Coverage for pygeodesy/triaxials.py: 95%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Triaxal ellipsoid classes L{JacobiConformal}, Jacobi's conformal projection, trancoded 

5from I{Charles Karney}'s C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/C++/ 

6doc/classGeographicLib_1_1JacobiConformal.html#details>} to pure Python, I{ordered} L{Triaxial} 

7and I{unordered} L{Triaxial_} and miscellaneous classes L{BetaOmega2Tuple}, L{BetaOmega3Tuple}, 

8L{Jacobi2Tuple} and L{TriaxialError}. 

9 

10@see: U{Geodesics on a triaxial ellipsoid<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid# 

11 Geodesics_on_a_triaxial_ellipsoid>} and U{Triaxial coordinate systems and their geometrical 

12 interpretation<https://www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

13 

14@var Triaxials.Amalthea: Triaxial(name='Amalthea', a=125000, b=73000, c=64000, e2ab=0.658944, e2bc=0.231375493, e2ac=0.737856, volume=2446253479595252, area=93239507787.490371704, area_p=93212299402.670425415) 

15@var Triaxials.Ariel: Triaxial(name='Ariel', a=581100, b=577900, c=577700, e2ab=0.01098327, e2bc=0.000692042, e2ac=0.011667711, volume=812633172614203904, area=4211301462766.580078125, area_p=4211301574065.829589844) 

16@var Triaxials.Earth: Triaxial(name='Earth', a=6378173.435, b=6378103.9, c=6356754.399999999, e2ab=0.000021804, e2bc=0.006683418, e2ac=0.006705077, volume=1083208241574987694080, area=510065911057441.0625, area_p=510065915922713.6875) 

17@var Triaxials.Enceladus: Triaxial(name='Enceladus', a=256600, b=251400, c=248300, e2ab=0.040119337, e2bc=0.024509841, e2ac=0.06364586, volume=67094551514082248, area=798618496278.596679688, area_p=798619018175.109863281) 

18@var Triaxials.Europa: Triaxial(name='Europa', a=1564130, b=1561230, c=1560930, e2ab=0.003704694, e2bc=0.000384275, e2ac=0.004087546, volume=15966575194402123776, area=30663773697323.51953125, area_p=30663773794562.45703125) 

19@var Triaxials.Io: Triaxial(name='Io', a=1829400, b=1819300, c=1815700, e2ab=0.011011391, e2bc=0.003953651, e2ac=0.014921506, volume=25313121117889765376, area=41691875849096.7421875, area_p=41691877397441.2109375) 

20@var Triaxials.Mars: Triaxial(name='Mars', a=3394600, b=3393300, c=3376300, e2ab=0.000765776, e2bc=0.009994646, e2ac=0.010752768, volume=162907283585817247744, area=144249140795107.4375, area_p=144249144150662.15625) 

21@var Triaxials.Mimas: Triaxial(name='Mimas', a=207400, b=196800, c=190600, e2ab=0.09960581, e2bc=0.062015624, e2ac=0.155444317, volume=32587072869017956, area=493855762247.691894531, area_p=493857714107.9375) 

22@var Triaxials.Miranda: Triaxial(name='Miranda', a=240400, b=234200, c=232900, e2ab=0.050915557, e2bc=0.011070811, e2ac=0.061422691, volume=54926187094835456, area=698880863325.756958008, area_p=698881306767.950317383) 

23@var Triaxials.Moon: Triaxial(name='Moon', a=1735550, b=1735324, c=1734898, e2ab=0.000260419, e2bc=0.000490914, e2ac=0.000751206, volume=21886698675223740416, area=37838824729886.09375, area_p=37838824733332.2265625) 

24@var Triaxials.Tethys: Triaxial(name='Tethys', a=535600, b=528200, c=525800, e2ab=0.027441672, e2bc=0.009066821, e2ac=0.036259685, volume=623086233855821440, area=3528073490771.394042969, area_p=3528074261832.738769531) 

25@var Triaxials.WGS84_35: Triaxial(name='WGS84_35', a=6378172, b=6378102, c=6356752.314245179, e2ab=0.00002195, e2bc=0.006683478, e2ac=0.006705281, volume=1083207319768789942272, area=510065621722018.125, area_p=510065626587483.3125) 

26''' 

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

28from __future__ import division as _; del _ # PYCHOK semicolon 

29 

30from pygeodesy.basics import isscalar, map1, _ValueError, _zip 

31from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, _EPS2e4, INT0, PI2, PI_3, PI4, \ 

32 _0_0, _0_5, _1_0, _N_2_0, float0_, isfinite, isnear1, \ 

33 _4_0 # PYCHOK used! 

34from pygeodesy.datums import Datum, Ellipsoid, Fmt, _spherical_datum, _WGS84 

35# from pygeodesy.ellipsoids import Ellipsoid # from .datums 

36# from pygeodesy.elliptic import Elliptic # ._MODS 

37# from pygeodesy.errors import _ValueError # from .basics 

38from pygeodesy.fmath import Fdot, fdot, fmean_, hypot, hypot_, _hypot21_, norm2 

39from pygeodesy.fsums import Fsum, fsum_, Property_RO 

40from pygeodesy.interns import NN, _a_, _b_, _beta_, _c_, _distant_, _height_, \ 

41 _inside_, _near_, _not_, _NL_, _NLATvar_, _NOTEQUAL_, \ 

42 _null_, _opposite_, _outside_, _SPACE_, _spherical_, \ 

43 _too_, _x_, _y_, _COMMA_ # PYCHOK used! 

44# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .vector3d 

45from pygeodesy.named import _NamedBase, _NamedEnum, _NamedEnumItem, \ 

46 _NamedTuple, _Pass, _lazyNamedEnumItem as _lazy 

47from pygeodesy.namedTuples import LatLon3Tuple, Vector3Tuple, Vector4Tuple 

48# from pygeodesy.props import Property_RO # from .fsums 

49# from pygeodesy.streprs import Fmt # from .datums 

50from pygeodesy.units import Degrees, Float, Height_, Meter, Meter2, Meter3, \ 

51 Radians, Radius 

52from pygeodesy.utily import asin1, atan2d, km2m, m2km, SinCos2, sincos2d_ 

53from pygeodesy.vector3d import _ALL_LAZY, _MODS, _otherV3d, Vector3d 

54 

55from math import atan2, fabs, sqrt 

56 

57__all__ = _ALL_LAZY.triaxials 

58__version__ = '23.04.14' 

59 

60_not_ordered_ = _not_('ordered') 

61_omega_ = 'omega' 

62_TRIPS = 537 # max 55, Eberly 1074? 

63 

64 

65class _ToNamedBase(_NamedBase): 

66 '''(INTERNAL) C{-.toDegrees}, C{-.toRadians} base. 

67 ''' 

68 def _toDegrees(self, a, b, *c, **toDMS_kwds): 

69 if toDMS_kwds: 

70 toDMS = _MODS.dms.toDMS 

71 a = toDMS(a.toDegrees(), **toDMS_kwds) 

72 b = toDMS(b.toDegrees(), **toDMS_kwds) 

73 elif isinstance(a, Degrees) and \ 

74 isinstance(b, Degrees): 

75 return self 

76 else: 

77 a, b = a.toDegrees(), b.toDegrees() 

78 return self.classof(a, b, *c, name=self.name) 

79 

80 def _toRadians(self, a, b, *c): 

81 return self if isinstance(a, Radians) and \ 

82 isinstance(b, Radians) else \ 

83 self.classof(a.toRadians(), b.toRadians(), 

84 *c, name=self.name) 

85 

86 

87class BetaOmega2Tuple(_NamedTuple, _ToNamedBase): 

88 '''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and 

89 longitude C{beta} and C{omega} both in C{Radians} (or 

90 C{Degrees}). 

91 ''' 

92 _Names_ = (_beta_, _omega_) 

93 _Units_ = (_Pass, _Pass) 

94 

95 def toDegrees(self, **toDMS_kwds): 

96 '''Convert this L{BetaOmega2Tuple} to C{Degrees} or C{toDMS}. 

