Coverage for pygeodesy/triaxials.py: 96%

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1 

2# -*- coding: utf-8 -*- 

3 

4u'''Triaxal ellipsoid classes I{ordered} L{Triaxial} and I{unordered} L{Triaxial_} and Jacobi 

5conformal projections L{JacobiConformal} and L{JacobiConformalSpherical}, transcoded from 

6I{Charles Karney}'s C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/C++/doc/ 

7classGeographicLib_1_1JacobiConformal.html#details>} to pure Python and miscellaneous classes 

8L{BetaOmega2Tuple}, L{BetaOmega3Tuple}, L{Jacobi2Tuple} and L{TriaxialError}. 

9 

10Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2023). For more information, 

11see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. 

12 

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

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

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

16 

17@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) 

18@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) 

19@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) 

20@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) 

21@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) 

22@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) 

23@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) 

24@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) 

25@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) 

26@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) 

27@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) 

28@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) 

29''' 

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

31from __future__ import division as _; del _ # PYCHOK semicolon 

32 

33from pygeodesy.basics import isLatLon, isscalar, _zip, _ValueError 

34from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INT0, PI2, PI_3, PI4, \ 

35 _EPS2e4, float0_, isfinite, isnear1, _0_0, _0_5, \ 

36 _1_0, _N_1_0, _N_2_0, _4_0 # PYCHOK used! 

37from pygeodesy.datums import Datum, _spherical_datum, _WGS84, Ellipsoid, _EWGS84, Fmt 

38# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums 

39# from pygeodesy.elliptic import Elliptic # _MODS 

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

41from pygeodesy.fmath import Fdot, fdot, fmean_, hypot, hypot_, norm2, sqrt0 

42from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1f_ 

43from pygeodesy.interns import NN, _a_, _b_, _beta_, _c_, _distant_, _finite_, \ 

44 _height_, _inside_, _near_, _negative_, _not_, \ 

45 _NOTEQUAL_, _null_, _opposite_, _outside_, _SPACE_, \ 

46 _spherical_, _too_, _x_, _y_ 

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

48from pygeodesy.named import _lazyNamedEnumItem as _lazy, _name__, _NamedEnum, \ 

49 _NamedEnumItem, _Pass 

50from pygeodesy.namedTuples import LatLon3Tuple, _NamedTupleTo, Vector3Tuple, \ 

51 Vector4Tuple 

52from pygeodesy.props import Property_RO, property_RO 

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

54from pygeodesy.units import Degrees, Float, Height_, Meter, Meter2, Meter3, \ 

55 Radians, Radius, Scalar_ 

56from pygeodesy.utily import asin1, atan2d, km2m, m2km, SinCos2, sincos2d_ 

57from pygeodesy.vector3d import _otherV3d, Vector3d, _ALL_LAZY, _MODS 

58 

59from math import atan2, fabs, sqrt 

60 

61__all__ = _ALL_LAZY.triaxials 

62__version__ = '24.06.24' 

63 

64_not_ordered_ = _not_('ordered') 

65_omega_ = 'omega' 

66_TRIPS = 269 # 48-55, Eberly 1074? 

67 

68 

69class _NamedTupleToX(_NamedTupleTo): # in .testNamedTuples 

70 '''(INTERNAL) Base class for L{BetaOmega2Tuple}, 

71 L{BetaOmega3Tuple} and L{Jacobi2Tuple}. 

72 ''' 

73 def _toDegrees(self, name, **toDMS_kwds): 

74 '''(INTERNAL) Convert C{self[0:2]} to L{Degrees} or C{toDMS}. 

75 ''' 

76 return self._toX3U(_NamedTupleTo._Degrees3, Degrees, name, *self, **toDMS_kwds) 

77 

78 def _toRadians(self, name): 

79 '''(INTERNAL) Convert C{self[0:2]} to L{Radians}. 

80 ''' 

81 return self._toX3U(_NamedTupleTo._Radians3, Radians, name, *self) 

82 

83 def _toX3U(self, _X3, U, name, a, b, *c, **kwds): 

84 a, b, s = _X3(self, a, b, **kwds) 

85 if s is None or name: 

86 n = self._name__(name) 

87 s = self.classof(a, b, *c, name=n).reUnit(U, U).toUnits() 

88 return s 

89 

90 

91class BetaOmega2Tuple(_NamedTupleToX): 

92 '''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and 

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

94 L{Degrees}). 

95 ''' 

96 _Names_ = (_beta_, _omega_) 

97 _Units_ = (_Pass, _Pass) 

98 

99 def toDegrees(self, name=NN, **toDMS_kwds): 

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

101 

102 @kwarg name: Optional name (C{str}), overriding this name. 

103 

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

105 C{omega} both in L{Degrees} or as L{toDMS} strings 

106 provided some B{C{toDMS_kwds}} keyword arguments are 

107 specified. 

