Coverage for pygeodesy/triaxials.py: 96%

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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 

30# from pygeodesy.basics import isscalar, _zip # from .fsums, .namedTuples, .streprs 

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

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

33 _4_0 # PYCHOK used! 

34from pygeodesy.datums import Datum, Ellipsoid, _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 .streprs 

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

39from pygeodesy.fsums import Fsum, fsum_, isscalar, 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 

49from pygeodesy.streprs import Fmt, _ValueError, _zip 

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

51from pygeodesy.utily import asin1, atan2d, km2m, m2km, sincos2, sincos2d, sincos2d_ 

52from pygeodesy.vector3d import _ALL_LAZY, _MODS, _otherV3d, Vector3d 

53 

54from math import atan2, fabs, sqrt 

55 

56__all__ = _ALL_LAZY.triaxials 

57__version__ = '23.04.02' 

58 

59_E = _WGS84.ellipsoid 

60_not_ordered_ = _not_('ordered') 

61_omega_ = 'omega' 

62_TRIPS = 537 # max 55, Eberly 1074? 

63_WGS84_35abc = _E.a + 35, _E.a - 35, _E.b 

64del _E 

65 

66 

67class _ToNamedBase(_NamedBase): 

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

69 ''' 

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

71 if toDMS_kwds: 

72 toDMS = _MODS.dms.toDMS 

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

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

75 elif isinstance(a, Degrees) and \ 

76 isinstance(b, Degrees): 

77 return self 

78 else: 

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

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

81 

82 def _toRadians(self, a, b, *c): 

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

84 isinstance(b, Radians) else \ 

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

86 *c, name=self.name) 

87 

88 

89class BetaOmega2Tuple(_NamedTuple, _ToNamedBase): 

90 '''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and 

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

92 C{Degrees}). 

93 ''' 

94 _Names_ = (_beta_, _omega_) 

95 _Units_ = (_Pass, _Pass) 

96 

97 def toDegrees(self, **toDMS_kwds): 

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

99 

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

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

102 or as an L{toDMS} string provided some 

103 B{C{toDMS_kwds}} are supplied. 

104 ''' 

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

106 

107 def toRadians(self): 

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

109 

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

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

112 ''' 

113 return _ToNamedBase._toRadians(self, *self) 

114 

115 

116class BetaOmega3Tuple(_NamedTuple, _ToNamedBase): 

117 '''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and 

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

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

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

121 in C{meter}, conventionally. 

122 ''' 

123 _Names_ = BetaOmega2Tuple._Names_ + (_height_,) 

124 _Units_ = BetaOmega2Tuple._Units_ + ( Meter,) 

125 

126 def toDegrees(self, **toDMS_kwds): 

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

128 

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

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

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

132 are supplied. 

133 ''' 

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

135 

136 def toRadians(self): 

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

138 

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

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

141 ''' 

142 return _ToNamedBase._toRadians(self, *self) 

143 

144 def to2Tuple(self): 

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

146 ''' 

147 return BetaOmega2Tuple(*self[:2]) 

148 

149 

150class Jacobi2Tuple(_NamedTuple, _ToNamedBase): 

151 '''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y} 

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

153 ''' 

154 _Names_ = (_x_, _y_) 

155 _Units_ = (_Pass, _Pass) 

156 

157 def toDegrees(self, **toDMS_kwds): 

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

159 

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

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

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

163 ''' 

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

165 

166 def toRadians(self): 

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

168 

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

170 and C{y} both in C{Radians}. 

171 ''' 

172 return _ToNamedBase._toRadians(self, *self) 

173 

174 

175class Triaxial_(_NamedEnumItem): 

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

177 

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

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

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

181 I{omega}=0. 

182 

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

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

185 

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

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

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

189 C{Degrees} if converted). 

190 ''' 

191 _ijk = _kji = None 

192 _unordered = True 

193 

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

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

196 

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

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

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

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

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

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

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

204 

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

206 ''' 

207 try: 

208 a = a_triaxial 

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

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

211 except (TypeError, ValueError) as x: 

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

213 if name: 

214 self.name = name 

215 

216 a, b, c = self._abc3 = t 

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

218 s, _, t = sorted(t) 

219 if not (isfinite(t) and s > 0): 

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

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

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

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

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

225 

226 def __str__(self): 

227 return self.toStr() 

228 

229 @Property_RO 

230 def a(self): 

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

232 ''' 

233 a, _, _ = self._abc3 

234 return a 

235 

236 @Property_RO 

237 def _a2b2(self): 

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

239 ''' 

240 a, b, _ = self._abc3 

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

242 

243 @Property_RO 

244 def _a2_b2(self): 

245 '''(INTERNAL) Get C{(a/b)**2}. 