97 

98 @return: L{BetaOmega2Tuple}C{(beta, omega)} with 

99 C{beta} and C{omega} both in C{Degrees} 

100 or as an L{toDMS} string provided some 

101 B{C{toDMS_kwds}} are supplied. 

102 ''' 

103 return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) 

104 

105 def toRadians(self): 

106 '''Convert this L{BetaOmega2Tuple} to C{Radians}. 

107 

108 @return: L{BetaOmega2Tuple}C{(beta, omega)} with 

109 C{beta} and C{omega} both in C{Radians}. 

110 ''' 

111 return _ToNamedBase._toRadians(self, *self) 

112 

113 

114class BetaOmega3Tuple(_NamedTuple, _ToNamedBase): 

115 '''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and 

116 longitude C{beta} and C{omega} both in C{Radians} (or C{Degrees}) 

117 and the C{height}, rather the (signed) I{distance} to the triaxial's 

118 surface (measured along the radial line to the triaxial's center) 

119 in C{meter}, conventionally. 

120 ''' 

121 _Names_ = BetaOmega2Tuple._Names_ + (_height_,) 

122 _Units_ = BetaOmega2Tuple._Units_ + ( Meter,) 

123 

124 def toDegrees(self, **toDMS_kwds): 

125 '''Convert this L{BetaOmega3Tuple} to C{Degrees} or C{toDMS}. 

126 

127 @return: L{BetaOmega3Tuple}C{(beta, omega, height)} with 

128 C{beta} and C{omega} both in C{Degrees} or as an 

129 L{toDMS} string provided some B{C{toDMS_kwds}} 

130 are supplied. 

131 ''' 

132 return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) 

133 

134 def toRadians(self): 

135 '''Convert this L{BetaOmega3Tuple} to C{Radians}. 

136 

137 @return: L{BetaOmega3Tuple}C{(beta, omega, height)} with 

138 C{beta} and C{omega} both in C{Radians}. 

139 ''' 

140 return _ToNamedBase._toRadians(self, *self) 

141 

142 def to2Tuple(self): 

143 '''Reduce this L{BetaOmega3Tuple} to a L{BetaOmega2Tuple}. 

144 ''' 

145 return BetaOmega2Tuple(*self[:2]) 

146 

147 

148class Jacobi2Tuple(_NamedTuple, _ToNamedBase): 

149 '''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y} 

150 projection, both in C{Radians} (or C{Degrees}). 

151 ''' 

152 _Names_ = (_x_, _y_) 

153 _Units_ = (_Pass, _Pass) 

154 

155 def toDegrees(self, **toDMS_kwds): 

156 '''Convert this L{Jacobi2Tuple} to C{Degrees} or C{toDMS}. 

157 

158 @return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y} 

159 both in C{Degrees} or as an L{toDMS} string 

160 provided some B{C{toDMS_kwds}} are supplied. 

161 ''' 

162 return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) 

163 

164 def toRadians(self): 

165 '''Convert this L{Jacobi2Tuple} to C{Radians}. 

166 

167 @return: L{Jacobi2Tuple}C{(x, y)} with C{x} 

168 and C{y} both in C{Radians}. 

169 ''' 

170 return _ToNamedBase._toRadians(self, *self) 

171 

172 

173class Triaxial_(_NamedEnumItem): 

174 '''I{Unordered} triaxial ellipsoid and base class. 

175 

176 Triaxial ellipsoids with right-handed semi-axes C{a}, C{b} and C{c}, oriented 

177 such that the large principal ellipse C{ab} is the equator I{Z}=0, I{beta}=0, 

178 while the small principal ellipse C{ac} is the prime meridian, plane I{Y}=0, 

179 I{omega}=0. 

180 

181 The four umbilic points, C{abs}(I{omega}) = C{abs}(I{beta}) = C{PI/2}, lie on 

182 the middle principal ellipse C{bc} in plane I{X}=0, I{omega}=C{PI/2}. 

183 

184 @note: I{Geodetic} C{lat}- and C{lon}gitudes are in C{degrees}, I{geodetic} 

185 C{phi} and C{lam}bda are in C{radians}, but I{ellipsoidal} lat- and 

186 longitude C{beta} and C{omega} are in C{Radians} by default (or in 

187 C{Degrees} if converted). 

188 ''' 

189 _ijk = _kji = None 

190 _unordered = True 

191 

192 def __init__(self, a_triaxial, b=None, c=None, name=NN): 

193 '''New I{unordered} L{Triaxial_}. 

194 

195 @arg a_triaxial: C{X} semi-axis (C{scalar}, conventionally in C{meter}) 

196 or an other L{Triaxial} or L{Triaxial_} instance. 

197 @kwarg b: C{Y} semi-axis (C{meter}, same units as B{C{a}}), required 

198 if C{B{a_triaxial} is scalar}, ignored otherwise. 

199 @kwarg c: C{Z} semi-axis (C{meter}, same units as B{C{a}}), required 

200 if C{B{a_triaxial} is scalar}, ignored otherwise. 

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

202 

203 @raise TriaxialError: Invalid semi-axis or -axes. 

204 ''' 

205 try: 

206 a = a_triaxial 

207 t = a._abc3 if isinstance(a, Triaxial_) else ( 

208 Radius(a=a), Radius(b=b), Radius(c=c)) 

209 except (TypeError, ValueError) as x: 

210 raise TriaxialError(a=a, b=b, c=c, cause=x) 

211 if name: 

212 self.name = name 

213 

214 a, b, c = self._abc3 = t 

215 if self._unordered: # == not isinstance(self, Triaxial) 

216 s, _, t = sorted(t) 

217 if not (isfinite(t) and s > 0): 

218 raise TriaxialError(a=a, b=b, c=c) # txt=_invalid_ 

219 elif not (isfinite(a) and a >= b >= c > 0): 

220 raise TriaxialError(a=a, b=b, c=c, txt=_not_ordered_) 

221 elif not (a > c and self._a2c2 > 0 and self.e2ac > 0): 

222 raise TriaxialError(a=a, c=c, e2ac=self.e2ac, txt=_spherical_) 

223 

224 def __str__(self): 

225 return self.toStr() 

226 

227 @Property_RO 

228 def a(self): 

229 '''Get the (largest) C{x} semi-axis (C{meter}, conventionally). 

230 ''' 

231 a, _, _ = self._abc3 

232 return a 

233 

234 @Property_RO 

235 def _a2b2(self): 

236 '''(INTERNAL) Get C{a**2 - b**2} == E_sub_e**2. 

237 ''' 

238 a, b, _ = self._abc3 

239 return ((a - b) * (a + b)) if a != b else _0_0 

240 

241 @Property_RO 

242 def _a2_b2(self): 

243 '''(INTERNAL) Get C{(a/b)**2}. 

244 ''' 

245 a, b, _ = self._abc3 

246 return (a / b)**2 if a != b else _1_0 

247 

248 @Property_RO 

249 def _a2c2(self): 

250 '''(INTERNAL) Get C{a**2 - c**2} == E_sub_x**2. 

251 ''' 

252 a, _, c = self._abc3 

253 return ((a - c) * (a + c)) if a != c else _0_0 

254 

255 @Property_RO 

256 def area(self): 

257 '''Get the surface area (C{meter} I{squared}). 

258 ''' 

259 c, b, a = sorted(self._abc3) 

260 if a > c: 

261 a = Triaxial(a, b, c).area if a > b else \ 

262 Ellipsoid(a, b=c).areax # a == b 

263 else: # a == c == b 

264 a = Meter2(area=a**2 * PI4) 

265 return a 

266 

267 def area_p(self, p=1.6075): 

268 '''I{Approximate} the surface area (C{meter} I{squared}). 

269 

270 @kwarg p: Exponent (C{scalar} > 0), 1.6 for near-spherical or 1.5849625007 

271 for "near-flat" triaxials. 

