Coverage for pygeodesy/rhumb/aux_.py: 96%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''A pure Python version of I{Karney}'s I{Auxiliary Latitudes}, 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 2.2+} renamed to L{RhumbAux} respectively L{RhumbLineAux}. 

8 

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

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

11a perpendicular geodesic or other rhumb line. 

12 

13For more details, see the U{GeographicLib<https://GeographicLib.SourceForge.io/C++/doc/index.html>} I{2.2} 

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>}, 

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

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

18 

19Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2022-2024) and licensed under the MIT/X11 

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

21 

22@note: C{S12} area calculations in classes L{RhumbAux} and L{RhumbLineAux} depend on class L{AuxDST} which 

23 requires U{numpy<https://PyPI.org/project/numpy>} to be installed, version 1.16 or newer. 

24 

25@note: Windows reserves file names U{AUX, COM[#], CON, LPT[#], NUL, PRN<https://learn.Microsoft.com/en-us/ 

26 windows/win32/fileio/naming-a-file#naming-conventions>} with and without extension. 

27''' 

28# make sure int/int division yields float quotient 

29from __future__ import division as _; del _ # PYCHOK semicolon 

30 

31from pygeodesy.auxilats.auxAngle import AuxMu, AuxPhi, hypot 

32from pygeodesy.auxilats.auxDLat import AuxDLat, _DClenshaw 

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

34from pygeodesy.auxilats.auxily import _Dlam, _Dp0Dpsi 

35from pygeodesy.auxilats._CX_Rs import _Rdict, _Rkey, _Rtuple 

36from pygeodesy.basics import copysign0, _reverange, _xkwds_get1 

37from pygeodesy.constants import EPS_2, MANT_DIG, PI4, isinf, \ 

38 _0_0, _4_0, _720_0, _log2, _over 

39# from pygeodesy.datums import _WGS84 # from .rhumb.bases 

40# from pygeodesy.errors import _xkwds_get1 # from .basics 

41from pygeodesy.karney import Caps, _polynomial 

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

43from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _ALL_MODS as _MODS 

44# from pygeodesy.props import Property_RO # from .rhumb.bases 

45from pygeodesy.rhumb.bases import RhumbBase, RhumbLineBase, \ 

46 Property_RO, _WGS84 

47 

48from math import ceil as _ceil, fabs, radians 

49 

50__all__ = _ALL_LAZY.rhumb_aux_ 

51__version__ = '25.01.15' 

52 

53# DIGITS = (sizeof(real) * 8) bits 

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

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

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

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

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

59# 

60# 64 = DIGITS for long double, 6 = 12 - _log2(64) 

61_Lbits = 1 << (int(_ceil(_log2(max(MANT_DIG, 64)))) + 6) 

62 

63 

64class RhumbAux(RhumbBase): 

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

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

67 

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

69 installed, version 1.16 or later. 

70 ''' 

71 

72 def __init__(self, a_earth=_WGS84, f=None, exact=True, **TMorder_name): # PYCHOK signature 

73 '''New C{RhumbAux}. 

74 

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

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

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

78 @kwarg f: The ellipsoid's flattening (C{scalar}), required if B{C{a_earth}} 

79 is C{scalar}, ignored otherwise. 

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

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

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

83 @kwarg TMorder_name: Optional C{B{name}=NN} (C{str}) and optional 

84 keyword argument C{B{TMorder}=6}, the order of the 

85 L{KTransverseMercator}, see property C{TMorder}. 

86 

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

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

89 

90 @raise RhumbError: Invalid B{C{a_earth}}, B{C{f}} or B{C{TMorder}}. 