246 ''' 

247 a, b, _ = self._abc3 

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

249 

250 @Property_RO 

251 def _a2c2(self): 

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

253 ''' 

254 a, _, c = self._abc3 

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

256 

257 @Property_RO 

258 def area(self): 

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

260 ''' 

261 c, b, a = sorted(self._abc3) 

262 if a > c: 

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

264 Ellipsoid(a, b=c).areax # a == b 

265 else: # a == c == b 

266 a = Meter2(area=a**2 * PI4) 

267 return a 

268 

269 def area_p(self, p=1.6075): 

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

271 

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

273 for "near-flat" triaxials. 

274 

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

276 ''' 

277 a, b, c = self._abc3 

278 if a == b == c: 

279 a *= a 

280 else: 

281 _p = pow 

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

283 return Meter2(area_p=a * PI4) 

284 

285 @Property_RO 

286 def b(self): 

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

288 ''' 

289 _, b, _ = self._abc3 

290 return b 

291 

292 @Property_RO 

293 def _b2c2(self): 

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

295 ''' 

296 _, b, c = self._abc3 

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

298 

299 @Property_RO 

300 def c(self): 

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

302 ''' 

303 _, _, c = self._abc3 

304 return c 

305 

306 @Property_RO 

307 def _c2_b2(self): 

308 '''(INTERNAL) Get C{(c/b)**2}. 

309 ''' 

310 _, b, c = self._abc3 

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

312 

313 @Property_RO 

314 def e2ab(self): 

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

316 ''' 

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

318 

319 @Property_RO 

320 def _1e2ab(self): 

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

322 ''' 

323 a, b, _ = self._abc3 

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

325 

326 @Property_RO 

327 def e2bc(self): 

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

329 ''' 

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

331 

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

333 

334 @Property_RO 

335 def e2ac(self): 

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

337 ''' 

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

339 

340 @Property_RO 

341 def _1e2ac(self): 

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

343 ''' 

344 a, _, c = self._abc3 

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

346 

347 @Property_RO 

348 def _Elliptic(self): 

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

350 ''' 

351 return _MODS.elliptic.Elliptic 

352 

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

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

355 from a Point-Of-View in space. 

356 

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

358 ''' 

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

360 

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

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

363 this triaxial's surface. 

364 

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

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

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

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

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

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

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

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

373 negative value to skip validation. 

374 

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

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

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

378 surface in C{meter}, conventionally. 

379 

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

381 convergence in root finding or validation failed. 

382 

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

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

385 DistancePointEllipseEllipsoid.pdf>}. 

386 ''' 

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

388 

389 i, h = None, v.length 

390 if h < EPS0: # EPS 

391 x = y = z = _0_0 

392 h -= min(self._abc3) # nearest 

393 elif r: # .isSpherical 

394 x, y, z = v.times(r / h).xyz 

395 h -= r 

396 else: 

397 x, y, z = v.xyz 

398 try: 

399 if normal: # perpendicular to triaxial 

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

401 else: # radially to triaxial's center 

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

403 h = v.minus_(x, y, z).length 

404 except Exception as e: 

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

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

407 h = -h # below the surface 

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

409 

410 @Property_RO 

411 def isOrdered(self): 

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

413 ''' 

414 a, b, c = self._abc3 

415 return bool(a >= b > c) # b > c! 