108 ''' 

109 return self._toDegrees(name, **toDMS_kwds) 

110 

111 def toRadians(self, **name): 

112 '''Convert this L{BetaOmega2Tuple} to L{Radians}. 

113 

114 @kwarg name: Optional C{B{name}=NN} (C{str}), overriding this name. 

115 

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

117 both in L{Radians}. 

118 ''' 

119 return self._toRadians(name) 

120 

121 

122class BetaOmega3Tuple(_NamedTupleToX): 

123 '''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and 

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

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

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

127 in C{meter}, conventionally. 

128 ''' 

129 _Names_ = BetaOmega2Tuple._Names_ + (_height_,) 

130 _Units_ = BetaOmega2Tuple._Units_ + ( Meter,) 

131 

132 def toDegrees(self, name=NN, **toDMS_kwds): 

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

134 

135 @kwarg name: Optional name (C{str}), overriding this name. 

136 

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

138 C{beta} and C{omega} both in L{Degrees} or as 

139 L{toDMS} strings provided some B{C{toDMS_kwds}} 

140 keyword arguments are specified. 

141 ''' 

142 return self._toDegrees(name, **toDMS_kwds) 

143 

144 def toRadians(self, **name): 

145 '''Convert this L{BetaOmega3Tuple} to L{Radians}. 

146 

147 @kwarg name: Optional C{B{name}=NN} (C{str}), overriding this name. 

148 

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

150 and C{omega} both in L{Radians}. 

151 ''' 

152 return self._toRadians(name) 

153 

154 def to2Tuple(self, **name): 

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

156 

157 @kwarg name: Optional C{B{name}=NN} (C{str}), overriding this name. 

158 ''' 

159 return BetaOmega2Tuple(*self[:2], name=self._name__(name)) 

160 

161 

162class Jacobi2Tuple(_NamedTupleToX): 

163 '''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y} 

164 projection, both in L{Radians} (or L{Degrees}). 

165 ''' 

166 _Names_ = (_x_, _y_) 

167 _Units_ = (_Pass, _Pass) 

168 

169 def toDegrees(self, name=NN, **toDMS_kwds): 

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

171 

172 @kwarg name: Optional name (C{str}), overriding this name. 

173 

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

175 in L{Degrees} or as L{toDMS} strings provided some 

176 B{C{toDMS_kwds}} keyword arguments are specified. 

177 ''' 

178 return self._toDegrees(name, **toDMS_kwds) 

179 

180 def toRadians(self, **name): 

181 '''Convert this L{Jacobi2Tuple} to L{Radians}. 

182 

183 @kwarg name: Optional C{B{name}=NN} (C{str}), overriding this name. 

184 

185 @return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y} both in L{Radians}. 

186 ''' 

187 return self._toRadians(name) 

188 

189 

190class Triaxial_(_NamedEnumItem): 

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

192 

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

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

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

196 I{omega}=0. 

197 

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

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

200 

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

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

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

204 L{Degrees} if converted). 

205 ''' 

206 _ijk = _kji = None 

207 _unordered = True 

208 

209 def __init__(self, a_triaxial, b=None, c=None, **name): 

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

211 

212 @arg a_triaxial: Large, C{X} semi-axis (C{scalar}, conventionally in 

213 C{meter}) or an other L{Triaxial} or L{Triaxial_} instance. 

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

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

216 @kwarg c: Small, C{Z} semi-axis (C{meter}, B{C{b}}). 

217 @kwarg name: Optional C{B{name}=NN} (C{str}). 

218 

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

220 ''' 

221 try: 

222 try: 

223 a = a_triaxial 

224 t = a._abc3 

225 except AttributeError: 

226 t = Radius(a=a), Radius(b=b), Radius(c=c) 

227 except (TypeError, ValueError) as x: 

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

229 if name: 

230 self.name = name 

231 

232 a, b, c = self._abc3 = t 

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

234 s, _, t = sorted(t) 

235 if not (isfinite(t) and s > 0): 

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

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

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

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

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

241 

242 def __str__(self): 

243 return self.toStr() 

244 

245 @Property_RO 

246 def a(self): 

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

248 ''' 

249 a, _, _ = self._abc3 

250 return a 

251 

252 @Property_RO 

253 def _a2b2(self): 

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

255 ''' 

256 a, b, _ = self._abc3 

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

258 

259 @Property_RO 

260 def _a2_b2(self): 

261 '''(INTERNAL) Get C{(a/b)**2}. 

