1

2# -*- coding: utf-8 -*-

3

4 u'''A pure Python version of I{Karney}'s I{elliptic functions}, I{Krüger series expansion}, C++

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

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

7 from I{GeographicLib version 2.0}, kept for backward compatibility.

8

9 Class L{RhumbLine} has been enhanced with methods C{Intersecant2}, C{Intersection} and C{PlumbTo} to

10 iteratively find the intersection of a rhumb line and a circle or an other rhumb line, respectively

11 a perpendicular geodesic or other rhumb line.

12

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

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

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

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

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

18

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

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

21'''

22# make sure int/int division yields float quotient

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

24

25 from pygeodesy.basics import copysign0, neg

26 from pygeodesy.constants import PI_2, _0_0s, _0_0, _0_5, _1_0, \

27 _2_0, _4_0, _720_0, _over, _1_over

28 from pygeodesy.datums import _WGS84

29# from pygeodesy.deprecated import RhumbOrder2Tuple # _MODS

30 from pygeodesy.errors import RhumbError, _Xorder

31 from pygeodesy.fmath import hypot, hypot1

32# from pygeodesy.fsums import fsum1f_ # _MODS

33# from pygeodesy.interns import NN # from .farney

34 from pygeodesy.karney import Caps, NN

35 from pygeodesy.ktm import KTransverseMercator, _Xs, \

36 _AlpCoeffs, _BetCoeffs # PYCHOK used!

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

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

39 from pygeodesy.rhumb.bases import RhumbBase, RhumbLineBase, _update_all_rls

40 from pygeodesy.utily import atan1, sincos2_

41

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

43

44__all__ = _ALL_LAZY.rhumb_ekx

45__version__ = '23.12.03'

46

47

48 class Rhumb(RhumbBase):

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

50 I{elliptic functions} or I{Krüger series expansion}

51

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

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

54 '''

55 _mRA = 6 # see .RAorder

56

57 def __init__(self, a_earth=_WGS84, f=None, exact=True, name=NN, **RA_TMorder):

58 '''New C{Rhumb}.

59

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

61 L{Ellipsoid2}, L{a_f2Tuple}, 2-tuple C{(a, f)}) or

62 the (equatorial) radius (C{meter}, conventionally).

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

64 C{scalar}, ignored otherwise.

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

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

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

68 C{exact} and C{TMorder}.

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

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

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

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

73

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

75 '''

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

77 if RA_TMorder:

78 self.orders(**RA_TMorder)

79

80 @Property_RO

81 def _A2(self): # Conformal2RectifyingCoeffs

82 m = self.TMorder

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

84

85 @Property_RO

86 def _B2(self): # Rectifying2ConformalCoeffs

87 m = self.TMorder

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

89

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

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

92

93 @deprecated_method

94 def Direct7(self, lat1, lon1, azi12, s12, outmask=Caps.LATITUDE_LONGITUDE_AREA): # PYCHOK no cover

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

96

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

98 '''

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

100

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

102 E = self.ellipsoid

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

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

105

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

107 if self.exact:

108 E = self.ellipsoid

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

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

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

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

113 else:

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

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

116 return r

117

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

119 E = self.ellipsoid

120 tbetx = E.f1 * tphix

121 tbety = E.f1 * tphiy

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

123 * _DfEt( tbetx, tbety, self._eF)

124 * _Datan(tbetx, tbety)) / E.L

125

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

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

128

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

130 E = self.ellipsoid

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

132 if self.exact:

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

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

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

136 else:

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

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

139 return r

140

141 @Property_RO

142 def _eF(self):

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

144 '''

145 # .k2 = 0.006739496742276434

146 return self.ellipsoid._elliptic_e12 # _MODS.elliptic.Elliptic(-self.ellipsoid._e12)

147

148 def _Inverse4(self, lon12, r, outmask):

149 '''(INTERNAL) See method C{RhumbBase.Inverse}.