416 

417 @Property_RO 

418 def isSpherical(self): 

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

420 ''' 

421 a, b, c = self._abc3 

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

423 

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

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

426 

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

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

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

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

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

432 

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

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

435 

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

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

438 ''' 

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

440 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered 

441 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2 

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

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

444 if n.length < EPS0: 

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

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

447 

448 @Property_RO 

449 def _normal3d(self): 

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

451 ''' 

452 d = max(self._abc3) 

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

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

455 

456 def _norm2(self, s, c, *a): 

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

458 ''' 

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

460 fabs(_hypot21_(s, c)) > EPS0: 

461 s, c = norm2(s, c) 

462 if a: 

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

464 return (s or _0_0), (c or _0_0) 

465 

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

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

468 

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

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

471 ''' 

472 ijk = self._order_ijk(**reverse) 

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

474 

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

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

477 

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

479 ''' 

480 ijk = self._order_ijk(**reverse) 

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

482 

483 @Property_RO 

484 def _ordered4(self): 

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

486 ''' 

487 def _order2(reverse, a, b, c): 

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

489 

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

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

492 ''' 

493 i, j, k = 0, 1, 2 

494 if a < b: 

495 a, b, i, j = b, a, j, i 

496 if a < c: 

497 a, c, i, k = c, a, k, i 

498 if b < c: 

499 b, c, j, k = c, b, k, j 

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

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

502 return (a, b, c), ijk 

503 

504 abc, T = self._abc3, self 

505 if not self.isOrdered: 

506 abc, ijk = _order2(False, *abc) 

507 if ijk: 

508 _, kji = _order2(True, *ijk) 

509 T = Triaxial_(*abc) 

510 T._ijk, T._kji = ijk, kji 

511 return abc + (T,) 

512 

513 def _order_ijk(self, reverse=False): 

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

515 ''' 

516 return self._kji if reverse else self._ijk 

517 

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

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

520 ''' 

521 def _rphi(a, b, sphi, cphi): 

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

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

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

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

526 

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

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

529 

530 a, b, c = self._abc3 

531 if a != b: 

532 a = _rphi(a, b, sb, cb) 

533 if a != c: 

534 c = _rphi(a, c, sa, ca) 

535 z, r = c * sa, c * ca 

536 x, y = r * cb, r * sb 

537 return x, y, z 

538 

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

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

541 

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

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

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

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

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

547 

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

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

550 if in- respectively outside this triaxial. 

551 

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

553 ''' 

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

555 

556 def _sqrt(self, x): 

557 '''(INTERNAL) Helper. 

558 ''' 

559 if x < 0: 

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

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

562 

563 def toEllipsoid(self, name=NN): 

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

565 

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

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

568 

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

570 

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

572 ''' 

573 a, b, c = self._abc3 

574 if a == b: # N = Z 

575 b = c 

576 elif b == c: # N = X 

577 a, b = b, a 

578 elif a != c: 

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

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

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

582 

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

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

585 

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

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

588 this triaxial's name. 

589 

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

591 ''' 

592 T = Triaxial_ 

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

594 if isinstance(self, JacobiConformal): 

595 t += JacobiConformal.xyQ2, 

596 t += T.volume, T.area 

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

598 

599 @Property_RO 

600 def volume(self): 

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

602 ''' 

603 a, b, c = self._abc3 

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

605 

606 

607class Triaxial(Triaxial_): 

608 '''I{Ordered} triaxial ellipsoid. 

609 

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

611 ''' 

612 _unordered = False 

613 

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

615 '''New I{ordered} L{Triaxial}. 

616 

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

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

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

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

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

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

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

624 

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

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

627 

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

629 ''' 

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

631 

632 @Property_RO 

633 def _a2b2_a2c2(self): 

634 ''' @see: Method C{forwardBetaOmega}. 

635 ''' 

636 return self._a2b2 / self._a2c2 

637 

638 @Property_RO 

639 def area(self): 

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

641 

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

643 ''' 

644 a, b, c = self._abc3 

645 if a != b: 

646 kp2, k2 = self._k2_kp2 # swapped! 

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

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

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

650 s = sqrt(s2) 

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

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

653 aE.fF(r) * c2 / s, floats=True) 

654 a = Meter2(area=a * b * PI2) 

655 else: # a == b > c 

656 a = Ellipsoid(a, b=c).areax 

657 return a 

658 

659 def _exyz3(self, u): 

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

661 ''' 

662 if u > 0: 

663 u2 = u**2 

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

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

666 else: 

667 x = y = u = _0_0 

668 return x, y, u 

669 

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

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

672 and height to cartesian. 

