Coverage for pygeodesy/rhumbaux.py: 95%

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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+} renamed as L{RhumbAux} respectively L{RhumbLineAux}. 

7 

8Class L{RhumbLineAux} 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 from an other point. 

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, needed only for I{exact} area calculations C{S12} in L{RhumbAux} 

23 and L{RhumbLineAux}. 

24''' 

25# make sure int/int division yields float quotient 

26from __future__ import division as _; del _ # PYCHOK semicolon 

27 

28from pygeodesy.auxilats.auxAngle import AuxMu, AuxPhi, hypot 

29from pygeodesy.auxilats.auxDLat import AuxDLat, _DClenshaw 

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

31from pygeodesy.auxilats.auxily import _Dlam, _Dp0Dpsi, _Ufloats 

32from pygeodesy.basics import copysign0, _reverange, _xkwds_get 

33from pygeodesy.constants import EPS_2, MANT_DIG, PI4, isinf, \ 

34 _0_0, _4_0, _720_0, _log2, _over 

35# from pygeodesy.datums import _WGS84 # from .karney 

36# from pygeodesy.errors import _xkwds_get # from .basics 

37from pygeodesy.karney import Caps, _polynomial, NN, _WGS84 

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

39# from pygeodesy.interns import NN # from .karney 

40from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

41# from pygeodesy.props import Property_RO # from .rhumbBase 

42from pygeodesy.rhumbBase import RhumbBase, RhumbLineBase, Property_RO 

43 

44from math import ceil as _ceil, fabs, radians 

45 

46__all__ = _ALL_LAZY.rhumbaux 

47__version__ = '23.10.27' 

48 

49# DIGITS = (sizeof(real) * 8) bits 

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

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 

59 

60class RhumbAux(RhumbBase): 

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

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

63 

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

65 installed, version 1.16 or later. 

66 ''' 

67 

68 def __init__(self, a_earth=_WGS84, f=None, exact=True, name=NN, **TMorder): # PYCHOK signature 

69 '''New C{rhumbaux.RhumbAux}. 

70 

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

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

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

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

75 C{scalar}, ignored otherwise. 

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

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

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

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

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

81 C{TMorder}. 

82 

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

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

85 

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

87 ''' 

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

89 if TMorder: 

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

91 

92 def areaux(self, **exact): 

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

94 

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

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

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

98 

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

100 

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

102 property C{f_max}. 

103 

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

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

106 

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

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

109 ''' 

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

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

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

113 return a 

114 

115 @Property_RO 

116 def _auxD(self): 

117 return AuxDLat(self.ellipsoid) 

118 

119 @Property_RO 

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

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

122 

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

124 xD = self._auxD 

125 r = xD.DRectifying(Phi1, Phi2) if self.exact else \ 

126 xD.CRectifying(Chi1, Chi2) 

127 if r: 

128 r = _over(r, xD.DIsometric(Phi1, Phi2) if self.exact else 

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

130 return r 

131 

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

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

134 ''' 

135 psi1, Chi1, Phi1 = self._psiChiPhi3(r.lat1) 

136 psi2, Chi2, Phi2 = self._psiChiPhi3(r.lat2) 

137 psi12 = psi2 - psi1 # radians 

138 lam12 = radians(lon12) 

139 if (outmask & Caps.DISTANCE): 

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

141 d = Phi2.toMu(self).toRadians 

142 d -= Phi1.toMu(self).toRadians 

143 s = fabs(d) 

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

145 s = hypot(lam12, psi12) 

146 if s: 

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

148 s *= self._rrm 

149 a = _over(s, self._mpd) 

150 r.set_(a12=copysign0(a, s), s12=s) 

151 return lam12, psi12, Chi1, Chi2 

152 

153 def _latPhi2(self, mu): 

154 Mu = AuxMu.fromDegrees(mu) 

155 Phi = Mu.toPhi(self) 

156 return Phi.toDegrees, Phi 

157 

158 @Property_RO 

159 def _mpd(self): # meter per degree 

160 return radians(self._rrm) # == self.ellipsoid._Lpd 

161 

162 def _psiChiPhi3(self, lat): 

163 Phi = AuxPhi.fromDegrees(lat) 

164 Chi = Phi.toChi(self) 

165 psi = Chi.toLambertianRadians 

166 return psi, Chi, Phi 

167 

168 @Property_RO 

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

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

171 _RAseries(self._auxD)) 

172 

173# _RhumbLine = RhumbLineAux # see further below 

174 

175 @Property_RO 

176 def _rrm(self): 

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

178 

179 _mpr = _rrm # meter per radian, see _mpd 

180 

181 def _S12d(self, Chix, Chiy, lon12): # degrees 

182 '''(INTERNAL) Compute the area C{S12} from C{._meanSinXi(Chix, Chiy) * .c2 * lon12}. 

