Coverage for pygeodesy/auxilats/auxDLat.py: 95%

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1# -*- coding: utf-8 -*- 

2 

3u'''Class L{AuxDLat} transcoded to Python from I{Karney}'s C++ class U{DAuxLatitude 

4<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1DAuxLatitude.html>} 

5in I{GeographicLib version 2.2+}. 

6 

7Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2022-2023) and licensed 

8under the MIT/X11 License. For more information, see the U{GeographicLib 

9<https://GeographicLib.SourceForge.io>} documentation. 

10''' 

11# make sure int/int division yields float quotient, see .basics 

12from __future__ import division as _; del _ # PYCHOK semicolon 

13 

14from pygeodesy.auxilats.auxily import Aux, _Datan, _Dasinh, _sc, _sn, AuxError 

15from pygeodesy.auxilats.auxLat import AuxLat, _ALL_DOCS 

16from pygeodesy.basics import map1, _reverange 

17from pygeodesy.constants import INF, NAN, isfinite, isinf, isnan, _over, \ 

18 _0_0, _0_5, _1_0, _2_0, _N_2_0, _3_0 

19from pygeodesy.elliptic import Elliptic as _Ef, Fsum 

20# from pygeodesy.errors import AuxError # from .auxilats.auxily 

21# from pygeodesy.fsums import Fsum # from .elliptic 

22# from pygeodesy.lazily import _ALL_DOCS # from .auxilats.auxLat 

23 

24from math import atan, atan2, cos, sin, sqrt 

25 

26__all__ = () 

27__version__ = '23.08.05' 

28 

29 

30class AuxDLat(AuxLat): 

31 '''Class to compute C{Divided Differences} of I{Auxiliary} 

32 latitudes and other C{Divided Differences} needed for 

33 L{RhumbAux} and L{RhumbLineAux} calculations. 

34 ''' 

35 

36 def _Datanhee(self, x, y): 

37 # atan(e*sn(tphi))/e: 

38 # Datan(e*sn(x),e*sn(y))*Dsn(x,y)/Datan(x,y) 

39 # asinh(e1*sn(fm1*tphi)): 

40 # Dasinh(e1*sn(fm1*x)), e1*sn(fm1*y)) * 

41 # e1 * Dsn(fm1*x, fm1*y) *fm1 / (e * Datan(x,y)) 

42 # = Dasinh(e1*sn(fm1*x)), e1*sn(fm1*y)) * 

43 # Dsn(fm1*x, fm1*y) / Datan(x,y) 

44 if self.f < 0: 

45 e = self._e 

46 r = _Datan(e * _sn(x), e * _sn(y)) 

47 else: 

48 x *= self._fm1 

49 y *= self._fm1 

50 e1 = self._e1 

51 r = _Dasinh(e1 * _sn(x), e1 * _sn(y)) 

52 return _Dsn(x, y) * r 

53 

54 def Dconvert(self, auxout, Zeta1, Zeta2): 

55 '''I{Divided Difference} of one auxiliary latitude wrt another. 

56 ''' 

57 auxin = Zeta1._AUX 

58 # assert Zeta2._AUX == auxin 

59 try: 

60 if auxin != auxout: 

61 cs = self._coeffs(auxout, auxin) 

62 # assert len(cs) == self.ALorder 

63 r = _DClenshaw(True, Zeta1, Zeta2, cs, self.ALorder) 

64 else: 

65 r = _1_0 

66 except AuxError: # no _coeffs 

67 r = NAN 

68 return r 

69 

70 def DE(self, X, Y): 

71 # We assume that X and Y are in [-90d, 90d] and 

72 # have the same sign. If not we would include 

73 # if (Xn.y() * Yn.y() < 0) 

74 # return d != 0 ? (E(X) - E(Y)) / d : 1 

75 # The general formula fails for x = y = 0d and 

76 # x = y = 90d. Probably this is fixable (the 

77 # formula works for other x = y. But let's 

78 # also stipulate that x != y. 

79 

80 # Make both y positive, so we can do the swap a <-> b trick 

81 sx, cx, x = X._yxr_normalized(True) 

82 sy, cy, y = Y._yxr_normalized(True) 

83 Dt, k2, d = _0_0, -self._e12, (y - x) 

84 # Switch prolate to oblate, then use formulas for k2 < 0 

85 if self.f < 0: # XXX and False? 