262 ''' 

263 a, b, _ = self._abc3 

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

265 

266 @Property_RO 

267 def _a2c2(self): 

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

269 ''' 

270 a, _, c = self._abc3 

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

272 

273 @Property_RO 

274 def area(self): 

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

276 ''' 

277 c, b, a = sorted(self._abc3) 

278 if a > c: 

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

280 Ellipsoid(a, b=c).areax # a == b 

281 else: # a == c == b 

282 a = Meter2(area=a**2 * PI4) 

283 return a 

284 

285 def area_p(self, p=1.6075): 

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

287 

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

289 for "near-flat" triaxials. 

290 

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

292 ''' 

293 a, b, c = self._abc3 

294 if a == b == c: 

295 a *= a 

296 else: 

297 _p = pow 

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

299 return Meter2(area_p=a * PI4) 

300 

301 @Property_RO 

302 def b(self): 

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

304 ''' 

305 _, b, _ = self._abc3 

306 return b 

307 

308 @Property_RO 

309 def _b2c2(self): 

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

311 ''' 

312 _, b, c = self._abc3 

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

314 

315 @Property_RO 

316 def c(self): 

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

318 ''' 

319 _, _, c = self._abc3 

320 return c 

321 

322 @Property_RO 

323 def _c2_b2(self): 

324 '''(INTERNAL) Get C{(c/b)**2}. 

325 ''' 

326 _, b, c = self._abc3 

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

328 

329 @Property_RO 

330 def e2ab(self): 

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

332 ''' 

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

334 

335 @Property_RO 

336 def _1e2ab(self): 

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

338 ''' 

339 a, b, _ = self._abc3 

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

341 

342 @Property_RO 

343 def e2ac(self): 

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

345 ''' 

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

347 

348 @Property_RO 

349 def _1e2ac(self): 

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

351 ''' 

352 a, _, c = self._abc3 

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

354 

355 @Property_RO 

356 def e2bc(self): 

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

358 ''' 

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

360 

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

362 

363 @property_RO 

364 def _Elliptic(self): 

365 '''(INTERNAL) Get class L{Elliptic}, I{once}. 

366 ''' 

367 Triaxial_._Elliptic = E = _MODS.elliptic.Elliptic # overwrite property_RO 

368 return E 

369 

370 def hartzell4(self, pov, los=False, **name): 

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

372 from a Point-Of-View in space. 

373 

374 @see: Function L{hartzell4<triaxials.hartzell4>} for further details. 

375 ''' 

376 return hartzell4(pov, los=los, tri_biax=self, **name) 

377 

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

379 '''Compute the projection on and the height above or below this triaxial's surface. 

380 

381 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, L{Ecef9Tuple}, 

382 L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

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

384 otherwise. 

385 @kwarg z: Z component (C{scalar}), like B{C{y}}. 

386 @kwarg normal: If C{True} the projection is the I{normal, plumb} to the surface of, 

387 otherwise the C{radial} line to the center of this triaxial (C{bool}). 

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

389 value to skip validation. 

390 @kwarg name: Optional C{B{name}="heigh4"} (C{str}). 

391 

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

393 and C{z} of the projection on or the intersection with and with the height 

394 C{h} above or below the triaxial's surface in C{meter}, conventionally. 

395 

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

397 root finding or validation failed. 

398 

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

400 <https://www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

401 ''' 

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

403 

404 i, h = None, v.length 

405 if h < EPS0: # EPS 

406 x = y = z = _0_0 

407 h -= min(self._abc3) # nearest 

408 elif r: # .isSpherical 

409 x, y, z = v.times(r / h).xyz 

410 h -= r 

411 else: 

412 x, y, z = v.xyz 

413 try: 

414 if normal: # plumb to surface 

415 x, y, z, h, i = _plumbTo5(x, y, z, self, eps=eps) 

416 else: # radial to center 

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

418 h = v.minus_(x, y, z).length 

419 except Exception as e: 

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

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

422 h = -h # below the surface 

423 n = _name__(name, name__=self.height4) 

424 return Vector4Tuple(x, y, z, h, iteration=i, name=n) 

425 

426 @Property_RO 

427 def isOrdered(self): 

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

429 ''' 

430 a, b, c = self._abc3 

431 return bool(a >= b > c) # b > c! 

432 

433 @Property_RO 

434 def isSpherical(self): 

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

436 ''' 

437 a, b, c = self._abc3 

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

439 

440 def _norm2(self, s, c, *a): 

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

442 ''' 

443 if fabs(_hypot21(s, c)) > EPS02: 

444 s, c = norm2(s, c) 

445 if a: 

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

447 return float0_(s, c) 

448 

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

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

451 

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

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

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

455 otherwise. 

456 @kwarg z: Z component (C{scalar}), like B{C{y}}. 

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

458 

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

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

461 

462 @raise TriaxialError: Zero length cartesian or vector. 