272 

273 @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Approximate_formula>}. 

274 ''' 

275 a, b, c = self._abc3 

276 if a == b == c: 

277 a *= a 

278 else: 

279 _p = pow 

280 a = _p(fmean_(_p(a * b, p), _p(a * c, p), _p(b * c, p)), _1_0 / p) 

281 return Meter2(area_p=a * PI4) 

282 

283 @Property_RO 

284 def b(self): 

285 '''Get the (middle) C{y} semi-axis (C{meter}, same units as B{C{a}}). 

286 ''' 

287 _, b, _ = self._abc3 

288 return b 

289 

290 @Property_RO 

291 def _b2c2(self): 

292 '''(INTERNAL) Get C{b**2 - c**2} == E_sub_y**2. 

293 ''' 

294 _, b, c = self._abc3 

295 return ((b - c) * (b + c)) if b != c else _0_0 

296 

297 @Property_RO 

298 def c(self): 

299 '''Get the (smallest) C{z} semi-axis (C{meter}, same units as B{C{a}}). 

300 ''' 

301 _, _, c = self._abc3 

302 return c 

303 

304 @Property_RO 

305 def _c2_b2(self): 

306 '''(INTERNAL) Get C{(c/b)**2}. 

307 ''' 

308 _, b, c = self._abc3 

309 return (c / b)**2 if b != c else _1_0 

310 

311 @Property_RO 

312 def e2ab(self): 

313 '''Get the C{ab} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (b/a)**2}. 

314 ''' 

315 return Float(e2ab=(_1_0 - self._1e2ab) or _0_0) 

316 

317 @Property_RO 

318 def _1e2ab(self): 

319 '''(INTERNAL) Get C{1 - e2ab} == C{(b/a)**2}. 

320 ''' 

321 a, b, _ = self._abc3 

322 return (b / a)**2 if a != b else _1_0 

323 

324 @Property_RO 

325 def e2bc(self): 

326 '''Get the C{bc} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/b)**2}. 

327 ''' 

328 return Float(e2bc=(_1_0 - self._1e2bc) or _0_0) 

329 

330 _1e2bc = _c2_b2 # C{1 - e2bc} == C{(c/b)**2} 

331 

332 @Property_RO 

333 def e2ac(self): 

334 '''Get the C{ac} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/a)**2}. 

335 ''' 

336 return Float(e2ac=(_1_0 - self._1e2ac) or _0_0) 

337 

338 @Property_RO 

339 def _1e2ac(self): 

340 '''(INTERNAL) Get C{1 - e2ac} == C{(c/a)**2}. 

341 ''' 

342 a, _, c = self._abc3 

343 return (c / a)**2 if a != c else _1_0 

344 

345 @Property_RO 

346 def _Elliptic(self): 

347 '''(INTERNAL) Get class L{Elliptic} once. 

348 ''' 

349 return _MODS.elliptic.Elliptic 

350 

351 def hartzell4(self, pov, los=None, name=NN): 

352 '''Compute the intersection of this triaxial's surface with a Line-Of-Sight 

353 from a Point-Of-View in space. 

354 

355 @see: Function L{pygeodesy.hartzell4} for further details. 

356 ''' 

357 return hartzell4(pov, los=los, tri_biax=self, name=name) 

358 

359 def height4(self, x_xyz, y=None, z=None, normal=True, eps=EPS): 

360 '''Compute the projection on and the height of a cartesian above or below 

361 this triaxial's surface. 

362 

363 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

364 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

365 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

366 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

367 @kwarg normal: If C{True} the projection is perpendicular to (the nearest 

368 point on) this triaxial's surface, otherwise the C{radial} 

369 line to this triaxial's center (C{bool}). 

370 @kwarg eps: Tolerance for root finding and validation (C{scalar}), use a 

371 negative value to skip validation. 

372 

373 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates 

374 C{x}, C{y} and C{z} of the projection on or the intersection 

375 with and with the height C{h} above or below the triaxial's 

376 surface in C{meter}, conventionally. 

377 

378 @raise TriaxialError: Non-cartesian B{C{xyz}}, invalid B{C{eps}}, no 

379 convergence in root finding or validation failed. 

380 

381 @see: Method L{Ellipsoid.height4} and I{Eberly}'s U{Distance from a Point 

382 to ... an Ellipsoid ...<https://www.GeometricTools.com/Documentation/ 

383 DistancePointEllipseEllipsoid.pdf>}. 

384 ''' 

385 v, r = _otherV3d_(x_xyz, y, z), self.isSpherical 

386 

387 i, h = None, v.length 

388 if h < EPS0: # EPS 

389 x = y = z = _0_0 

390 h -= min(self._abc3) # nearest 

391 elif r: # .isSpherical 

392 x, y, z = v.times(r / h).xyz 

393 h -= r 

394 else: 

395 x, y, z = v.xyz 

396 try: 

397 if normal: # perpendicular to triaxial 

398 x, y, z, h, i = _normalTo5(x, y, z, self, eps=eps) 

399 else: # radially to triaxial's center 

400 x, y, z = self._radialTo3(z, hypot(x, y), y, x) 

401 h = v.minus_(x, y, z).length 

402 except Exception as e: 

403 raise TriaxialError(x=x, y=y, z=z, cause=e) 

404 if h > 0 and self.sideOf(v, eps=EPS0) < 0: 

405 h = -h # below the surface 

406 return Vector4Tuple(x, y, z, h, iteration=i, name=self.height4.__name__) 

407 

408 @Property_RO 

409 def isOrdered(self): 

410 '''Is this triaxial I{ordered} and I{not spherical} (C{bool})? 

411 ''' 

412 a, b, c = self._abc3 

413 return bool(a >= b > c) # b > c! 

414 

415 @Property_RO 

416 def isSpherical(self): 

417 '''Is this triaxial I{spherical} (C{Radius} or INT0)? 

418 ''' 

419 a, b, c = self._abc3 

420 return a if a == b == c else INT0 

421 

422 def normal3d(self, x_xyz, y=None, z=None, length=_1_0): 

423 '''Get a 3-D vector perpendicular to at a cartesian on this triaxial's surface. 

424 

425 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

426 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

427 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

428 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

429 @kwarg length: Optional length and in-/outward direction (C{scalar}). 

430 

431 @return: A C{Vector3d(x_, y_, z_)} normalized to B{C{length}}, pointing 

432 in- or outward for neg- respectively positive B{C{length}}. 

433 

434 @note: Cartesian location C{(B{x}, B{y}, B{z})} must be on this triaxial's 

435 surface, use method L{Triaxial.sideOf} to validate. 

436 ''' 

437 # n = 2 * (x / a2, y / b2, z / c2) 

438 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered 

439 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2 

440 # == unit(x, y / _1e2ab, z / _1e2ac).times(length) 

441 n = self._normal3d.times_(*_otherV3d_(x_xyz, y, z).xyz) 

442 if n.length < EPS0: 

443 raise TriaxialError(x=x_xyz, y=y, z=z, txt=_null_) 

444 return n.times(length / n.length) 

445 

446 @Property_RO 

447 def _normal3d(self): 

448 '''(INTERNAL) Get M{Vector3d((d/a)**2, (d/b)**2, (d/c)**2)}, M{d = max(a, b, c)}. 

449 ''' 

450 d = max(self._abc3) 

451 t = tuple(((d / x)**2 if x != d else _1_0) for x in self._abc3) 

452 return Vector3d(*t, name=self.normal3d.__name__) 

453 

454 def _norm2(self, s, c, *a): 

455 '''(INTERNAL) Normalize C{s} and C{c} iff not already. 

456 ''' 

457 if fabs(s) > _1_0 or fabs(c) > _1_0 or \ 

458 fabs(_hypot21_(s, c)) > EPS0: 

459 s, c = norm2(s, c) 

460 if a: 

461 s, c = norm2(s * self.b, c * a[0]) 

462 return float0_(s, c) 

463 

464 def _order3(self, *abc, **reverse): # reverse=False 

465 '''(INTERNAL) Un-/Order C{a}, C{b} and C{c}. 