673 

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

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

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

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

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

679 

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

681 

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

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

684 ''' 

685 if height: 

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

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

688 else: 

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

690 if z: # and x and y: 

691 sa, ca = _SinCos2(beta) 

692 sb, cb = _SinCos2(omega) 

693 

694 r = self._a2b2_a2c2 

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

696 y *= ca * sb 

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

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

699 

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

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

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

703 

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

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

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

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

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

709 

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

711 

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

713 

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

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

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

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

718 ''' 

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

720 return Vector3Tuple(*t, name=name) 

721 

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

723 '''Project a cartesian on this triaxial. 

724 

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

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

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

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

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

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

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

732 

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

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

735 ''' 

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

737 _ = t.rename(name) 

738 return t 

739 

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

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

742 

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

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

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

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

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

748 

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

750 

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

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

753 ''' 

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

755 

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

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

758 

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

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

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

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

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

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

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

766 

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

768 

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

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

771 ''' 

772 sa, ca = self._norm2(slat, clat) 

773 sb, cb = self._norm2(slon, clon) 

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

775 

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

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

778 ''' 

779 ca_x_sb = ca * sb 

780 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2 

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

782 N = self.a / self._sqrt(t) # prime vertical 

783 x = (h + N) * ca * cb 

784 y = (h + N * self._1e2ab) * ca_x_sb 

785 z = (h + N * self._1e2ac) * sa 

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

787 

788 @Property_RO 

789 def _k2_kp2(self): 

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

791 ''' 

792 # k2 = a2b2 / a2c2 * c2_b2 

793 # kp2 = b2c2 / a2c2 * a2_b2 

794 # b2 = b**2 

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

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

797 # aE = Elliptic(kp2, 0, k2, 1) 

798 return (self._a2b2 / self._a2c2 * self._c2_b2, 

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

800 

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

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

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

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

805 ''' 

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

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

808 

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

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

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

812 

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

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

815 

816 x = self.a * cb * self._sqrt(x) 

817 y = self.b * ca * sb 

818 z = self.c * sa * self._sqrt(z) 

819 return x, y, z 

820 

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

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

823 and height. 

824 

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

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

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

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

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

830 

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

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

833 units as this triaxial's axes. 

834 

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

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

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

838 ''' 

839 v = _otherV3d_(x_xyz, y, z) 

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

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

842 

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

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

845 

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

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

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

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

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

851 as the axes). 

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

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

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

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

856 

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

858 

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

860 

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

862 ''' 

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

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

865 if s: # PYCHOK no cover 

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

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

868 

869 if h: 

870 if normal: 

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

872 elif v.length > EPS0: 

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

874 return v.xyz # Vector3Tuple 

875 

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

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

878 

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

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

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

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

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

884 

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

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

887 as this triaxial's axes. 

888 

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

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

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

892 ''' 

893 v = _otherV3d_(x_xyz, y, z) 

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

895 self._1e2bc, # == 1 - e_sub_y**2 

896 _1_0) 

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

898 return LatLon3Tuple(*t, name=name) 

899 

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

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

902 ''' 

903 x, y, z = s.xyz 

904 d = hypot( x, y) 

905 a = atan2_(z, d) 

906 b = atan2_(y, x) 

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

908 return a, b, h 

909 

910 

911class JacobiConformal(Triaxial): 

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

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

914 

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

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

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

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

919 not supported}). 

920 

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

922 licensed under the MIT/X11 License. 

923 

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

925 

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

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

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

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

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

931 ''' 

932# @Property_RO 

933# def ab(self): 

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

935# ''' 

936# return self._ab 

937 

938# @Property_RO 

939# def bc(self): 

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

941# ''' 

942# return self._bc 

943 

944 @Property_RO 

945 def _xE(self): 

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

947 ''' 

948 k2, kp2 = self._k2_kp2 

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

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

951 

952 def xR(self, omega): 

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

954 

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

956 

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

958 ''' 

959 return self.xR_(*_SinCos2(omega)) 

960 

961 def xR_(self, somega, comega): 

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

963 

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

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

966 

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

968 ''' 

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

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

971 

972 @Property_RO 

973 def xyQ2(self): 

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

975 ''' 

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

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

978 name=JacobiConformal.xyQ2.name) 

979 

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

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

982 

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

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

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

986 

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

988 ''' 

989 return self.xyR2_(*(_SinCos2(beta) + _SinCos2(omega)), 

990 name=name or self.xyR2.__name__) 

991 

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

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

994 

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

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

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

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

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

1000 

1001 @return: A L{Jacobi2Tuple}C{(x, y)}. 