463 

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

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

466 ''' 

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

468 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered 

469 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2 

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

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

472 if n.length < EPS0: 

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

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

475 

476 @Property_RO 

477 def _normal3d(self): 

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

479 ''' 

480 d = max(self._abc3) 

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

482 return Vector3d(*t, name__=self.normal3d) 

483 

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

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

486 

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

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

489 ''' 

490 ijk = self._order_ijk(**reverse) 

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

492 

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

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

495 

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

497 ''' 

498 ijk = self._order_ijk(**reverse) 

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

500 

501 @Property_RO 

502 def _ordered4(self): 

503 '''(INTERNAL) Helper for C{_hartzell3} and C{_plumbTo5}. 

504 ''' 

505 def _order2(reverse, a, b, c): 

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

507 

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

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

510 ''' 

511 i, j, k = 0, 1, 2 # range(3) 

512 if a < b: 

513 a, b, i, j = b, a, j, i 

514 if a < c: 

515 a, c, i, k = c, a, k, i 

516 if b < c: 

517 b, c, j, k = c, b, k, j 

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

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

520 return (a, b, c), ijk 

521 

522 abc, T = self._abc3, self 

523 if not self.isOrdered: 

524 abc, ijk = _order2(False, *abc) 

525 if ijk: 

526 _, kji = _order2(True, *ijk) 

527 T = Triaxial_(*abc) 

528 T._ijk, T._kji = ijk, kji 

529 return abc + (T,) 

530 

531 def _order_ijk(self, reverse=False): 

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

533 ''' 

534 return self._kji if reverse else self._ijk 

535 

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

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

538 ''' 

539 def _rphi(a, b, sphi, cphi): 

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

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

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

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

544 

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

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

547 

548 a, b, c = self._abc3 

549 if a != b: 

550 a = _rphi(a, b, sb, cb) 

551 if a != c: 

552 c = _rphi(a, c, sa, ca) 

553 z, r = c * sa, c * ca 

554 x, y = r * cb, r * sb 

555 return x, y, z 

556 

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

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

559 

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

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

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

563 ignored otherwise. 

564 @kwarg z: Z component (C{scalar}), like B{C{y}}. 

565 @kwarg eps: Near-surface tolerance (C{scalar}, distance I{squared}). 

566 

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

568 within tolerance B{C{eps}}, otherwise a signed, radial, normalized 

569 distance I{squared} (C{float}), negative or positive for in- 

570 respectively outside this triaxial. 

571 

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

573 ''' 

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

575 

576 def toEllipsoid(self, **name): 

577 '''Convert this triaxial to an L{Ellipsoid}, provided 2 axes match. 

578 

579 @kwarg name: Optional, overriding C{B{name}=NN} (C{str})=. 

580 

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

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

583 

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

585 

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

587 ''' 

588 a, b, c = self._abc3 

589 if a == b: 

590 b = c # N = c-Z 

591 elif b == c: # N = a-X 

592 a, b = b, a 

593 elif a != c: # N = b-Y 

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

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

596 return Ellipsoid(a, b=b, name=self._name__(name)) 

597 

598 def toStr(self, prec=9, **name): # PYCHOK signature 

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

600 

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

602 @kwarg name: Optional, overriding C{B{name}=NN} (C{str}) 

603 or C{None} to exclude this triaxial's name. 

604 

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

606 ''' 

607 T = Triaxial_ 

608 t = T.a, 

609 J = JacobiConformalSpherical 

610 t += (J.ab, J.bc) if isinstance(self, J) else (T.b, T.c) 

611 t += T.e2ab, T.e2bc, T.e2ac 

612 J = JacobiConformal 

613 if isinstance(self, J): 

614 t += J.xyQ2, 

615 t += T.volume, T.area 

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

617 

618 @Property_RO 

619 def volume(self): 

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

621 ''' 

622 a, b, c = self._abc3 

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

624 

625 

626class Triaxial(Triaxial_): 

627 '''I{Ordered} triaxial ellipsoid. 

628 

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

630 ''' 

631 _unordered = False 

632 

633 def __init__(self, a_triaxial, b=None, c=None, **name): 

634 '''New I{ordered} L{Triaxial}. 

635 

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

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

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

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

640 @kwarg c: Smallest semi-axis (C{meter}, like B{C{b}}). 

641 @kwarg name: Optional C{B{name}=NN} (C{str}). 

642 

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

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

645 

646 @raise TriaxialError: Semi-axes unordered, spherical or invalid. 

647 ''' 

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

649 

650 @Property_RO 

651 def _a2b2_a2c2(self): 

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

653 ''' 

654 return self._a2b2 / self._a2c2 

655 

656 @Property_RO 

657 def area(self): 

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

659 

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

661 ''' 

662 a, b, c = self._abc3 

663 if a != b: 

664 kp2, k2 = self._k2_kp2 # swapped! 