466 

467 @return: 3-Tuple C{(a, b, c)} ordered by or un-ordered 

468 (reverse-ordered) C{ijk} if C{B{reverse}=True}. 

469 ''' 

470 ijk = self._order_ijk(**reverse) 

471 return _getitems(abc, *ijk) if ijk else abc 

472 

473 def _order3d(self, v, **reverse): # reverse=False 

474 '''(INTERNAL) Un-/Order a C{Vector3d}. 

475 

476 @return: Vector3d(x, y, z) un-/ordered. 

477 ''' 

478 ijk = self._order_ijk(**reverse) 

479 return v.classof(*_getitems(v.xyz, *ijk)) if ijk else v 

480 

481 @Property_RO 

482 def _ordered4(self): 

483 '''(INTERNAL) Helper for C{_hartzell3d2} and C{_normalTo5}. 

484 ''' 

485 def _order2(reverse, a, b, c): 

486 '''(INTERNAL) Un-Order C{a}, C{b} and C{c}. 

487 

488 @return: 2-Tuple C{((a, b, c), ijk)} with C{a} >= C{b} >= C{c} 

489 and C{ijk} a 3-tuple with the initial indices. 

490 ''' 

491 i, j, k = 0, 1, 2 # range(3) 

492 if a < b: 

493 a, b, i, j = b, a, j, i 

494 if a < c: 

495 a, c, i, k = c, a, k, i 

496 if b < c: 

497 b, c, j, k = c, b, k, j 

498 # reverse (k, j, i) since (a, b, c) is reversed-sorted 

499 ijk = (k, j, i) if reverse else (None if i < j < k else (i, j, k)) 

500 return (a, b, c), ijk 

501 

502 abc, T = self._abc3, self 

503 if not self.isOrdered: 

504 abc, ijk = _order2(False, *abc) 

505 if ijk: 

506 _, kji = _order2(True, *ijk) 

507 T = Triaxial_(*abc) 

508 T._ijk, T._kji = ijk, kji 

509 return abc + (T,) 

510 

511 def _order_ijk(self, reverse=False): 

512 '''(INTERNAL) Get the un-/order indices. 

513 ''' 

514 return self._kji if reverse else self._ijk 

515 

516 def _radialTo3(self, sbeta, cbeta, somega, comega): 

517 '''(INTERNAL) I{Unordered} helper for C{.height4}. 

518 ''' 

519 def _rphi(a, b, sphi, cphi): 

520 # <https://WikiPedia.org/wiki/Ellipse#Polar_form_relative_to_focus> 

521 # polar form: radius(phi) = a * b / hypot(a * sphi, b * cphi) 

522 return (b / hypot(sphi, b / a * cphi)) if a > b else ( 

523 (a / hypot(cphi, a / b * sphi)) if a < b else a) 

524 

525 sa, ca = self._norm2(sbeta, cbeta) 

526 sb, cb = self._norm2(somega, comega) 

527 

528 a, b, c = self._abc3 

529 if a != b: 

530 a = _rphi(a, b, sb, cb) 

531 if a != c: 

532 c = _rphi(a, c, sa, ca) 

533 z, r = c * sa, c * ca 

534 x, y = r * cb, r * sb 

535 return x, y, z 

536 

537 def sideOf(self, x_xyz, y=None, z=None, eps=EPS4): 

538 '''Is a cartesian above, below or on the surface of this triaxial? 

539 

540 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

541 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

542 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

543 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

544 @kwarg eps: Near surface tolerance(C{scalar}). 

545 

546 @return: C{INT0} if C{(B{x}, B{y}, B{z})} is near this triaxial's surface 

547 within tolerance B{C{eps}}, otherwise a neg- or positive C{float} 

548 if in- respectively outside this triaxial. 

549 

550 @see: Methods L{Triaxial.height4} and L{Triaxial.normal3d}. 

551 ''' 

552 return _sideOf(_otherV3d_(x_xyz, y, z).xyz, self._abc3, eps=eps) 

553 

554 def _sqrt(self, x): 

555 '''(INTERNAL) Helper. 

556 ''' 

557 if x < 0: 

558 raise TriaxialError(Fmt.PAREN(sqrt=x)) 

559 return _0_0 if x < EPS02 else sqrt(x) 

560 

561 def toEllipsoid(self, name=NN): 

562 '''Convert this triaxial to an L{Ellipsoid}, provided C{a == b} or C{b == c}. 

563 

564 @return: An L{Ellipsoid} with north along this C{Z} axis if C{a == b}, 

565 this C{Y} axis if C{a == c} or this C{X} axis if C{b == c}. 

566 

567 @raise TriaxialError: This C{a != b}, C{b != c} and C{c != a}. 

568 

569 @see: Method L{Ellipsoid.toTriaxial}. 

570 ''' 

571 a, b, c = self._abc3 

572 if a == b: # N = Z 

573 b = c 

574 elif b == c: # N = X 

575 a, b = b, a 

576 elif a != c: 

577 t = _SPACE_(_a_, _NOTEQUAL_, _b_, _NOTEQUAL_, _c_) 

578 raise TriaxialError(a=a, b=b, c=c, txt=t) 

579 return Ellipsoid(a, b=b, name=name or self.name) 

580 

581 def toStr(self, prec=9, name=NN, **unused): # PYCHOK signature 

582 '''Return this C{Triaxial} as a string. 

583 

584 @kwarg prec: Precision, number of decimal digits (0..9). 

585 @kwarg name: Override name (C{str}) or C{None} to exclude 

586 this triaxial's name. 

587 

588 @return: This C{Triaxial}'s attributes (C{str}). 

589 ''' 

590 T = Triaxial_ 

591 t = T.a, T.b, T.c, T.e2ab, T.e2bc, T.e2ac 

592 if isinstance(self, JacobiConformal): 

593 t += JacobiConformal.xyQ2, 

594 t += T.volume, T.area 

595 return self._instr(name, prec, props=t, area_p=self.area_p()) 

596 

597 @Property_RO 

598 def volume(self): 

599 '''Get the volume (C{meter**3}), M{4 / 3 * PI * a * b * c}. 

600 ''' 

601 a, b, c = self._abc3 

602 return Meter3(volume=a * b * c * PI_3 * _4_0) 

603 

604 

605class Triaxial(Triaxial_): 

606 '''I{Ordered} triaxial ellipsoid. 

607 

608 @see: L{Triaxial_} for more information. 

609 ''' 

610 _unordered = False 

611 

612 def __init__(self, a_triaxial, b=None, c=None, name=NN): 

613 '''New I{ordered} L{Triaxial}. 

614 

615 @arg a_triaxial: Largest semi-axis (C{scalar}, conventionally in C{meter}) 

616 or an other L{Triaxial} or L{Triaxial_} instance. 

617 @kwarg b: Middle semi-axis (C{meter}, same units as B{C{a}}), required 

618 if C{B{a_triaxial} is scalar}, ignored otherwise. 

619 @kwarg c: Smallest semi-axis (C{meter}, same units as B{C{a}}), required 

620 if C{B{a_triaxial} is scalar}, ignored otherwise. 

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

622 

623 @note: The semi-axes must be ordered as C{B{a} >= B{b} >= B{c} > 0} and 

624 must be ellipsoidal, C{B{a} > B{c}}. 

625 

626 @raise TriaxialError: Semi-axes not ordered, spherical or invalid. 

627 ''' 

628 Triaxial_.__init__(self, a_triaxial, b=b, c=c, name=name) 

629 

630 @Property_RO 

631 def _a2b2_a2c2(self): 

632 '''@see: Methods C{.forwardBetaOmega} and C{._k2_kp2}. 

