '''

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

340 if (outmask & Cs.LATITUDE_LONGITUDE_AREA):

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

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

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

344 Mu2 = AuxMu.fromDegrees(mu2)

345 Phi2 = Mu2.toPhi(R)

346 lat2 = Phi2.toDegrees

347 if x90: # PYCHOK no cover

348 lon2 = NAN

349 if (outmask & Cs.AREA):

350 r.set_(S12=NAN)

351 else:

352 Chi2 = Phi2.toChi(R)

353 Chi1 = self._Chi1

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

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

356 if (outmask & Cs.AREA):

357 S = R._c2SinXi(Chi1, Chi2)

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

359 if (outmask & Cs.LONGITUDE):

360 if (outmask & Cs.LONG_UNROLL):

361 lon2 += self.lon1

362 else:

363 lon2 = _norm180(self._lon12 + lon2)

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

365 if (outmask & Cs.LATITUDE):

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

367 if (outmask & Cs.LONGITUDE):

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

369 return r

370

371# @Property_RO

372# def _psi1(self):

373# return self._Chi1.toLambertianRadians

374

375

376 def _RAintegrate(auxD):

377 # Compute coefficients by Fourier transform of integrand

378 L = 2

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

380 f = auxD._qIntegrand

381 c = fft.transform(f)

382 # assert L < _Lbits

383 while L < _Lbits:

384 fft.reset(L)

385 c = fft.refine(f, c)

386 L *= 2 # == len(c)

387 # assert len(c) == L

388 pP, k = [], -1

389 for j in range(1, L + 1):

390 # Compute Fourier coefficients of integral

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

392 if fabs(p) > EPS_2:

393 k = -1 # run interrupted

394 else:

395 if k < 0:

396 k = 1 # mark as first small value

397 if (j - k) >= ((j + 7) // 8):

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

399 return pP[:j]

400 pP.append(p)

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

402

403

404 def _RAseries(auxD):

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

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

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

408 d = n = auxD._n

409 i = 0

410 pP = []

411 aL = auxD.ALorder

412 Cs = _RACoeffs[aL]

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

414 _p = _MODS.karney._polynomial

415 for m in _reverange(aL): # order

416 j = i + m + 1

417 pP.append(_p(n, Cs, i, j) * d)

418 d *= n

419 i = j

420 # assert i == len(pP)

421 return pP

422

423

424_f, _u = float, _Ufloats()

425_RACoeffs = { # Rhumb Area Coefficients in matrix Q

426 4: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4

427 596 / _f(2025), -398 / _f(945), 22 / _f(45), -1 / _f(3),

428 1543 / _f(4725), -118 / _f(315), 1 / _f(5),

429 152 / _f(945), -17 / _f(315),

430 5 / _f(252)),

431 5: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5

432 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45),

433 -1 / _f(3),

434 -24562 / _f(155925), 1543 / _f(4725), -118 / _f(315), 1 / _f(5),

435 -38068 / _f(155925), 152 / _f(945), -17 / _f(315),

436 -752 / _f(10395), 5 / _f(252),

437 -101 / _f(17325)),

438 6: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6

439 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025),

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

441 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725),

442 -118 / _f(315), 1 / _f(5),

443 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945),

444 -17 / _f(315),

445 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252),

446 62464 / _f(2027025), -101 / _f(17325),

447 11537 / _f(4054050)),

448 7: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7

449 -565017322 / _f(1915538625), 138734126 / _f(638512875),

450 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45),

451 -1 / _f(3),

452 -1969276 / _f(58046625), 17749373 / _f(425675250), -24562 / _f(155925),

453 1543 / _f(4725), -118 / _f(315), 1 / _f(5),

454 -58573784 / _f(638512875), 1882432 / _f(8513505), -38068 / _f(155925),

455 152 / _f(945), -17 / _f(315),

456 -6975184 / _f(42567525), 268864 / _f(2027025), -752 / _f(10395),

457 5 / _f(252),

458 -112832 / _f(1447875), 62464 / _f(2027025), -101 / _f(17325),

459 -4096 / _f(289575), 11537 / _f(4054050),

460 -311 / _f(525525)),

461 8: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8

462 188270561816 / _f(488462349375), -565017322 / _f(1915538625),

463 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025),

464 -398 / _f(945), 22 / _f(45), -1 / _f(3),

465 2332829602 / _f(23260111875), -1969276 / _f(58046625),

466 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725),

467 -118 / _f(315), 1 / _f(5),

468 -41570288 / _f(930404475), -58573784 / _f(638512875),

469 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945),

470 -17 / _f(315),

471 1538774036 / _f(10854718875), -6975184 / _f(42567525),

472 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252),

473 436821248 / _f(3618239625), -112832 / _f(1447875),

474 62464 / _f(2027025), -101 / _f(17325),

475 3059776 / _f(80405325), -4096 / _f(289575), 11537 / _f(4054050),

476 4193792 / _f(723647925), -311 / _f(525525),

477 1097653 / _f(1929727800))

478}

479 del _f, _u, _Ufloats

480

481

482__all__ += _ALL_DOCS(Caps, RhumbAux, RhumbLineAux)

483

484 if __name__ == '__main__':

485

486 from pygeodesy.lazily import printf

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

488

489 def _re(fmt, r3, x3):

490 e3 = []

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

492 e = fabs(r - x) / fabs(x)

493 e3.append('%.g' % (e,))

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

495

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

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

498 printf('# %r\n', rhumb)

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

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

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

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

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

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

505

506# % python3 -m pygeodesy.rhumbaux

507

508# 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)

509

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

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

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