183 ''' 

184 pP, xD = self._RA, self._auxD 

185 

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

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

188 

189 dD = xD.DParametric(Phix, Phiy) if self.exact else \ 

190 xD.CParametric(Chix, Chiy) 

191 if dD: 

192 dD = _over(dD, xD.DIsometric(Phix, Phiy) if self.exact else 

193 _Dlam(tx, ty)) # not Lambertian! 

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

195 Phiy.toBeta(self).normalized, 

196 pP, min(len(pP), xD.ALorder)) # Fsum 

197 dD += _Dp0Dpsi(tx, ty) 

198 dD *= self._c2 * lon12 

199 return float(dD) 

200 

201 

202class RhumbLineAux(RhumbLineBase): 

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

204 

205 Class C{RhumbLineAux} facilitates the determination of points 

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

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

208 

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

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

211 area C{S12} under the rhumb line. 

212 

213 Method C{RhumbLineAux.intersection2} finds the intersection 

214 between two rhumb lines. 

215 

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

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

218 ''' 

219 _Rhumb = RhumbAux # rhumbaux.RhumbAux 

220 

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

222 '''New C{rhumbaux.RhumbLineAux}. 

223 

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

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

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

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

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

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

230 values specifying the required capabilities. Include 

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

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

233 ''' 

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

235 

236 @Property_RO 

237 def _Chi1(self): 

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

239 

240 @Property_RO 

241 def _mu1(self): 

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

243 ''' 

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

245 

246 def _mu2lat(self, mu): 

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

248 ''' 

249 lat, _ = self.rhumb._latPhi2(mu) 

250 return lat 

251 

252 @Property_RO 

253 def _Phi1(self): 

254 return AuxPhi.fromDegrees(self.lat1) 

255 

256 def _Position4(self, a12, mu2, *unused): # PYCHOK s12, mu2 

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

258 ''' 

259 R = self.rhumb 

260 lat2, Phi2 = R._latPhi2(mu2) 

261 Chi2 = Phi2.toChi(R) 

262 Chi1 = self._Chi1 

263 lon2 = self._salp * a12 

264 if lon2: 

265 m = R._DMu_DPsi(self._Phi1, Phi2, Chi1, Chi2) 

266 lon2 = _over(lon2, m) 

267 return lat2, lon2, Chi1, Chi2 

268 

269# @Property_RO 

270# def _psi1(self): 

271# return self._Chi1.toLambertianRadians 

272 

273RhumbAux._RhumbLine = RhumbLineAux # PYCHOK see RhumbBase._RhumbLine 

274 

275 

276def _RAintegrate(auxD): 

277 # Compute coefficients by Fourier transform of integrand 

278 L = 2 

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

280 f = auxD._qIntegrand 

281 c_ = fft.transform(f) 

282 pP = [] 

283 _P = pP.append 

284 # assert L < _Lbits 

285 while L < _Lbits: 

286 L = fft.reset(L) * 2 

287 c_ = fft.refine(f, c_, _0_0) # sentine[L] 

288 # assert len(c_) == L + 1 

289 pP[:], k = [], -1 

290 for j in range(1, L + 1): 

291 # Compute Fourier coefficients of integral 

292 p = (c_[j - 1] + c_[j]) / (_4_0 * j) 

293 if fabs(p) > EPS_2: 

294 k = -1 # run interrupted 

295 else: 

296 if k < 0: 

297 k = 1 # mark as first small value 

298 if (j - k) >= ((j + 7) // 8): 

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

300 return pP[:j] # break while L loop 

301 _P(-p) 

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

303 

304 

305def _RAseries(auxD): 

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

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

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

309 d = n = auxD._n 

310 i = 0 

311 aL = auxD.ALorder 

312 Cs = _RACoeffs[aL] 

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

314 pP = [] 