633 ''' 

634 return self._a2b2 / self._a2c2 

635 

636 @Property_RO 

637 def area(self): 

638 '''Get the surface area (C{meter} I{squared}). 

639 

640 @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Surface_area>}. 

641 ''' 

642 a, b, c = self._abc3 

643 if a != b: 

644 kp2, k2 = self._k2_kp2 # swapped! 

645 aE = self._Elliptic(k2, _0_0, kp2, _1_0) 

646 c2 = self._1e2ac # cos(phi)**2 == (c/a)**2 

647 s2 = self. e2ac # sin(phi)**2 == 1 - c2 

648 s = sqrt(s2) 

649 r = asin1(s) # phi == atan2(sqrt(c2), s) 

650 b *= fsum_(aE.fE(r) * s, c / a * c / b, 

651 aE.fF(r) * c2 / s, floats=True) 

652 a = Meter2(area=a * b * PI2) 

653 else: # a == b > c 

654 a = Ellipsoid(a, b=c).areax 

655 return a 

656 

657 def _exyz3(self, u): 

658 '''(INTERNAL) Helper for C{.forwardBetOmg}. 

659 ''' 

660 if u > 0: 

661 u2 = u**2 

662 x = u * self._sqrt(_1_0 + self._a2c2 / u2) 

663 y = u * self._sqrt(_1_0 + self._b2c2 / u2) 

664 else: 

665 x = y = u = _0_0 

666 return x, y, u 

667 

668 def forwardBetaOmega(self, beta, omega, height=0, name=NN): 

669 '''Convert I{ellipsoidal} lat- and longitude C{beta}, C{omega} 

670 and height to cartesian. 

671 

672 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

673 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

674 @arg height: Height above or below the ellipsoid's surface (C{meter}, same 

675 units as this triaxial's C{a}, C{b} and C{c} semi-axes). 

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

677 

678 @return: A L{Vector3Tuple}C{(x, y, z)}. 

679 

680 @see: Method L{Triaxial.reverseBetaOmega} and U{Expressions (23-25)<https:// 

681 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

682 ''' 

683 if height: 

684 h = Height_(height=height, low=-self.c, Error=TriaxialError) 

685 x, y, z = self._exyz3(h + self.c) 

686 else: 

687 x, y, z = self._abc3 # == self._exyz3(self.c) 

688 if z: # and x and y: 

689 sa, ca = SinCos2(beta) 

690 sb, cb = SinCos2(omega) 

691 

692 r = self._a2b2_a2c2 

693 x *= cb * self._sqrt(ca**2 + r * sa**2) 

694 y *= ca * sb 

695 z *= sa * self._sqrt(_1_0 - r * cb**2) 

696 return Vector3Tuple(x, y, z, name=name) 

697 

698 def forwardBetaOmega_(self, sbeta, cbeta, somega, comega, name=NN): 

699 '''Convert I{ellipsoidal} lat- and longitude C{beta} and C{omega} 

700 to cartesian coordinates I{on the triaxial's surface}. 

701 

702 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

703 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

704 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

705 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

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

707 

708 @return: A L{Vector3Tuple}C{(x, y, z)} on the surface. 

709 

710 @raise TriaxialError: This triaxial is near-spherical. 

711 

712 @see: Method L{Triaxial.reverseBetaOmega}, U{Triaxial ellipsoid coordinate 

713 system<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid# 

714 Triaxial_ellipsoid_coordinate_system>} and U{expressions (23-25)<https:// 

715 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

716 ''' 

717 t = self._radialTo3(sbeta, cbeta, somega, comega) 

718 return Vector3Tuple(*t, name=name) 

719 

720 def forwardCartesian(self, x_xyz, y=None, z=None, name=NN, **normal_eps): 

721 '''Project a cartesian on this triaxial. 

722 

723 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

724 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

725 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

726 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

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

728 @kwarg normal_eps: Optional keyword arguments C{B{normal}=True} and 

729 C{B{eps}=EPS}, see method L{Triaxial.height4}. 

730 

731 @see: Method L{Triaxial.height4} for further information and method 

732 L{Triaxial.reverseCartesian} to reverse the projection. 

733 ''' 

734 t = self.height4(x_xyz, y, z, **normal_eps) 

735 _ = t.rename(name) 

736 return t 

737 

738 def forwardLatLon(self, lat, lon, height=0, name=NN): 

739 '''Convert I{geodetic} lat-, longitude and heigth to cartesian. 

740 

741 @arg lat: Geodetic latitude (C{degrees}). 

742 @arg lon: Geodetic longitude (C{degrees}). 

743 @arg height: Height above the ellipsoid (C{meter}, same units 

744 as this triaxial's C{a}, C{b} and C{c} axes). 

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

746 

747 @return: A L{Vector3Tuple}C{(x, y, z)}. 

748 

749 @see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https:// 

750 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

751 ''' 

752 return self._forwardLatLon3(height, name, *sincos2d_(lat, lon)) 

753 

754 def forwardLatLon_(self, slat, clat, slon, clon, height=0, name=NN): 

755 '''Convert I{geodetic} lat-, longitude and heigth to cartesian. 

756 

757 @arg slat: Geodetic latitude C{sin(lat)} (C{scalar}). 

758 @arg clat: Geodetic latitude C{cos(lat)} (C{scalar}). 

759 @arg slon: Geodetic longitude C{sin(lon)} (C{scalar}). 

760 @arg clon: Geodetic longitude C{cos(lon)} (C{scalar}). 

761 @arg height: Height above the ellipsoid (C{meter}, same units 

762 as this triaxial's axes C{a}, C{b} and C{c}). 

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

764 

765 @return: A L{Vector3Tuple}C{(x, y, z)}. 

766 

767 @see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https:// 

768 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

769 ''' 

770 sa, ca = self._norm2(slat, clat) 

771 sb, cb = self._norm2(slon, clon) 

772 return self._forwardLatLon3(height, name, sa, ca, sb, cb) 

773 

774 def _forwardLatLon3(self, h, name, sa, ca, sb, cb): 

775 '''(INTERNAL) Helper for C{.forwardLatLon} and C{.forwardLatLon_}. 

776 ''' 

777 ca_x_sb = ca * sb 

778 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2 

779 t = fsum_(_1_0, -self.e2ac * sa**2, -self.e2ab * ca_x_sb**2, floats=True) 

780 n = self.a / self._sqrt(t) # prime vertical 

781 x = (h + n) * ca * cb 

782 y = (h + n * self._1e2ab) * ca_x_sb 

783 z = (h + n * self._1e2ac) * sa 

784 return Vector3Tuple(x, y, z, name=name) 

785 

786 @Property_RO 

787 def _k2_kp2(self): 

788 '''(INTERNAL) Get C{k2} and C{kp2} for C{._xE}, C{._yE} and C{.area}. 

789 ''' 

790 # k2 = a2b2 / a2c2 * c2_b2 

791 # kp2 = b2c2 / a2c2 * a2_b2 

792 # b2 = b**2 

793 # xE = Elliptic(k2, -a2b2 / b2, kp2, a2_b2) 

794 # yE = Elliptic(kp2, +b2c2 / b2, k2, c2_b2) 

795 # aE = Elliptic(kp2, 0, k2, 1) 

796 return (self._a2b2_a2c2 * self._c2_b2, 

797 self._b2c2 / self._a2c2 * self._a2_b2) 

798 

799 def _radialTo3(self, sbeta, cbeta, somega, comega): 

800 '''(INTERNAL) Convert I{ellipsoidal} lat- and longitude C{beta} and 

801 C{omega} to cartesian coordinates I{on the triaxial's surface}, 

802 also I{ordered} helper for C{.height4}. 