1002 ''' 

1003 return Jacobi2Tuple(self.xR_(somega, comega), 

1004 self.yR_(sbeta, cbeta), 

1005 name=name or self.xyR2_.__name__) 

1006 

1007 @Property_RO 

1008 def _yE(self): 

1009 '''(INTERNAL) Get the x-elliptic function. 

1010 ''' 

1011 kp2, k2 = self._k2_kp2 # swapped! 

1012 # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc 

1013 return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2) 

1014 

1015 def yR(self, beta): 

1016 '''Compute a Jacobi Conformal C{y} projection. 

1017 

1018 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

1019 

1020 @return: The C{y} projection (C{Radians}). 

1021 ''' 

1022 return self.yR_(*_SinCos2(beta)) 

1023 

1024 def yR_(self, sbeta, cbeta): 

1025 '''Compute a Jacobi Conformal C{y} projection. 

1026 

1027 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

1028 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

1029 

1030 @return: The C{y} projection (C{Radians}). 

1031 ''' 

1032 s, c = self._norm2(sbeta, cbeta, self.c) 

1033 return Radians(y=self._yE.fPi(s, c) * self._c2_b2) 

1034 

1035 

1036class TriaxialError(_ValueError): 

1037 '''Raised for L{Triaxial} issues. 

1038 ''' 

1039 pass # ... 

1040 

1041 

1042class Triaxials(_NamedEnum): 

1043 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class 

1044 to accommodate the L{_LazyNamedEnumItem} properties. 

1045 ''' 

1046 def _Lazy(self, *abc, **name): 

1047 '''(INTERNAL) Instantiate the C{Triaxial}. 

1048 ''' 

1049 a, b, c = map(km2m, abc) 

1050 return Triaxial(a, b, c, **name) 

1051 

1052Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton 

1053'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.''' 

1054# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in km 

1055# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf> 

1056# <https://link.Springer.com/article/10.1007/s00190-022-01650-9> 

1057Triaxials._assert( # a (km) b (km) c (km) planet 

1058 Amalthea = _lazy('Amalthea', 125.0, 73.0, 64), # Jupiter 

1059 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus 

1060 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544), 

1061 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn 

1062 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter 

1063 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter 

1064 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3), 

1065 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn 

1066 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus 

1067 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth 

1068 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn 

1069 WGS84_35 = _lazy('WGS84_35', *map(m2km, _WGS84_35abc))) 

1070 

1071del _WGS84_35abc 

1072 

1073 

1074def _getitems(items, *indices): 

1075 '''(INTERNAL) Get the C{items} at the given I{indices}. 

1076 

1077 @return: C{Type(items[i] for i in indices)} with 

1078 C{Type = type(items)}, any C{type} having 

1079 the special method C{__getitem__}. 

1080 ''' 

1081 return type(items)(map(items.__getitem__, indices)) 

1082 

1083 

1084def _hartzell3d2(pov, los, Tun): # MCCABE 13 in .formy.hartzell 

1085 '''(INTERNAL) Hartzell's "Satellite Lin-of-Sight Intersection ...", 

1086 formula for I{un-/ordered} triaxials. 