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

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

667 s = sqrt(self.e2ac) # sin(phi)**2 = 1 - c2 

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

669 b *= fsum1f_(aE.fE(r) * s, c / a * c / b, 

670 aE.fF(r) * c2 / s) 

671 a = Meter2(area=a * b * PI2) 

672 else: # a == b > c 

673 a = Ellipsoid(a, b=c).areax 

674 return a 

675 

676 def _exyz3(self, u): 

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

678 ''' 

679 if u > 0: 

680 u2 = u**2 

681 x = u * sqrt0(_1_0 + self._a2c2 / u2, Error=TriaxialError) 

682 y = u * sqrt0(_1_0 + self._b2c2 / u2, Error=TriaxialError) 

683 else: 

684 x = y = u = _0_0 

685 return x, y, u 

686 

687 def forwardBetaOmega(self, beta, omega, height=0, **name): 

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

689 and height to cartesian. 

690 

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

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

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

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

695 @kwarg name: Optional C{B{name}=NN} (C{str}). 

696 

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

698 

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

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

701 ''' 

702 if height: 

703 h = self._Height(height) 

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

705 else: 

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

707 if z: # and x and y: 

708 sa, ca = SinCos2(beta) 

709 sb, cb = SinCos2(omega) 

710 

711 r = self._a2b2_a2c2 

712 x *= cb * sqrt0(ca**2 + r * sa**2, Error=TriaxialError) 

713 y *= ca * sb 

714 z *= sa * sqrt0(_1_0 - r * cb**2, Error=TriaxialError) 

715 return Vector3Tuple(x, y, z, **name) 

716 

717 def forwardBetaOmega_(self, sbeta, cbeta, somega, comega, **name): 

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

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

720 

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

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

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

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

725 @kwarg name: Optional C{B{name}=NN} (C{str}). 

726 

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

728 

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

730 

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

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

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

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

735 ''' 

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

737 return Vector3Tuple(*t, **name) 

738 

739 def forwardCartesian(self, x_xyz, y=None, z=None, **normal_eps_name): 

740 '''Project a cartesian on this triaxial. 

741 

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

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

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

745 ignored otherwise. 

746 @kwarg z: Z component (C{scalar}), like B{C{y}}. 

747 @kwarg normal_eps_name: Optional keyword arguments C{B{normal}=True}, 

748 C{B{eps}=EPS} and overriding C{B{name}="height4"} (C{str}), 

749 see method L{Triaxial.height4}. 

750 

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

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

753 ''' 

754 return self.height4(x_xyz, y, z, **normal_eps_name) 

755 

756 def forwardLatLon(self, lat, lon, height=0, **name): 

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

758 

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

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

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

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

763 @kwarg name: Optional C{B{name}=NN} (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 return self._forwardLatLon3(height, name, *sincos2d_(lat, lon)) 

771 

772 def forwardLatLon_(self, slat, clat, slon, clon, height=0, **name): 

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

774 

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

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

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

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

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

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

781 @kwarg name: Optional C{B{name}=NN} (C{str}). 

782 

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

784 

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

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

787 ''' 

788 sa, ca = self._norm2(slat, clat) 

789 sb, cb = self._norm2(slon, clon) 

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

791 

792 def _forwardLatLon3(self, height, name, sa, ca, sb, cb): # name always **name 

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

794 ''' 

795 ca_x_sb = ca * sb 

796 h = self._Height(height) 

797 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2 

798 t = fsumf_(_1_0, -self.e2ac * sa**2, -self.e2ab * ca_x_sb**2) 

799 n = self.a / sqrt0(t, Error=TriaxialError) # prime vertical 

800 x = (h + n) * ca * cb 

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

802 z = (h + n * self._1e2ac) * sa 

803 return Vector3Tuple(x, y, z, **name) 

804 

805 def _Height(self, height): 

806 '''(INTERNAL) Validate a C{height}. 

807 ''' 

808 return Height_(height=height, low=-self.c, Error=TriaxialError) 

809 

810 @Property_RO 

811 def _k2_kp2(self): 

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

813 ''' 

814 # k2 = a2b2 / a2c2 * c2_b2 

815 # kp2 = b2c2 / a2c2 * a2_b2 

816 # b2 = b**2 

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

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

819 # aE = Elliptic(kp2, 0, k2, 1) 

820 return (self._a2b2_a2c2 * self._c2_b2, 

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

822 

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

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

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

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

827 ''' 

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

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

830 

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

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

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

834 

835 x2 = _Fsumf_(_1_0, -b2_a2 * sa**2, c2_a2 * ca**2).fover(a2c2_a2) 

836 z2 = _Fsumf_(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2) 

837 

838 x, y, z = self._abc3 

839 x *= cb * sqrt0(x2, Error=TriaxialError) 

840 y *= ca * sb 

841 z *= sa * sqrt0(z2, Error=TriaxialError) 

842 return x, y, z 

843 

844 def reverseBetaOmega(self, x_xyz, y=None, z=None, **name): 

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

846 and height. 