803 ''' 

804 sa, ca = self._norm2(sbeta, cbeta) 

805 sb, cb = self._norm2(somega, comega) 

806 

807 b2_a2 = self._1e2ab # == (b/a)**2 

808 c2_a2 = -self._1e2ac # == -(c/a)**2 

809 a2c2_a2 = self. e2ac # (a**2 - c**2) / a**2 == 1 - (c/a)**2 

810 

811 x = Fsum(_1_0, -b2_a2 * sa**2, c2_a2 * ca**2).fover(a2c2_a2) 

812 z = Fsum(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2) 

813 

814 x = self.a * cb * self._sqrt(x) 

815 y = self.b * ca * sb 

816 z = self.c * sa * self._sqrt(z) 

817 return x, y, z 

818 

819 def reverseBetaOmega(self, x_xyz, y=None, z=None, name=NN): 

820 '''Convert cartesian to I{ellipsoidal} lat- and longitude, C{beta}, C{omega} 

821 and height. 

822 

823 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

824 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

825 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

826 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

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

828 

829 @return: A L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and 

830 C{omega} in C{Radians} and (radial) C{height} in C{meter}, same 

831 units as this triaxial's axes. 

832 

833 @see: Methods L{Triaxial.forwardBetaOmega} and L{Triaxial.forwardBetaOmega_} 

834 and U{Expressions (21-22)<https://www.Topo.Auth.GR/wp-content/uploads/ 

835 sites/111/2021/12/09_Panou.pdf>}. 

836 ''' 

837 v = _otherV3d_(x_xyz, y, z) 

838 a, b, h = self._reverseLatLon3(v, atan2, v, self.forwardBetaOmega_) 

839 return BetaOmega3Tuple(Radians(beta=a), Radians(omega=b), h, name=name) 

840 

841 def reverseCartesian(self, x_xyz, y=None, z=None, h=0, normal=True, eps=_EPS2e4, name=NN): 

842 '''"Unproject" a cartesian on to a cartesion I{off} this triaxial's surface. 

843 

844 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

845 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

846 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

847 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

848 @arg h: Height above or below this triaxial's surface (C{meter}, same units 

849 as the axes). 

850 @kwarg normal: If C{True} the height is C{normal} to the surface, otherwise 

851 C{radially} to the center of this triaxial (C{bool}). 

852 @kwarg eps: Tolerance for surface test (C{scalar}). 

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

854 

855 @return: A L{Vector3Tuple}C{(x, y, z)}. 

856 

857 @raise TrialError: Cartesian C{(x, y, z)} not on this triaxial's surface. 

858 

859 @see: Methods L{Triaxial.forwardCartesian} and L{Triaxial.height4}. 

860 ''' 

861 v = _otherV3d_(x_xyz, y, z, name=name) 

862 s = _sideOf(v.xyz, self._abc3, eps=eps) 

863 if s: # PYCHOK no cover 

864 t = _SPACE_((_inside_ if s < 0 else _outside_), self.toRepr()) 

865 raise TriaxialError(eps=eps, sideOf=s, x=v.x, y=v.y, z=v.z, txt=t) 

866 

867 if h: 

868 if normal: 

869 v = v.plus(self.normal3d(*v.xyz, length=h)) 

870 elif v.length > EPS0: 

871 v = v.times(_1_0 + (h / v.length)) 

872 return v.xyz # Vector3Tuple 

873 

874 def reverseLatLon(self, x_xyz, y=None, z=None, name=NN): 

875 '''Convert cartesian to I{geodetic} lat-, longitude and height. 

876 

877 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

878 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

879 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

880 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

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

882 

883 @return: A L{LatLon3Tuple}C{(lat, lon, height)} with C{lat} and C{lon} 

884 in C{degrees} and (radial) C{height} in C{meter}, same units 

885 as this triaxial's axes. 

886 

887 @see: Methods L{Triaxial.forwardLatLon} and L{Triaxial.forwardLatLon_} 

888 and U{Expressions (4-5)<https://www.Topo.Auth.GR/wp-content/uploads/ 

889 sites/111/2021/12/09_Panou.pdf>}. 

890 ''' 

891 v = _otherV3d_(x_xyz, y, z) 

892 s = v.times_(self._1e2ac, # == 1 - e_sub_x**2 

893 self._1e2bc, # == 1 - e_sub_y**2 

894 _1_0) 

895 t = self._reverseLatLon3(s, atan2d, v, self.forwardLatLon_) 

896 return LatLon3Tuple(*t, name=name) 

897 

898 def _reverseLatLon3(self, s, atan2_, v, forward_): 

899 '''(INTERNAL) Helper for C{.reverseBetOmg} and C{.reverseLatLon}. 

900 ''' 

901 x, y, z = s.xyz 

902 d = hypot( x, y) 

903 a = atan2_(z, d) 

904 b = atan2_(y, x) 

905 h = v.minus_(*forward_(z, d, y, x)).length 

906 return a, b, h 

907 

908 

909class JacobiConformal(Triaxial): 

910 '''This is a conformal projection of a triaxial ellipsoid to a plane in which the 

911 C{X} and C{Y} grid lines are straight. 

912 

913 Ellipsoidal coordinates I{beta} and I{omega} are converted to Jacobi Conformal 

914 I{y} respectively I{x} separately. Jacobi's coordinates have been multiplied 

915 by C{sqrt(B{a}**2 - B{c}**2) / (2 * B{b})} so that the customary results are 

916 returned in the case of an ellipsoid of revolution (or a sphere, I{currently 

917 not supported}). 

918 

919 Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2020) and 

920 licensed under the MIT/X11 License. 

921 

922 @note: This constructor can not be used to specify a sphere. 

923 

924 @see: L{Triaxial}, C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/ 

925 C++/doc/classGeographicLib_1_1JacobiConformal.html#details>}, U{Jacobi's conformal 

926 projection<https://GeographicLib.SourceForge.io/C++/doc/jacobi.html>} and Jacobi, 

927 C. G. J. I{U{Vorlesungen über Dynamik<https://Books.Google.com/books? 

928 id=ryEOAAAAQAAJ&pg=PA212>}}, page 212ff, 

929 ''' 

930# @Property_RO 

931# def ab(self): 

932# '''Get relative magnitude C{ab} (C{None} or C{meter}, same units as B{C{a}}). 

933# ''' 

934# return self._ab 

935 

936# @Property_RO 

937# def bc(self): 

938# '''Get relative magnitude C{bc} (C{None} or C{meter}, same units as B{C{a}}). 

939# ''' 

940# return self._bc 

941 

942 @Property_RO 

943 def _xE(self): 

944 '''(INTERNAL) Get the x-elliptic function. 

945 ''' 

946 k2, kp2 = self._k2_kp2 

947 # -a2b2 / b2 == (b2 - a2) / b2 == 1 - a2 / b2 == 1 - a2_b2 

948 return self._Elliptic(k2, _1_0 - self._a2_b2, kp2, self._a2_b2) 

949 

950 def xR(self, omega): 

951 '''Compute a Jacobi Conformal C{x} projection. 

952 

953 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

954 

955 @return: The C{x} projection (C{Radians}). 

956 ''' 

957 return self.xR_(*SinCos2(omega)) 

958 

959 def xR_(self, somega, comega): 

960 '''Compute a Jacobi Conformal C{x} projection. 

961 

962 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

963 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

964 

965 @return: The C{x} projection (C{Radians}). 

966 ''' 

967 s, c = self._norm2(somega, comega, self.a) 

968 return Radians(x=self._xE.fPi(s, c) * self._a2_b2) 

969 

970 @Property_RO 

971 def xyQ2(self): 

972 '''Get the Jacobi Conformal quadrant size (L{Jacobi2Tuple}C{(x, y)}). 

973 ''' 

974 return Jacobi2Tuple(Radians(x=self._a2_b2 * self._xE.cPi), 

975 Radians(y=self._c2_b2 * self._yE.cPi), 

976 name=JacobiConformal.xyQ2.name) 

977 

978 def xyR2(self, beta, omega, name=NN): 

979 '''Compute a Jacobi Conformal C{x} and C{y} projection. 

980 

981 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

982 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

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

984 

985 @return: A L{Jacobi2Tuple}C{(x, y)}. 