150 '''

151 E, Cs = self.ellipsoid, Caps

152 psi1 = E.auxIsometric(r.lat1)

153 psi2 = E.auxIsometric(r.lat2)

154 psi12 = psi2 - psi1 # degrees

155 if (outmask & Cs.DISTANCE):

156 a = s = hypot(lon12, psi12)

157 if a:

158 a *= self._DIsometric2Rectifyingd(psi2, psi1)

159 s = self._mpd * a # == E._Lpd

160 a = copysign0(a, s)

161 r.set_(a12=a, s12=s)

162

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

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

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

166 lon12=lon12, psi1=psi1, exact=self.exact,

167 psi12=psi12, psi2=psi2)

168 return lon12, psi12, psi1, psi2

169

170 @deprecated_method

171 def Inverse7(self, lat1, lon1, azi12, s12, outmask=Caps.AZIMUTH_DISTANCE_AREA): # PYCHOK no cover

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

173

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

175 '''

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

177

178 @Property_RO

179 def _mpd(self): # meter per degree

180 return self.ellipsoid._Lpd

181

182 @Property_RO

183 def _mpr(self): # meter per radian

184 return self.ellipsoid._Lpr # degrees(._Lpd)

185

186 @deprecated_method

187 def orders(self, RAorder=None, TMorder=None): # PYCHOK no cover

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

189

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

191

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

193 or 8).

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

195 5, 6, 7 or 8).

196

197 @return: DEPRECATED L{RhumbOrder2Tuple}C{(RAorder, TMorder)}

198 with the previous C{RAorder} and C{TMorder} setting.

199 '''

200 t = _MODS.deprecated.classes.RhumbOrder2Tuple(self.RAorder, self.TMorder)

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

202 self.RAorder = RAorder

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

204 self.TMorder = TMorder

205 return t

206

207 @Property_RO

208 def _RA2(self):

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

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

211 m = self.RAorder

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

213

214 @Property

215 def RAorder(self):

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

217 '''

218 return self._mRA

219

220 @RAorder.setter # PYCHOK setter!

221 def RAorder(self, order):

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

223 '''

224 m = _Xorder(_RACoeffs, RhumbError, RAorder=order)

225 if self._mRA != m:

226 _update_all_rls(self)

227 self._mRA = m

228

229# _RhumbLine = RhumbLine # see further below

230

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

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

233 '''

234 S = (self.ellipsoid.areax if self.exact else

235 self.ellipsoid.area) * lon12 / _720_0

236 if S:

237 x, y = radians(psi1), radians(psi2) # _meanSinXi(x, y)

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

239 if self.f:

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

241 S *= s

242 return S

243

244

245 class RhumbLine(RhumbLineBase):

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

247

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

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

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

251 '''

252 _Rhumb = Rhumb # rhumb.ekx.Rhumb

253

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

255 '''New C{RhumbLine}.

256

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

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

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

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

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

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

263 values specifying the required capabilities. Include

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

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

266 '''

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

268

269 @Property_RO

270 def _dpm12(self): # PYCHOK no cover

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

272 '''

273 r = self._salp

274 if r:

275 r = _over(r, self.ellipsoid.circle4(self.lat1).radius)

276 return r

277

278 @Property_RO

279 def _mu1(self):

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

281 '''

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

283

284 def _mu2lat(self, mu):

285 '''(INTERNAL) Get the inverse I{rectifying auxiliary} latitude (C{degrees}).

286 '''

287 return self.ellipsoid.auxRectifying(mu, inverse=True)

288

289 def _Position4(self, unused, mu2, s12, mu12):

290 '''(INTERNAL) See method C{RhumbLineBase._Position}.

291 '''

292 psi1 = psi2 = self._psi1

293 if mu12: # self._calp != 0

294 lat2 = self._mu2lat(mu2)

295 psi12 = self.rhumb._DRectifying2Isometricd(mu2, self._mu1) * mu12

296 lon2 = self._talp * psi12

297 psi2 += psi12

298 else: # meridional

299 lat2 = self.lat1

300 lon2 = self._dpm12 * s12

301 return lat2, lon2, psi1, psi2

302

303 @Property_RO

304 def _psi1(self):

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

306 '''

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

308

309 @property_RO

310 def RAorder(self):

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

312 '''

313 return self.rhumb.RAorder

314

315 Rhumb._RhumbLine = RhumbLine # PYCHOK see RhumbBase._RhumbLine

316

317

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

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

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

321# 7-28 (1999). U{ACM<https://DL.ACM.org/doi/pdf/10.1145/334714.334716> and @see

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

323

324 def _Dasinh(x, y):

325 hx = hypot1(x)

326 d = x - y

327 if d:

328 hx *= y

329 hy = x * hypot1(y)

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

331 r = asinh(t) / d

332 else:

333 r = _1_0 / hx

334 return r

335

336

337 def _Datan(x, y):

338 xy = x * y

339 r = xy + _1_0

340 d = x - y

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

342 r = (atan1(d, r) if xy > -r else (atan1(x) - atan1(y))) / d

343 else:

344 r = _1_over(r)

345 return r

346

347

348 def _Dcosh(x, y):

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

350

351

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

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

354 e = _1_0 - E.e2 * x * y

355 if e: # assert not isnear0(e)

356 d = x - y

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

358 return e

359

360

361 def _DfEt(tx, ty, eF): # tangents

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

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

364 d = x - y

365 if (x * y) > 0:

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

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

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

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

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

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

372 # = t = d * Dt

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

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

375 # holds if x*y > 0):

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

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

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

379 sx, cx, sy, cy = sincos2_(x, y)

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

381 eF.fDelta(sx, cx) * sy)

382 D = (sx + sy) * _Dsin(x, y) / D

383 t = D * d

384 t2 = _1_0 + t**2

385 D *= _2_0 / t2

386 s = D * d

387 if s:

388 c = (t + _1_0) * (_1_0 - t) / t2

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

390 r = D * (r - eF.k2 * sx * sy)

391 elif d:

392 r = (eF.fE(x) - eF.fE(y)) / d

393 return r

394

395

396 def _Dgd(x, y):

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

398

399

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

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

402

403

404 def _Dlog(x, y):

405 d = (x - y) * _0_5

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

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

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

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

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

411 return (asinh(d / sqrt(x * y)) / d) if d else _1_over(x)

412

413

414 def _Dsin(x, y):

415 return _Dsincos(x, y, sin, cos)

416

417

418 def _Dsincos(x, y, sin_, cos_):

419 r = cos_((x + y) * _0_5)

420 d = (x - y) * _0_5

421 if d:

422 r *= sin_(d) / d

423 return r

424

425

426 def _Dsinh(x, y):

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

428

429

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

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

432

433

434 def _Dtant(dxy, tx, ty):

435 txy = tx * ty

436 r = txy + _1_0

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

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

439 return r

440

441

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

443 # get inverse auxiliary lats in radians and tangents

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

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

446 return phix, phiy, tan(phix), tan(phiy)

447

448

449 def _gd(x):

450 return atan(sinh(x))

451

452

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

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

455 # C[0] is ignored and n >= 0

456 # Use Clenshaw summation to evaluate

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

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

459 # where

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

461 # SC = sinp ? sin : cos

462 # CS = sinp ? cos : sin

463 # ...

464 d, _neg = (x - y), neg

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

466 sd = (sd / d) if d else _1_0

467 s = _neg(sp * sd) # negative

468 # 2x2 matrices in row-major order

469 a1 = s * d**2

470 a2 = s * _4_0

471 a0 = a3 = _2_0 * cp * cd # m

472 b2 = b1 = _0_0s(4)

473 if n > 0:

474 b1 = C[n], _0_0, _0_0, C[n]

475

476 _fsum = _MODS.fsums.fsum1f_

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

478 b1, b2, Cj = b2, b1, C[j]

479 # b1 = a * b2 - b1 + C[j] * I

480 m0, m1, m2, m3 = b2

481 n0, n1, n2, n3 = map(_neg, b1)

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

483 _fsum(a0 * m1, a1 * m3, n1),

484 _fsum(a2 * m0, a3 * m2, n2),

485 _fsum(a2 * m1, a3 * m3, n3, Cj))

486 # Here are the full expressions for m and s

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

488 # (1, 0, cd * cp, -2 * sd * sp)

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

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

491 cd *= b1[2]

492 sd *= b1[3] * _2_0

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

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

495 return s

496

497

498_RACoeffs = { # Generated by Maxima on 2015-05-15 08:24:04-04:00

499 4: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4

500 691, 7860, -20160, 18900, 0, 56700, # R[0]/n^0, polynomial(n), order 4

501 1772, -5340, 6930, -4725, 14175, # R[1]/n^1, polynomial(n), order 3

502 -1747, 1590, -630, 4725, # PYCHOK R[2]/n^2, polynomial(n), order 2

503 104, -31, 315, # R[3]/n^3, polynomial(n), order 1

504 -41, 420), # PYCHOK R[4]/n^4, polynomial(n), order 0, count = 20

505 5: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5

506 -79036, 22803, 259380, -665280, 623700, 0, 1871100, # PYCHOK R[0]/n^0, polynomial(n), order 5