847 

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

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

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

851 ignored otherwise. 

852 @kwarg z: Z component (C{scalar}), like B{C{y}}. 

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

854 

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

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

857 units as this triaxial's axes. 

858 

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

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

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

862 ''' 

863 v = _otherV3d_(x_xyz, y, z) 

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

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

866 

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

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

869 

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

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

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

873 ignored otherwise. 

874 @kwarg z: Z component (C{scalar}), like B{C{y}}. 

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

876 as the axes). 

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

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

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

880 @kwarg name: Optional C{B{name}=NN} (C{str}). 

881 

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

883 

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

885 

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

887 ''' 

888 v = _otherV3d_(x_xyz, y, z, **name) 

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

890 if s: # PYCHOK no cover 

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

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

893 

894 if h: 

895 if normal: 

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

897 elif v.length > EPS0: 

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

899 return v.xyz # Vector3Tuple 

900 

901 def reverseLatLon(self, x_xyz, y=None, z=None, **name): 

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

903 

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

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

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

907 ignored otherwise. 

908 @kwarg z: Z component (C{scalar}), like B{C{y}}. 

909 @kwarg name: Optional C{B{name}=NN} (C{str}). 

910 

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

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

913 as this triaxial's axes. 

914 

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

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

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

918 ''' 

919 v = _otherV3d_(x_xyz, y, z) 

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

921 self._1e2bc, # == 1 - e_sub_y**2 

922 _1_0) 

923 a, b, h = self._reverseLatLon3(s, atan2d, v, self.forwardLatLon_) 

924 return LatLon3Tuple(Degrees(lat=a), Degrees(lon=b), h, **name) 

925 

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

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

928 ''' 

929 x, y, z = s.xyz 

930 d = hypot( x, y) 

931 a = atan2_(z, d) 

932 b = atan2_(y, x) 

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

934 return a, b, h 

935 

936 

937class JacobiConformal(Triaxial): 

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

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

940 

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

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

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

944 returned in the case of an ellipsoid of revolution. 

945 

946 Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2014-2023) and 

947 licensed under the MIT/X11 License. 

948 

949 @note: This constructor can I{not be used to specify a sphere}, see alternate 

950 L{JacobiConformalSpherical}. 

951 

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

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

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

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

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

957 ''' 

958 

959 @Property_RO 

960 def _xE(self): 

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

962 ''' 

963 k2, kp2 = self._k2_kp2 

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

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

966 

967 def xR(self, omega): 

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

969 

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

971 

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

973 ''' 

974 return self.xR_(*SinCos2(omega)) 

975 

976 def xR_(self, somega, comega): 

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

978 

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

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

981 

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

983 ''' 

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

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

986 

987 @Property_RO 

988 def xyQ2(self): 

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

990 ''' 

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

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

993 name=JacobiConformal.xyQ2.name) 

994 

995 def xyR2(self, beta, omega, **name): 

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

997 

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

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

1000 @kwarg name: Optional, overriding C{B{name}="xyR2"} (C{str}). 

1001 

1002 @return: A L{Jacobi2Tuple}C{(x, y)}. 

1003 ''' 

1004 return self.xyR2_(*(SinCos2(beta) + SinCos2(omega)), 

1005 name=_name__(name, name__=self.xyR2)) 

1006 

1007 def xyR2_(self, sbeta, cbeta, somega, comega, **name): 

1008 '''Compute a Jacobi Conformal C{x} and C{y} projection. 

1009 

1010 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

1011 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

1012 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

1013 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

1014 @kwarg name: Optional, overriding C{B{name}="xyR2_"} (C{str}). 

1015 

1016 @return: A L{Jacobi2Tuple}C{(x, y)}. 

1017 ''' 

1018 return Jacobi2Tuple(self.xR_(somega, comega), 

1019 self.yR_(sbeta, cbeta), 

1020 name=_name__(name, name__=self.xyR2_)) 

1021 

1022 @Property_RO 

1023 def _yE(self): 

1024 '''(INTERNAL) Get the x-elliptic function. 

1025 ''' 

1026 kp2, k2 = self._k2_kp2 # swapped! 

1027 # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc 

1028 return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2) 

1029 

1030 def yR(self, beta): 

1031 '''Compute a Jacobi Conformal C{y} projection. 

1032 

1033 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

1034 

1035 @return: The C{y} projection (L{Radians}). 

1036 ''' 

1037 return self.yR_(*SinCos2(beta)) 

1038 

1039 def yR_(self, sbeta, cbeta): 

1040 '''Compute a Jacobi Conformal C{y} projection. 