986 ''' 

987 return self.xyR2_(*(SinCos2(beta) + SinCos2(omega)), 

988 name=name or self.xyR2.__name__) 

989 

990 def xyR2_(self, sbeta, cbeta, somega, comega, name=NN): 

991 '''Compute a Jacobi Conformal C{x} and C{y} projection. 

992 

993 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

994 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

995 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

996 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

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

998 

999 @return: A L{Jacobi2Tuple}C{(x, y)}. 

1000 ''' 

1001 return Jacobi2Tuple(self.xR_(somega, comega), 

1002 self.yR_(sbeta, cbeta), 

1003 name=name or self.xyR2_.__name__) 

1004 

1005 @Property_RO 

1006 def _yE(self): 

1007 '''(INTERNAL) Get the x-elliptic function. 

1008 ''' 

1009 kp2, k2 = self._k2_kp2 # swapped! 

1010 # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc 

1011 return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2) 

1012 

1013 def yR(self, beta): 

1014 '''Compute a Jacobi Conformal C{y} projection. 

1015 

1016 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

1017 

1018 @return: The C{y} projection (C{Radians}). 

1019 ''' 

1020 return self.yR_(*SinCos2(beta)) 

1021 

1022 def yR_(self, sbeta, cbeta): 

1023 '''Compute a Jacobi Conformal C{y} projection. 

1024 

1025 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

1026 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

1027 

1028 @return: The C{y} projection (C{Radians}). 

1029 ''' 

1030 s, c = self._norm2(sbeta, cbeta, self.c) 

1031 return Radians(y=self._yE.fPi(s, c) * self._c2_b2) 

1032 

1033 

1034class TriaxialError(_ValueError): 

1035 '''Raised for L{Triaxial} issues. 

1036 ''' 

1037 pass # ... 

1038 

1039 

1040class Triaxials(_NamedEnum): 

1041 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class 

1042 to accommodate the L{_LazyNamedEnumItem} properties. 

1043 ''' 

1044 def _Lazy(self, *abc, **name): 

1045 '''(INTERNAL) Instantiate the C{Triaxial}. 

1046 ''' 

1047 a, b, c = map(km2m, abc) 

1048 return Triaxial(a, b, c, **name) 

1049 

1050Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton 

1051'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.''' 

1052# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in km 

1053# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf> 

1054# <https://link.Springer.com/article/10.1007/s00190-022-01650-9> 

1055_E = _WGS84.ellipsoid 

1056Triaxials._assert( # a (km) b (km) c (km) planet 

1057 Amalthea = _lazy('Amalthea', 125.0, 73.0, 64), # Jupiter 

1058 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus 

1059 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544), 

1060 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn 

1061 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter 

1062 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter 

1063 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3), 

1064 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn 

1065 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus 

1066 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth 

1067 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn 

1068 WGS84_35 = _lazy('WGS84_35', *map1(m2km, _E.a + 35, _E.a - 35, _E.b))) 

1069del _E 

1070 

1071 

1072def _getitems(items, *indices): 

1073 '''(INTERNAL) Get the C{items} at the given I{indices}. 

1074 

1075 @return: C{Type(items[i] for i in indices)} with 

1076 C{Type = type(items)}, any C{type} having 

1077 the special method C{__getitem__}. 

1078 ''' 

1079 return type(items)(map(items.__getitem__, indices)) 

1080 

1081 

1082def _hartzell3d2(pov, los, Tun): # MCCABE 13 in .ellipsoidal.hartzell4, .formy.hartzell 

1083 '''(INTERNAL) Hartzell's "Satellite Line-of-Sight Intersection ...", 

1084 formula for I{un-/ordered} triaxials. 

1085 ''' 

1086 a, b, c, T = Tun._ordered4 

1087 

1088 a2 = a**2 # largest, factored out 

1089 b2, p2 = (b**2, T._1e2ab) if b != a else (a2, _1_0) 

1090 c2, q2 = (c**2, T._1e2ac) if c != a else (a2, _1_0) 

1091 

1092 p3 = T._order3d(_otherV3d(pov=pov)) 

1093 u3 = T._order3d(_otherV3d(los=los)) if los else p3.negate() 

1094 u3 = u3.unit() # unit vector, opposing signs 

1095 

1096 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz 

1097 ux, vy, wz = u3.times_(p3).xyz 

1098 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

1099 

1100 t = (p2 * c2), c2, b2 

1101 m = fdot(t, u2, v2, w2) # a2 factored out 

1102 if m < EPS0: # zero or near-null LOS vector 

1103 raise _ValueError(_near_(_null_)) 

1104 

1105 r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

1106 -w2 * y2, b2 * u2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2, 

1107 -w2 * x2 * p2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True) 

1108 if r > 0: # a2 factored out 

1109 r = sqrt(r) * b * c # == a * a * b * c / a2 

1110 elif r < 0: # LOS pointing away from or missing the triaxial 

1111 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

1112 

1113 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

1114 if d > 0: # POV inside or LOS missing, outside the triaxial 

1115 s = fsum_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0, floats=True) # like _sideOf 

1116 raise _ValueError(_outside_ if s > 0 else _inside_) 

1117 elif fsum_(x2, y2, z2, floats=True) < d**2: # d past triaxial's center 

1118 raise _ValueError(_too_(_distant_)) 

1119 

1120 v = p3.minus(u3.times(d)) # Vector3d 

1121 h = p3.minus(v).length # distance to triaxial 

1122 return T._order3d(v, reverse=True), h 

1123 

1124 

1125def hartzell4(pov, los=None, tri_biax=_WGS84, name=NN): 

1126 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight 

1127 from a Point-Of-View outside. 

1128 

1129 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple} 

1130 or L{Vector3d}). 

1131 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Vector3d}) or 

1132 C{None} to point to the tri-/biaxial's center. 

1133 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal}) 

1134 or biaxial ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2}, 

1135 L{a_f2Tuple} or C{scalar} radius, conventionally in C{meter}). 

1136 @kwarg name: Optional name (C{str}). 

1137 

1138 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h} 

1139 the distance from B{C{pov}} to C{(x, y, z)} along B{C{los}}, all in 

1140 C{meter}, conventionally. 

1141 

1142 @raise TriaxialError: Null B{C{pov}} or B{C{los}}, or B{C{pov}} is inside the 

1143 tri-/biaxial or B{C{los}} points outside the tri-/biaxial 

1144 or points in an opposite direction. 

1145 

1146 @raise TypeError: Invalid B{C{pov}} or B{C{los}}. 

1147 

1148 @see: Function L{pygeodesy.hartzell}, L{pygeodesy.tyr3d} for B{C{los}} and 

1149 U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell. 

1150 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

1151 ''' 

1152 if isinstance(tri_biax, Triaxial_): 

1153 T = tri_biax 

1154 else: 

1155 D = tri_biax if isinstance(tri_biax, Datum) else \ 

1156 _spherical_datum(tri_biax, name=hartzell4.__name__) 

1157 T = D.ellipsoid._triaxial 

1158 

1159 try: 

1160 v, h = _hartzell3d2(pov, los, T) 

1161 except Exception as x: 

1162 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x) 

1163 return Vector4Tuple(v.x, v.y, v.z, h, name=name or hartzell4.__name__) 

1164 

1165 

1166def _normalTo4(x, y, a, b, eps=EPS): # MCCABE 14 

1167 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}. 