91 ''' 

92 RhumbBase.__init__(self, a_earth, f, exact, TMorder_name) 

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_get1(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 _DMu_DPsi(self, Phi1, Phi2, Chi1, Chi2): 

126 xD = self._auxD 

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

128 xD.CRectifying(Chi1, Chi2) 

129 if r: 

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

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

132 return r 

133 

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

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

136 ''' 

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

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

139 psi12 = psi2 - psi1 # radians 

140 lam12 = radians(lon12) 

141 if (outmask & Caps.DISTANCE): 

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

143 s = fabs(Phi2.toMu(self).toRadians - 

144 Phi1.toMu(self).toRadians) 

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

146 s = hypot(lam12, psi12) 

147 if s: 

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

149 s *= self._rrm 

150 a = _over(s, self._mpd) 

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

152 return lam12, psi12, Chi1, Chi2 

153 

154 def _latPhi2(self, mu): 

155 Mu = AuxMu.fromDegrees(mu) 

156 Phi = Mu.toPhi(self) 

157 return Phi.toDegrees, Phi 

158 

159 @Property_RO 

160 def _mpd(self): # meter per degree 

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

162 

163 def _psiChiPhi3(self, lat): 

164 Phi = AuxPhi.fromDegrees(lat) 

165 Chi = Phi.toChi(self) 

166 psi = Chi.toLambertianRadians 

167 return psi, Chi, Phi 

168 

169 @Property_RO 

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

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

172 _RAseries(self._auxD)) 

173 

174# _RhumbLine = RhumbLineAux # see further below 

175 

176 @Property_RO 

177 def _rrm(self): 

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

179 

180 _mpr = _rrm # meter per radian, see _mpd 

181 

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

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

184 ''' 

185 pP, xD = self._RA, self._auxD 

186 

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

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

189 

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

191 xD.CParametric(Chix, Chiy) 

192 if dD: 

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

194 _Dlam(tx, ty)) # not Lambertian! 

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

196 Phiy.toBeta(self).normalized, 

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

198 dD += _Dp0Dpsi(tx, ty) 

199 dD *= self._c2 * lon12 

200 return float(dD) 

201 

202 

203class RhumbLineAux(RhumbLineBase): 

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

205 

206 Class C{RhumbLineAux} facilitates the determination of points 

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

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

209 ''' 

210 _Rhumb = RhumbAux # rhumb.aux_.RhumbAux 

211 

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

213 '''New C{RhumbLineAux}. 

214 

215 @arg rhumb: The rhumb reference (L{RhumbAux}). 

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

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

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

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

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

221 values specifying the required capabilities. Include 

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

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

224 ''' 

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

226 

227 @Property_RO 

228 def _Chi1(self): 

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

230 

231 @Property_RO 

232 def _mu1(self): 

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

234 ''' 

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

236 

237 def _mu2lat(self, mu): 

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

239 ''' 

240 lat, _ = self.rhumb._latPhi2(mu) 

241 return lat 

242 

243 @Property_RO 

244 def _Phi1(self): 

245 return AuxPhi.fromDegrees(self.lat1) 

246 

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

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

249 ''' 

250 R = self.rhumb 

251 lat2, Phi2 = R._latPhi2(mu2) 

252 Chi2 = Phi2.toChi(R) 

253 Chi1 = self._Chi1 

254 lon2 = self._salp * a12 

255 if lon2: 

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

257 lon2 = _over(lon2, m) 

258 return lat2, lon2, Chi1, Chi2 

259 

260# @Property_RO 

261# def _psi1(self): 

262# return self._Chi1.toLambertianRadians 

263 

264RhumbAux._RhumbLine = RhumbLineAux # PYCHOK see RhumbBase._RhumbLine 

265 

266 

267def _RAintegrate(auxD): 

268 # Compute coefficients by Fourier transform of integrand 

269 L = 2 

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

271 f = auxD._qIntegrand 

272 c_ = fft.transform(f) 

273 pP = [] 

274 _P = pP.append 

275 # assert L < _Lbits 

276 while L < _Lbits: 

277 L = fft.reset(L) * 2 

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

279 # assert len(c_) == L + 1 

280 pP[:], k = [], -1 

281 for j in range(1, L + 1): 