1087 ''' 

1088 a, b, c, T = Tun._ordered4 

1089 

1090 a2 = a**2 # largest, factored out 

1091 b2, p2 = (b**2, (b / a)**2) if b != a else (a2, _1_0) 

1092 c2, q2 = c**2, (c / a)**2 

1093 

1094 p3 = T._order3d(_otherV3d(pov=pov)) 

1095 u3 = T._order3d(_otherV3d(los=los)) if los else p3.negate() 

1096 u3 = u3.unit() # unit vector, opposing signs 

1097 

1098 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz 

1099 ux, vy, wz = u3.times_(p3).xyz 

1100 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

1101 

1102 t = (p2 * c2), c2, b2 

1103 m = fdot(t, u2, v2, w2) # a2 factored out 

1104 if m < EPS0: # zero or near-null LOS vector 

1105 raise _ValueError(_near_(_null_)) 

1106 

1107 r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

1108 -w2 * y2, b2 * u2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2, 

1109 -w2 * x2 * p2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True) 

1110 if r > 0: # a2 factored out 

1111 r = sqrt(r) * b * c # == a * a * b * c / a2 

1112 elif r < 0: # LOS pointing away from or missing the triaxial 

1113 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

1114 

1115 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

1116 if d > 0: # POV inside or LOS missing, outside the triaxial 

1117 s = fsum_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0, floats=True) # like _sideOf 

1118 raise _ValueError(_outside_ if s > 0 else _inside_) 

1119 elif fsum_(x2, y2, z2, floats=True) < d**2: # d past triaxial's center 

1120 raise _ValueError(_too_(_distant_)) 

1121 

1122 v = p3.minus(u3.times(d)) # Vector3d 

1123 h = p3.minus(v).length # distance to triaxial 

1124 return T._order3d(v, reverse=True), h 

1125 

1126 

1127def hartzell4(pov, los=None, tri_biax=_WGS84, name=NN): 

1128 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight 

1129 from a Point-Of-View outside. 

1130 

1131 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple} 

1132 or L{Vector3d}). 

1133 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Vector3d}) or 

1134 C{None} to point to the tri-/biaxial's center. 

1135 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal}) 

1136 or biaxial ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2}, 

1137 L{a_f2Tuple} or C{scalar} radius in C{meter}). 

1138 @kwarg name: Optional name (C{str}). 

1139 

1140 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h} 

1141 the distance from B{C{pov}} to C{(x, y, z)} along B{C{los}}. 

1142 

1143 @raise TriaxialError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}} is inside 

1144 the tri-/biaxial or B{C{los}} points outside the 

1145 tri-/biaxial or points in an opposite direction. 

1146 

1147 @raise TypeError: Invalid B{C{pov}} or B{C{los}}. 

1148 

1149 @see: Function L{pygeodesy.hartzell}, L{pygeodesy.tyr3d} for B{C{los}} and 

1150 U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell. 

1151 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

1152 ''' 

1153 if isinstance(tri_biax, Triaxial_): 

1154 T = tri_biax 

1155 else: 

1156 D = tri_biax if isinstance(tri_biax, Datum) else \ 

1157 _spherical_datum(tri_biax, name=hartzell4.__name__) 

1158 E = D.ellipsoid 

1159 T = Triaxial_(E.a, E.a, E.b, name=E.name) 

1160 

1161 try: 

1162 v, h = _hartzell3d2(pov, los, T) 

1163 except Exception as x: 

1164 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x) 

1165 return Vector4Tuple(v.x, v.y, v.z, h, name=name or hartzell4.__name__) 

1166 

1167 

1168def _normalTo4(x, y, a, b, eps=EPS): # MCCABE 14 

1169 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}. 

1170 

1171 @see: Function C{pygeodesy.ellipsoids._normalTo3} and I{Eberly}'s U{Distance 

1172 from a Point to ... an Ellipsoid ...<https://www.GeometricTools.com/ 

1173 Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1174 ''' 

1175 def _root2d(r, u, v, g, eps): 

1176 # robust root finder 

1177 _1, __2 = _1_0, _0_5 

1178 _a, _h2 = fabs, _hypot21_ 

1179 u *= r 

1180 t0 = v - _1 

1181 t1 = _0_0 if g < 0 else _h2(u, v) 

1182 for i in range(1, _TRIPS): 

1183 t = (t0 + t1) * __2 

1184 if t in (t0, t1) or _a(t0 - t1) < eps: 

1185 break 

1186 g = _h2(u / (t + r), v / (t + _1)) 

1187 if g > 0: 

1188 t0 = t 

1189 elif g < 0: 

1190 t1 = t 

1191 else: 