1041 

1042 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

1043 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

1044 

1045 @return: The C{y} projection (L{Radians}). 

1046 ''' 

1047 s, c = self._norm2(sbeta, cbeta, self.c) 

1048 return Radians(y=self._yE.fPi(s, c) * self._c2_b2) 

1049 

1050 

1051class JacobiConformalSpherical(JacobiConformal): 

1052 '''An alternate, I{spherical} L{JacobiConformal} projection. 

1053 

1054 @see: L{JacobiConformal} for other and more details. 

1055 ''' 

1056 _ab = _bc = 0 

1057 

1058 def __init__(self, radius_triaxial, ab=0, bc=0, **name): 

1059 '''New L{JacobiConformalSpherical}. 

1060 

1061 @arg radius_triaxial: Radius (C{scalar}, conventionally in 

1062 C{meter}) or an other L{JacobiConformalSpherical}, 

1063 L{JacobiConformal} or ordered L{Triaxial}. 

1064 @kwarg ab: Relative magnitude of C{B{a} - B{b}} (C{meter}, 

1065 same units as C{scalar B{radius}}. 

1066 @kwarg bc: Relative magnitude of C{B{b} - B{c}} (C{meter}, 

1067 same units as C{scalar B{radius}}. 

1068 @kwarg name: Optional C{B{name}=NN} (C{str}). 

1069 

1070 @raise TriaxialError: Invalid B{C{radius_triaxial}}, negative 

1071 B{C{ab}}, negative B{C{bc}} or C{(B{ab} 

1072 + B{bc})} not positive. 

1073 

1074 @note: If B{C{radius_triaxial}} is a L{JacobiConformalSpherical} 

1075 and if B{C{ab}} and B{C{bc}} are both zero or C{None}, 

1076 the B{C{radius_triaxial}}'s C{ab}, C{bc}, C{a}, C{b} 

1077 and C{c} are copied. 

1078 ''' 

1079 try: 

1080 r = radius_triaxial 

1081 if isinstance(r, Triaxial): # ordered only 

1082 t = r._abc3 

1083 j = isinstance(r, JacobiConformalSpherical) and not bool(ab or bc) 

1084 else: 

1085 t = (Radius(radius=r),) * 3 

1086 j = False 

1087 self._ab = r.ab if j else Scalar_(ab=ab) # low=0 

1088 self._bc = r.bc if j else Scalar_(bc=bc) # low=0 

1089 if (self.ab + self.bc) <= 0: 

1090 raise ValueError(_negative_) 

1091 a, _, c = self._abc3 = t 

1092 if not (a >= c and isfinite(self._a2b2) 

1093 and isfinite(self._a2c2)): 

1094 raise ValueError(_not_(_finite_)) 

1095 except (TypeError, ValueError) as x: 

1096 raise TriaxialError(radius_triaxial=r, ab=ab, bc=bc, cause=x) 

1097 if name: 

1098 self.name = name 

1099 

1100 @Property_RO 

1101 def ab(self): 

1102 '''Get relative magnitude C{ab} (C{meter}, same units as B{C{a}}). 

1103 ''' 

1104 return self._ab 

1105 

1106 @Property_RO 

1107 def _a2b2(self): 

1108 '''(INTERNAL) Get C{a**2 - b**2} == ab * (a + b). 

1109 ''' 

1110 a, b, _ = self._abc3 

1111 return self.ab * (a + b) 

1112 

1113 @Property_RO 

1114 def _a2c2(self): 

1115 '''(INTERNAL) Get C{a**2 - c**2} == a2b2 + b2c2. 

1116 ''' 

1117 return self._a2b2 + self._b2c2 

1118 

1119 @Property_RO 

1120 def bc(self): 

1121 '''Get relative magnitude C{bc} (C{meter}, same units as B{C{a}}). 

1122 ''' 

1123 return self._bc 

1124 

1125 @Property_RO 

1126 def _b2c2(self): 

1127 '''(INTERNAL) Get C{b**2 - c**2} == bc * (b + c). 

1128 ''' 

1129 _, b, c = self._abc3 

1130 return self.bc * (b + c) 

1131 

1132 @Property_RO 

1133 def radius(self): 

1134 '''Get radius (C{meter}, conventionally). 

1135 ''' 

1136 return self.a 

1137 

1138 

1139class TriaxialError(_ValueError): 

1140 '''Raised for L{Triaxial} issues. 

1141 ''' 

1142 pass # ... 

1143 

1144 

1145class Triaxials(_NamedEnum): 

1146 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class 

1147 to accommodate the L{_LazyNamedEnumItem} properties. 

1148 ''' 

1149 def _Lazy(self, *abc, **name): 

1150 '''(INTERNAL) Instantiate the C{Triaxial}. 