1168 

1169 @see: Function C{pygeodesy.ellipsoids._normalTo3} and I{Eberly}'s U{Distance 

1170 from a Point to ... an Ellipsoid ...<https://www.GeometricTools.com/ 

1171 Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1172 ''' 

1173 def _root2d(r, u, v, g, eps): 

1174 # robust root finder 

1175 _1, __2 = _1_0, _0_5 

1176 _a, _h2 = fabs, _hypot21_ 

1177 u *= r 

1178 t0 = v - _1 

1179 t1 = _0_0 if g < 0 else _h2(u, v) 

1180 for i in range(1, _TRIPS): 

1181 t = (t0 + t1) * __2 

1182 if t in (t0, t1) or _a(t0 - t1) < eps: 

1183 break 

1184 g = _h2(u / (t + r), v / (t + _1)) 

1185 if g > 0: 

1186 t0 = t 

1187 elif g < 0: 

1188 t1 = t 

1189 else: 

1190 break 

1191 else: # PYCHOK no cover 

1192 e = _a(t0 - t1) 

1193 t = _root2d.__name__ 

1194 raise _ValueError(Fmt.no_convergence(e, eps), txt=t) 

1195 return t, i 

1196 

1197 if a < b: 

1198 b, a, d, i = _normalTo4(y, x, b, a, eps=eps) 

1199 return a, b, d, i 

1200 

1201 if not (isfinite(a) and b > 0): 

1202 raise _ValueError(a=a, b=b) 

1203 

1204 i = None 

1205 if y: 

1206 if x: 

1207 u = fabs(x / a) 

1208 v = fabs(y / b) 

1209 g = _hypot21_(u, v) 

1210 if g: 

1211 r = (a / b)**2 

1212 t, i = _root2d(r, u, v, g, eps) 

1213 a = x / (t / r + _1_0) 

1214 b = y / (t + _1_0) 

1215 d = hypot(x - a, y - b) 

1216 else: # on the ellipse 

1217 a, b, d = x, y, _0_0 

1218 else: # x == 0 

1219 if y < 0: 

1220 b = -b 

1221 a, d = x, fabs(y - b) 

1222 

1223 else: # y == 0 

1224 n = a * x 

1225 d = (a + b) * (a - b) 

1226 if d > fabs(n): # PYCHOK no cover 

1227 r = n / d 

1228 a *= r 

1229 b *= sqrt(_1_0 - r**2) 

1230 d = hypot(x - a, b) 

1231 else: 

1232 if x < 0: 

1233 a = -a 

1234 b, d = y, fabs(x - a) 

1235 return a, b, d, i 

1236 

1237 

1238def _normalTo5(x, y, z, Tun, eps=EPS): # MCCABE 24 

1239 '''(INTERNAL) Nearest point on and distance to an I{un- or ordered} triaxial. 

1240 

1241 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https:// 

1242 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1243 ''' 

1244 def _root3d(r, s, u, v, w, g, eps): 

1245 # robust root finder 

1246 _1, __2 = _1_0, _0_5 

1247 _a, _h2 = fabs, _hypot21_ 

1248 u *= r 

1249 v *= s 

1250 t0 = w - _1 

1251 t1 = _0_0 if g < 0 else _h2(u, v, w) 

1252 for i in range(1, _TRIPS): 

1253 t = (t0 + t1) * __2 

1254 if t in (t0, t1) or _a(t0 - t1) < eps: 

1255 break 

1256 g = _h2(u / (t + r), v / (t + s), w / (t + _1)) 

1257 if g > 0: 

1258 t0 = t 

1259 elif g < 0: 

1260 t1 = t 

1261 else: 

1262 break 

1263 else: # PYCHOK no cover 

1264 e = _a(t0 - t1) 

1265 t = _root3d.__name__ 

1266 raise _ValueError(Fmt.no_convergence(e, eps), txt=t) 

1267 return t, i 

1268 

1269 a, b, c, T = Tun._ordered4 

1270 if Tun is not T: # T is ordered, Tun isn't 

1271 t = T._order3(x, y, z) + (T,) 

1272 a, b, c, d, i = _normalTo5(*t, eps=eps) 

1273 return T._order3(a, b, c, reverse=True) + (d, i) 

1274 

1275 if not (isfinite(a) and c > 0): 

1276 raise _ValueError(a=a, b=b, c=c) 

1277 

1278 if eps > 0: 

1279 val = max(eps * 1e8, EPS) 

1280 else: # no validation 

1281 val, eps = 0, -eps 

1282 

1283 i = None 

1284 if z: 

1285 if y: 

1286 if x: 

1287 u = fabs(x / a) 

1288 v = fabs(y / b) 

1289 w = fabs(z / c) 

1290 g = _hypot21_(u, v, w) 

1291 if g: 

1292 r = T._1e2ac # (c / a)**2 

1293 s = T._1e2bc # (c / b)**2 

1294 t, i = _root3d(_1_0 / r, _1_0 / s, u, v, w, g, eps) 

1295 a = x / (t * r + _1_0) 

1296 b = y / (t * s + _1_0) 

1297 c = z / (t + _1_0) 

1298 d = hypot_(x - a, y - b, z - c) 

1299 else: # on the ellipsoid 

1300 a, b, c, d = x, y, z, _0_0 

1301 else: # x == 0 

1302 a = x # 0 

1303 b, c, d, i = _normalTo4(y, z, b, c, eps=eps) 

1304 elif x: # y == 0 

1305 b = y # 0 

1306 a, c, d, i = _normalTo4(x, z, a, c, eps=eps) 

1307 else: # x == y == 0 

1308 if z < 0: 

1309 c = -c 

1310 a, b, d = x, y, fabs(z - c) 

1311 

1312 else: # z == 0 

1313 t = False 

1314 n = a * x 

1315 d = T._a2c2 # (a + c) * (a - c) 

1316 if d > fabs(n): 

1317 u = n / d 

1318 n = b * y 

1319 d = T._b2c2 # (b + c) * (b - c) 

1320 if d > fabs(n): 

1321 v = n / d 

1322 n = _hypot21_(u, v) 

1323 if n < 0: 

1324 a *= u 

1325 b *= v 

1326 c *= sqrt(-n) 

1327 d = hypot_(x - a, y - b, c) 

1328 t = True 

1329 if not t: 

1330 c = z # 0 

1331 a, b, d, i = _normalTo4(x, y, a, b, eps=eps) 

1332 

1333 if val > 0: # validate 

1334 e = T.sideOf(a, b, c, eps=val) 

1335 if e: # not near the ellipsoid's surface 

1336 raise _ValueError(a=a, b=b, c=c, d=d, 

1337 sideOf=e, eps=val) 

1338 if d: # angle of delta and normal vector 

1339 m = Vector3d(x, y, z).minus_(a, b, c) 

1340 if m.euclid > val: 

1341 m = m.unit() 

1342 n = T.normal3d(a, b, c) 

1343 e = n.dot(m) # n.negate().dot(m) 

1344 if not isnear1(fabs(e), eps1=val): 

1345 raise _ValueError(n=n, m=m, 

1346 dot=e, eps=val) 

1347 return a, b, c, d, i 

1348 

1349 

1350def _otherV3d_(x_xyz, y, z, name=NN): 

1351 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}. 

1352 ''' 

1353 return Vector3d(x_xyz, y, z, name=name) if isscalar(x_xyz) else \ 

1354 _otherV3d(x_xyz=x_xyz) 

1355 

1356 

1357def _sideOf(xyz, abc, eps=EPS): # in .formy 

1358 '''(INTERNAL) Helper for C{_hartzell3d2}, M{.sideOf} and M{.reverseCartesian}. 

1359 

1360 @return: M{sum((x / a)**2 for x, a in zip(xyz, abc)) - 1} or C{INT0}, 

1361 ''' 

1362 s = _hypot21_(*((x / a) for x, a in _zip(xyz, abc) if a)) # strict=True 

1363 return s if fabs(s) > eps else INT0 

1364 

1365 

1366if __name__ == '__main__': 

1367 

1368 from pygeodesy import printf 

1369 

1370 # __doc__ of this file, force all into registery 

1371 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_) 

1372 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t)) 

1373 

1374# **) MIT License 

1375# 

1376# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1377# 

1378# Permission is hereby granted, free of charge, to any person obtaining a 

1379# copy of this software and associated documentation files (the "Software"), 

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1384# 

1385# The above copyright notice and this permission notice shall be included 

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1387# 

1388# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

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