507 41662, 58476, -176220, 228690, -155925, 467775, # PYCHOK R[1]/n^1, polynomial(n), order 4

508 18118, -57651, 52470, -20790, 155925, # PYCHOK R[2]/n^2, polynomial(n), order 3

509 -23011, 17160, -5115, 51975, # PYCHOK R[3]/n^3, polynomial(n), order 2

510 5480, -1353, 13860, # PYCHOK R[4]/n^4, polynomial(n), order 1

511 -668, 5775), # PYCHOK R[5]/n^5, polynomial(n), order 0, count = 27

512 6: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6

513 128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0, 2554051500, # R[0]/n^0, polynomial(n), order 6

514 -114456994, 56868630, 79819740, -240540300, 312161850, -212837625, 638512875, # PYCHOK R[1]/n^1, polynomial(n), order 5

515 51304574, 24731070, -78693615, 71621550, -28378350, 212837625, # R[2]/n^2, polynomial(n), order 4

516 1554472, -6282003, 4684680, -1396395, 14189175, # R[3]/n^3, polynomial(n), order 3

517 -4913956, 3205800, -791505, 8108100, # PYCHOK R[4]/n^4, polynomial(n), order 2

518 1092376, -234468, 2027025, # R[5]/n^5, polynomial(n), order 1

519 -313076, 2027025), # PYCHOK R[6]/n^6, polynomial(n), order 0, count = 35

520 7: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7

521 -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600, 2554051500, 0, 7662154500, # PYCHOK R[0]/n^0, polynomial(n), order 7

522 258618446, -343370982, 170605890, 239459220, -721620900, 936485550, -638512875, 1915538625, # PYCHOK R[1]/n^1, polynomial(n), order 6

523 -248174686, 153913722, 74193210, -236080845, 214864650, -85135050, 638512875, # PYCHOK R[2]/n^2, polynomial(n), order 5

524 114450437, 23317080, -94230045, 70270200, -20945925, 212837625, # PYCHOK R[3]/n^3, polynomial(n), order 4

525 15445736, -103193076, 67321800, -16621605, 170270100, # PYCHOK R[4]/n^4, polynomial(n), order 3

526 -27766753, 16385640, -3517020, 30405375, # PYCHOK R[4]/n^4, polynomial(n), order 3

527 4892722, -939228, 6081075, # PYCHOK R[4]/n^4, polynomial(n), order 3

528 -3189007, 14189175), # PYCHOK R[7]/n^7, polynomial(n), order 0, count = 44

529 8: ( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8

530 71374704821, -161769749880, 196369790040, -165062734200, 47622925350, 541702161000, -1389404016000, 1302566265000, 0, 3907698795000, # R[0]/n^0, polynomial(n), order 8

531 -13691187484, 65947703730, -87559600410, 43504501950, 61062101100, -184013329500, 238803815250, -162820783125, 488462349375, # PYCHOK R[1]/n^1, polynomial(n), order 7

532 30802104839, -63284544930, 39247999110, 18919268550, -60200615475, 54790485750, -21709437750, 162820783125, # R[2]/n^2, polynomial(n), order 6

533 -8934064508, 5836972287, 1189171080, -4805732295, 3583780200, -1068242175, 10854718875, # PYCHOK R[3]/n^3, polynomial(n), order 5

534 50072287748, 3938662680, -26314234380, 17167059000, -4238509275, 43418875500, # R[4]/n^4, polynomial(n), order 4

535 359094172, -9912730821, 5849673480, -1255576140, 10854718875, # R[5]/n^5, polynomial(n), order 3

536 -16053944387, 8733508770, -1676521980, 10854718875, # PYCHOK R[6]/n^6, polynomial(n), order 2

537 930092876, -162639357, 723647925, # R[7]/n^7, polynomial(n), order 1

538 -673429061, 1929727800) # PYCHOK R[8]/n^8, polynomial(n), order 0, count = 54

539}

540

541__all__ += _ALL_DOCS(Caps)

542

543# **) MIT License

544#

545# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.

546#

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

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

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

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

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

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

553#

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

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

556#

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

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

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563# OTHER DEALINGS IN THE SOFTWARE.