86 sx, cx = cx, sx 

87 sy, cy = cy, sy 

88 d, k2 = -d, self._e2 

89 # See DLMF: Eqs (19.11.2) and (19.11.4) letting 

90 if sx and sy: 

91 t = _sxk2y(sx, sy, k2) + _sxk2y(sy, sx, k2) 

92 Dt = _over(_Dsin(x, y) * (sx + sy), t * (cx + cy)) 

93 t = d * Dt 

94 t2 = _1_0 + t**2 

95 Dt *= _2_0 / t2 

96 sk2 = (d * Dt)**2 * k2 

97 d2 = _1_0 - sk2 

98 c2 = ((_1_0 - t) * (_1_0 + t) / t2)**2 if t else _1_0 

99 # E(z)/sin(z) 

100 E_s = (_Ef.fRF(c2, d2, _1_0) - 

101 _Ef.fRD(c2, d2, _1_0, _3_0) * sk2) 

102 Dt *= E_s - k2 * sx * sy 

103 return Dt 

104 

105 def DIsometric(self, Phi1, Phi2): 

106 '''I{Divided Difference} of the isometric wrt the geographic latitude. 

107 ''' 

108 tx, ty = Phi1.tan, Phi2.tan 

109 if isnan(ty) or isnan(tx): # PYCHOK no cover 

110 r = NAN 

111 elif isinf(ty) or isinf(tx): # PYCHOK no cover 

112 r = INF 

113 else: # psi = asinh(tan(Phi)) - e^2 * atanhee(tan(Phi)) 

114 r = self._Datanhee(tx, ty) * self._e2 

115 r = _over(_Dasinh(tx, ty) - r, _Datan(tx, ty)) 

116 return r 

117 

118 def DParametric(self, Phi1, Phi2): 

119 '''I{Divided Difference} of the parametric wrt the geographic latitude. 

120 ''' 

121 fm1, e2m1 = self._fm1, self._e2m1 

122 tx, ty = Phi1.tan, Phi2.tan 

123 # DbetaDphi = Datan(fm1*tx, fm1*ty) * fm1 / Datan(tx, ty) 

124 # Datan(x, y) = 1/(1 + x^2), for x = y 

125 # = (atan(y) - atan(x)) / (y-x), for x*y < 0 

126 # = atan( (y-x) / (1 + x*y) ) / (y-x), for x*y > 0 

127 txy = tx * ty 

128 if txy < 0 or (isinf(ty) and not tx): 

129 _a = atan 

130 r = _over(_a(fm1 * ty) - _a(fm1 * tx), _a(ty) - _a(tx)) 

131 elif tx == ty: # includes tx = ty = inf 

132 if txy > 1: # == tx**2 

133 txy = _1_0 / txy 

134 r = txy + e2m1 

135 else: 

136 r = txy * e2m1 + _1_0 

137 r = _over(fm1 * (txy + _1_0), r) 

138 else: 

139 if txy > 1: 

140 tx = _1_0 / tx 

141 ty = _1_0 / ty 

142 txy = tx * ty 

143 t = txy + e2m1 

144 else: 

145 t = txy * e2m1 + _1_0 

146 r = ty - tx 

147 r = _over(atan2(r * fm1, t), atan2(r, _1_0 + txy)) 

148 return r 

149 

150 def DParametricZ(self, Zeta1, Zeta2): 

151 '''Short for C{.Dconvert(Aux.BETA, Zeta1, Zeta2)}. 

152 ''' 

153 return self.Dconvert(Aux.BETA, Zeta1, Zeta2) 

154 

155 def DRectifying(self, Phi1, Phi2): 

156 '''I{Divided Difference} of the rectifying wrt the geographic latitude. 

157 ''' 

158 # Stipulate that Phi1 and Phi2 are in [-90d, 90d] 

159 x, y = Phi1.toRadians, Phi2.toRadians 

160 if y == x: # isnear0 

161 Mu1 = self.Rectifying(Phi1, diff=True) 

162 tphi1, r = Phi1.tan, Mu1.diff 

163 if isfinite(tphi1): 

164 r *= _over(_sc(tphi1), _sc(Mu1.tan))**2 

165 else: # PYCHOK no cover 

166 r = _over(_1_0, r) 

167 elif (x * y) < 0: 

168 r = _over(self.Rectifying(Phi2).toRadians - 

169 self.Rectifying(Phi1).toRadians, y - x) 

170 else: 

171 r = _over(self.b, self.RectifyingRadius(True)) 

172 r *= self.DE(*map1(self.Parametric, Phi1, Phi2)) 

173 r *= self.DParametric(Phi1, Phi2) 

174 return r # or INF or NAN 

175 

176 def DRectifyingZ(self, Zeta1, Zeta2): 

177 '''Short for C{.Dconvert(Aux.MU, Zeta1, Zeta2)}. 