1192 break 

1193 else: # PYCHOK no cover 

1194 e = _a(t0 - t1) 

1195 t = _root2d.__name__ 

1196 raise _ValueError(Fmt.no_convergence(e, eps), txt=t) 

1197 return t, i 

1198 

1199 if a < b: 

1200 b, a, d, i = _normalTo4(y, x, b, a, eps=eps) 

1201 return a, b, d, i 

1202 

1203 if not (isfinite(a) and b > 0): 

1204 raise _ValueError(a=a, b=b) 

1205 

1206 i = None 

1207 if y: 

1208 if x: 

1209 u = fabs(x / a) 

1210 v = fabs(y / b) 

1211 g = _hypot21_(u, v) 

1212 if g: 

1213 r = (a / b)**2 

1214 t, i = _root2d(r, u, v, g, eps) 

1215 a = x / (t / r + _1_0) 

1216 b = y / (t + _1_0) 

1217 d = hypot(x - a, y - b) 

1218 else: # on the ellipse 

1219 a, b, d = x, y, _0_0 

1220 else: # x == 0 

1221 if y < 0: 

1222 b = -b 

1223 a, d = x, fabs(y - b) 

1224 

1225 else: # y == 0 

1226 n = a * x 

1227 d = (a + b) * (a - b) 

1228 if d > fabs(n): # PYCHOK no cover 

1229 r = n / d 

1230 a *= r 

1231 b *= sqrt(_1_0 - r**2) 

1232 d = hypot(x - a, b) 

1233 else: 

1234 if x < 0: 

1235 a = -a 

1236 b, d = y, fabs(x - a) 

1237 return a, b, d, i 

1238 

1239 

1240def _normalTo5(x, y, z, Tun, eps=EPS): # MCCABE 24 

1241 '''(INTERNAL) Nearest point on and distance to an I{un- or ordered} triaxial. 

1242 

1243 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https:// 

1244 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1245 ''' 

1246 def _root3d(r, s, u, v, w, g, eps): 

1247 # robust root finder 

1248 _1, __2 = _1_0, _0_5 

1249 _a, _h2 = fabs, _hypot21_ 

1250 u *= r 

1251 v *= s 

1252 t0 = w - _1 

1253 t1 = _0_0 if g < 0 else _h2(u, v, w) 

1254 for i in range(1, _TRIPS): 

1255 t = (t0 + t1) * __2 

1256 if t in (t0, t1) or _a(t0 - t1) < eps: 

1257 break 

1258 g = _h2(u / (t + r), v / (t + s), w / (t + _1)) 

1259 if g > 0: 

1260 t0 = t 

1261 elif g < 0: 

1262 t1 = t 

1263 else: 

1264 break 

1265 else: # PYCHOK no cover 

1266 e = _a(t0 - t1) 

1267 t = _root3d.__name__ 

1268 raise _ValueError(Fmt.no_convergence(e, eps), txt=t) 

1269 return t, i 

1270 

1271 a, b, c, T = Tun._ordered4 

1272 if Tun is not T: # T is ordered, Tun isn't 

1273 t = T._order3(x, y, z) + (T,) 

1274 a, b, c, d, i = _normalTo5(*t, eps=eps) 

1275 return T._order3(a, b, c, reverse=True) + (d, i) 

1276 

1277 if not (isfinite(a) and c > 0): 

1278 raise _ValueError(a=a, b=b, c=c) 

1279 

1280 if eps > 0: 

1281 val = max(eps * 1e8, EPS) 

1282 else: # no validation 

1283 val, eps = 0, -eps 

1284 

1285 i = None 

1286 if z: 

1287 if y: 

1288 if x: 

1289 u = fabs(x / a) 

1290 v = fabs(y / b) 

1291 w = fabs(z / c) 

1292 g = _hypot21_(u, v, w) 

1293 if g: 

1294 r = T._1e2ac # (c / a)**2 

1295 s = T._1e2bc # (c / b)**2 

1296 t, i = _root3d(_1_0 / r, _1_0 / s, u, v, w, g, eps) 

1297 a = x / (t * r + _1_0) 

1298 b = y / (t * s + _1_0) 

1299 c = z / (t + _1_0) 