315 _P = pP.append 

316 _p = _polynomial 

317 for m in _reverange(aL): # order 

318 j = i + m + 1 

319 _P(_p(n, Cs, i, j) * d) 

320 d *= n 

321 i = j 

322 # assert i == len(pP) 

323 return pP 

324 

325 

326_f, _u = float, _Ufloats() 

327_RACoeffs = { # Rhumb Area Coefficients in matrix Q 

328 4: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4 

329 596 / _f(2025), -398 / _f(945), 22 / _f(45), -1 / _f(3), 

330 1543 / _f(4725), -118 / _f(315), 1 / _f(5), 

331 152 / _f(945), -17 / _f(315), 

332 5 / _f(252)), 

333 5: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5 

334 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45), 

335 -1 / _f(3), 

336 -24562 / _f(155925), 1543 / _f(4725), -118 / _f(315), 1 / _f(5), 

337 -38068 / _f(155925), 152 / _f(945), -17 / _f(315), 

338 -752 / _f(10395), 5 / _f(252), 

339 -101 / _f(17325)), 

340 6: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6 

341 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025), 

342 -398 / _f(945), 22 / _f(45), -1 / _f(3), 

343 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725), 

344 -118 / _f(315), 1 / _f(5), 

345 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945), 

346 -17 / _f(315), 

347 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252), 

348 62464 / _f(2027025), -101 / _f(17325), 

349 11537 / _f(4054050)), 

350 7: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7 

351 -565017322 / _f(1915538625), 138734126 / _f(638512875), 

352 -102614 / _f(467775), 596 / _f(2025), -398 / _f(945), 22 / _f(45), 

353 -1 / _f(3), 

354 -1969276 / _f(58046625), 17749373 / _f(425675250), -24562 / _f(155925), 

355 1543 / _f(4725), -118 / _f(315), 1 / _f(5), 

356 -58573784 / _f(638512875), 1882432 / _f(8513505), -38068 / _f(155925), 

357 152 / _f(945), -17 / _f(315), 

358 -6975184 / _f(42567525), 268864 / _f(2027025), -752 / _f(10395), 

359 5 / _f(252), 

360 -112832 / _f(1447875), 62464 / _f(2027025), -101 / _f(17325), 

361 -4096 / _f(289575), 11537 / _f(4054050), 

362 -311 / _f(525525)), 

363 8: _u( # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8 

364 188270561816 / _f(488462349375), -565017322 / _f(1915538625), 

365 138734126 / _f(638512875), -102614 / _f(467775), 596 / _f(2025), 

366 -398 / _f(945), 22 / _f(45), -1 / _f(3), 

367 2332829602 / _f(23260111875), -1969276 / _f(58046625), 

368 17749373 / _f(425675250), -24562 / _f(155925), 1543 / _f(4725), 

369 -118 / _f(315), 1 / _f(5), 

370 -41570288 / _f(930404475), -58573784 / _f(638512875), 

371 1882432 / _f(8513505), -38068 / _f(155925), 152 / _f(945), 

372 -17 / _f(315), 

373 1538774036 / _f(10854718875), -6975184 / _f(42567525), 

374 268864 / _f(2027025), -752 / _f(10395), 5 / _f(252), 

375 436821248 / _f(3618239625), -112832 / _f(1447875), 

376 62464 / _f(2027025), -101 / _f(17325), 

377 3059776 / _f(80405325), -4096 / _f(289575), 11537 / _f(4054050), 

378 4193792 / _f(723647925), -311 / _f(525525), 

379 1097653 / _f(1929727800)) 

380} 

381del _f, _u, _Ufloats 

382 

383__all__ += _ALL_DOCS(Caps, RhumbAux, RhumbLineAux) 

384 

385if __name__ == '__main__': 

386 

387 from pygeodesy import printf, RhumbAux # PYCHOK RhumbAux is __main__.RhumbAux 

388 from pygeodesy.basics import _zip 

389 

390 def _re(fmt, r3, x3): 

391 e3 = [] 

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

393 e = fabs(r - x) / fabs(x) 

394 e3.append('%.g' % (e,)) 

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

396 

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

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

399 printf('# %r\n', rhumb) 

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

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

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

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

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

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

406 

407# % python3 -m pygeodesy.rhumbaux 

408 

409# 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) 

410 

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

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

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