178 ''' 

179 return self.Dconvert(Aux.MU, Zeta1, Zeta2) 

180 

181 

182def _DClenshaw(sinp, Zeta1, Zeta2, cs, K): 

183 '''(INTERNAL) I{Divided Difference} of L{AuxLat._Clenshaw}. 

184 

185 @return: C{Fsum} if sinp otherwise a C{float}. 

186 ''' 

187 s1, c1, r1 = Zeta1._yxr_normalized(False) 

188 s2, c2, r2 = Zeta2._yxr_normalized(False) 

189 Delta = r2 - r1 

190 # Evaluate (Clenshaw(sinp, szeta2, czeta2, cs, K) - 

191 # Clenshaw(sinp, szeta1, czeta1, cs, K)) / Delta 

192 # or f = sin if sinp else cos 

193 # sum(cs[k] * (f((2*k+2) * Zeta2) - 

194 # f((2*k+2) * Zeta2))) / Delta 

195 # 

196 # Delta is EITHER 1, giving the plain difference OR (Zeta2 - Zeta1) 

197 # in radians, giving the I{Divided Difference}. Other values will 

198 # produce nonsense. 

199 # 

200 # Suffices a and b denote [1,1], [2,1] elements of matrix/vector 

201 cp = cm = c2 * c1 

202 t = s2 * s1 

203 cp -= t # not + 

204 cm += t # not - 

205 

206 sp = s2 * c1 

207 t = c2 * s1 

208 smd = ((sin(Delta) / Delta) if Delta != _1_0 else 

209 (sp - t)) if Delta else _1_0 

210 sp += t 

211 

212 xa = cp * cm * _2_0 

213 xb = sp * smd * _N_2_0 

214 xD = xb * Delta**2 

215 

216 if isfinite(xD) and isfinite(xb) and isfinite(xa): 

217 U0a, U1a = Fsum(), Fsum() 

218 U0b, U1b = Fsum(), Fsum() 

219 else: # XXX avoid Fsum(NAN) exceptions 

220 U0a = U1a = U0b = U1b = _0_0 

221 for k in _reverange(K): # assert len(cs) == K 

222 # t = x . U0 - U1 + cs[k] * I 

223 U1a -= U0a * xa + U0b * xD + cs[k] 

224 U1b -= U0a * xb + U0b * xa 

225 U1a, U0a = U0a, -U1a 

226 U1b, U0b = U0b, -U1b 

227 # F0a = (sp if sinp else cp) * cm 

228 # F0b = (cp if sinp else -sp) * smd 

229 # Fm1a = 0 if sinp else 1 # Fm1b = 0 

230 # return (U0b * F0a + U0a * F0b - U1b * Fm1a) * 2 

231 if sinp: 

232 U1b = _0_0 

233 else: 

234 sp, cp = cp, -sp 

235 U0b *= sp * cm 

236 U0a *= cp * smd 

237 U0a += U0b 

238 U0a -= U1b 

239 U0a *= _2_0 

240 return float(U0a) if sinp else U0a # Fsum 

241 

242 

243def _Dsin(x, y): # see also .rhumbx._Dsin 

244 r = cos((x + y) * _0_5) 

245 d = (x - y) * _0_5 

246 if d: 

247 r *= sin(d) / d 

248 return r 

249 

250 

251def _Dsn(x, y): 

252 # (sn(y) - sn(x)) / (y - x) 

253 if x != y: 

254 snx, sny = map1(_sn, x, y) 

255 if (x * y) > 0: 

256 scx, scy = map1(_sc, x, y) 

257 r = _over((snx / scy) + (sny / scx), 

258 (snx + sny) * scy * scx) 

259 else: 

260 r = (sny - snx) / (y - x) 

261 elif x: 

262 r = _1_0 / (_sc(x) * (x**2 + _1_0)) # == 1 / sqrt3(x**2 + 1) 

263 else: 

264 r = _1_0 

265 return r 

266 

267 

268def _sxk2y(sx, sy, k2): 

269 # .DE helper 

270 if sx: 

271 try: 

272 sx *= sqrt(_1_0 - sy**2 * k2) 

273 except ValueError: # domain error 

274 sx = NAN 

275 return sx 

276 

277 

278__all__ += _ALL_DOCS(AuxDLat) 

279 

280# **) MIT License 

281# 

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

283# 

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

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

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

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

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

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

290# 

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

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

293# 

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

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

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

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

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

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

300# OTHER DEALINGS IN THE SOFTWARE.