1300 d = hypot_(x - a, y - b, z - c) 

1301 else: # on the ellipsoid 

1302 a, b, c, d = x, y, z, _0_0 

1303 else: # x == 0 

1304 a = x # 0 

1305 b, c, d, i = _normalTo4(y, z, b, c, eps=eps) 

1306 elif x: # y == 0 

1307 b = y # 0 

1308 a, c, d, i = _normalTo4(x, z, a, c, eps=eps) 

1309 else: # x == y == 0 

1310 if z < 0: 

1311 c = -c 

1312 a, b, d = x, y, fabs(z - c) 

1313 

1314 else: # z == 0 

1315 t = False 

1316 n = a * x 

1317 d = T._a2c2 # (a + c) * (a - c) 

1318 if d > fabs(n): 

1319 u = n / d 

1320 n = b * y 

1321 d = T._b2c2 # (b + c) * (b - c) 

1322 if d > fabs(n): 

1323 v = n / d 

1324 n = _hypot21_(u, v) 

1325 if n < 0: 

1326 a *= u 

1327 b *= v 

1328 c *= sqrt(-n) 

1329 d = hypot_(x - a, y - b, c) 

1330 t = True 

1331 if not t: 

1332 c = z # 0 

1333 a, b, d, i = _normalTo4(x, y, a, b, eps=eps) 

1334 

1335 if val > 0: # validate 

1336 e = T.sideOf(a, b, c, eps=val) 

1337 if e: # not near the ellipsoid's surface 

1338 raise _ValueError(a=a, b=b, c=c, d=d, 

1339 sideOf=e, eps=val) 

1340 if d: # angle of delta and normal vector 

1341 m = Vector3d(x, y, z).minus_(a, b, c) 

1342 if m.euclid > val: 

1343 m = m.unit() 

1344 n = T.normal3d(a, b, c) 

1345 e = n.dot(m) # n.negate().dot(m) 

1346 if not isnear1(fabs(e), eps1=val): 

1347 raise _ValueError(n=n, m=m, 

1348 dot=e, eps=val) 

1349 return a, b, c, d, i 

1350 

1351 

1352def _otherV3d_(x_xyz, y, z, name=NN): 

1353 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}. 

1354 ''' 

1355 return Vector3d(x_xyz, y, z, name=name) if isscalar(x_xyz) else \ 

1356 _otherV3d(x_xyz=x_xyz) 

1357 

1358 

1359def _sideOf(xyz, abc, eps=EPS): # in .formy 

1360 '''(INTERNAL) Helper for C{_hartzell3d2}, M{.sideOf} and M{.reverseCartesian}. 

1361 

1362 @return: M{sum((x / a)**2 for x, a in zip(xyz, abc)) - 1} or C{INT0}, 

1363 ''' 

1364 s = _hypot21_(*((x / a) for x, a in _zip(xyz, abc) if a)) # strict=True 

1365 return s if fabs(s) > eps else INT0 

1366 

1367 

1368def _SinCos2(x): 

1369 '''Get C{sin} and C{cos} of C{x} in C{Degrees}, C{Radians} or {radians}. 

1370 ''' 

1371 return sincos2d(x) if isinstance(x, Degrees) else ( 

1372 sincos2(x) if isinstance(x, Radians) else 

1373 sincos2(float(x))) # assume C{radians} 

1374 

1375 

1376if __name__ == '__main__': 

1377 

1378 from pygeodesy import printf 

1379 

1380 # __doc__ of this file, force all into registery 

1381 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_) 

1382 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t)) 

1383 

1384# **) MIT License 

1385# 

1386# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1387# 

1388# Permission is hereby granted, free of charge, to any person obtaining a 

1389# copy of this software and associated documentation files (the "Software"), 

1390# to deal in the Software without restriction, including without limitation 

1391# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1392# and/or sell copies of the Software, and to permit persons to whom the 

1393# Software is furnished to do so, subject to the following conditions: 

1394# 

1395# The above copyright notice and this permission notice shall be included 

1396# in all copies or substantial portions of the Software. 

1397# 

1398# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1399# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1400# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1401# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1402# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1403# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1404# OTHER DEALINGS IN THE SOFTWARE.