282 # Compute Fourier coefficients of integral 

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

284 if fabs(p) > EPS_2: 

285 k = -1 # run interrupted 

286 else: 

287 if k < 0: 

288 k = 1 # mark as first small value 

289 if (j - k) >= ((j + 7) // 8): 

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

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

292 _P(-p) 

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

294 

295 

296def _RAseries(auxD): 

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

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

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

300 d = n = auxD._n 

301 i = 0 

302 aL = auxD.ALorder 

303 Cs = _RACoeffs[aL] 

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

305 pP = [] 

306 _p = _polynomial 

307 for m in _reverange(aL): # order 

308 j = i + m + 1 

309 pP.append(_p(n, Cs, i, j) * d) 

310 d *= n 

311 i = j 

312 # assert i == len(pP) 

313 return pP 

314 

315 

316_RACoeffs = _Rdict(110, # Rhumb Area Coefficients in matrix Q 

317 _Rtuple(_Rkey(4), 10, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 4 

318 '596/2025, -398/945, 22/45, -1/3', 

319 '1543/4725, -118/315, 1/5', 

320 '152/945, -17/315', 

321 '5/252'), 

322 _Rtuple(_Rkey(5), 15, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 5 

323 '-102614/467775, 596/2025, -398/945, 22/45, -1/3', 

324 '-24562/155925, 1543/4725, -118/315, 1/5', 

325 '-38068/155925, 152/945, -17/315', 

326 '-752/10395, 5/252', 

327 '-101/17325'), 

328 _Rtuple(_Rkey(6), 21, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 6 

329 '138734126/638512875, -102614/467775, 596/2025, -398/945, 22/45, -1/3', 

330 '17749373/425675250, -24562/155925, 1543/4725, -118/315, 1/5', 

331 '1882432/8513505, -38068/155925, 152/945, -17/315', 

332 '268864/2027025, -752/10395, 5/252', 

333 '62464/2027025, -101/17325', 

334 '11537/4054050'), 

335 _Rtuple(_Rkey(7), 28, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 7 

336 '-565017322/1915538625, 138734126/638512875, -102614/467775, 596/2025, -398/945, 22/45, -1/3', 

337 '-1969276/58046625, 17749373/425675250, -24562/155925, 1543/4725, -118/315, 1/5', 

338 '-58573784/638512875, 1882432/8513505, -38068/155925, 152/945, -17/315', 

339 '-6975184/42567525, 268864/2027025, -752/10395, 5/252', 

340 '-112832/1447875, 62464/2027025, -101/17325', 

341 '-4096/289575, 11537/4054050', 

342 '-311/525525'), 

343 _Rtuple(_Rkey(8), 36, # GEOGRAPHICLIB_RHUMBAREA_ORDER == 8 

344 '188270561816/488462349375, -565017322/1915538625, 138734126/638512875, -102614/467775, 596/2025, -398/945, 22/45, -1/3', 

345 '2332829602/23260111875, -1969276/58046625, 17749373/425675250, -24562/155925, 1543/4725, -118/315, 1/5', 

346 '-41570288/930404475, -58573784/638512875, 1882432/8513505, -38068/155925, 152/945, -17/315', 

347 '1538774036/10854718875, -6975184/42567525, 268864/2027025, -752/10395, 5/252', 

348 '436821248/3618239625, -112832/1447875, 62464/2027025, -101/17325', 

349 '3059776/80405325, -4096/289575, 11537/4054050', 

350 '4193792/723647925, -311/525525', 

351 '1097653/1929727800') 

352) 

353del _Rdict, _Rkey, _Rtuple 

354 

355__all__ += _ALL_DOCS(Caps) 

356 

357# **) MIT License 

358# 

359# Copyright (C) 2023-2025 -- mrJean1 at Gmail -- All Rights Reserved. 

360# 

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

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

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

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

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

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

367# 

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

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

370# 

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

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

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

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

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

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

377# OTHER DEALINGS IN THE SOFTWARE.