1151 ''' 

1152 a, b, c = map(km2m, abc) 

1153 return Triaxial(a, b, c, **name) 

1154 

1155Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton 

1156'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.''' 

1157# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in Km 

1158# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf> 

1159# <https://link.Springer.com/article/10.1007/s00190-022-01650-9> 

1160_abc84_35 = (_EWGS84.a + 35), (_EWGS84.a - 35), _EWGS84.b 

1161Triaxials._assert( # a (Km) b (Km) c (Km) planet 

1162 Amalthea = _lazy('Amalthea', 125.0, 73.0, 64), # Jupiter 

1163 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus 

1164 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544), 

1165 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn 

1166 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter 

1167 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter 

1168 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3), 

1169 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn 

1170 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus 

1171 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth 

1172 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn 

1173 WGS84_35 = _lazy('WGS84_35', *map(m2km, _abc84_35))) 

1174del _abc84_35, _EWGS84 

1175 

1176 

1177def _getitems(items, *indices): 

1178 '''(INTERNAL) Get the C{items} at the given I{indices}. 

1179 

1180 @return: C{Type(items[i] for i in indices)} with 

1181 C{Type = type(items)}, any C{type} having 

1182 the special method C{__getitem__}. 

1183 ''' 

1184 return type(items)(map(items.__getitem__, indices)) 

1185 

1186 

1187def _hartzell3(pov, los, Tun): # in .Ellipsoid.hartzell4, .formy.hartzell 

1188 '''(INTERNAL) Hartzell's "Satellite Line-of-Sight Intersection ...", 

1189 formula from a Point-Of-View to an I{un-/ordered} Triaxial. 

1190 ''' 

1191 def _toUvwV3d(los, pov): 

1192 try: # pov must be LatLon or Cartesian if los is a Los 

1193 v = los.toUvw(pov) 

1194 except (AttributeError, TypeError): 

1195 v = _otherV3d(los=los) 

1196 return v 

1197 

1198 p3 = _otherV3d(pov=pov.toCartesian() if isLatLon(pov) else pov) 

1199 if los is True: # normal 

1200 a, b, c, d, i = _plumbTo5(p3.x, p3.y, p3.z, Tun) 

1201 return type(p3)(a, b, c), d, i 

1202 

1203 u3 = p3.negate() if los is False or los is None else _toUvwV3d(los, pov) 

1204 

1205 a, b, c, T = Tun._ordered4 

1206 

1207 a2 = a**2 # largest, factored out 

1208 b2, p2 = (b**2, T._1e2ab) if b != a else (a2, _1_0) 

1209 c2, q2 = (c**2, T._1e2ac) if c != a else (a2, _1_0) 

1210 

1211 p3 = T._order3d(p3) 

1212 u3 = T._order3d(u3).unit() # unit vector, opposing signs 

1213 

1214 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz 

1215 ux, vy, wz = u3.times_(p3).xyz 

1216 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

1217 

1218 t = (p2 * c2), c2, b2 

1219 m = fdot(t, u2, v2, w2) # a2 factored out 

1220 if m < EPS0: # zero or near-null LOS vector 

1221 raise _ValueError(_near_(_null_)) 

1222 

1223 r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

1224 -w2 * y2, -u2 * y2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2, 

1225 -w2 * x2 * p2, b2 * u2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2) 

1226 if r > 0: # a2 factored out 

1227 r = sqrt(r) * b * c # == a * a * b * c / a2 

1228 elif r < 0: # LOS pointing away from or missing the triaxial 

1229 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

1230 

1231 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

1232 if d > 0: # POV inside or LOS outside or missing the triaxial 

1233 s = fsumf_(_N_1_0, x2 / a2, y2 / b2, z2 / c2) # like _sideOf 

1234 raise _ValueError(_outside_ if s > 0 else _inside_) 

1235 elif fsum1f_(x2, y2, z2) < d**2: # d past triaxial's center 

1236 raise _ValueError(_too_(_distant_)) 

1237 

1238 v = p3.minus(u3.times(d)) # cartesian type(pov) or Vector3d 

1239 h = p3.minus(v).length # distance to pov == -d 

1240 return T._order3d(v, reverse=True), h, None 

1241 

1242 

1243def hartzell4(pov, los=False, tri_biax=_WGS84, **name): 

1244 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight 

1245 from a Point-Of-View outside. 

1246 

1247 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple} 

1248 C{LatLon} or L{Vector3d}). 

1249 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Los}, L{Vector3d}), 

1250 C{True} for the I{normal, plumb} onto the surface or C{False} or 

1251 C{None} to point to the center of the tri-/biaxial. 

1252 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal} or 

1253 L{JacobiConformalSpherical}) or biaxial ellipsoid (L{Datum}, 

1254 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} radius, 

1255 conventionally in C{meter}). 

1256 @kwarg name: Optional, overriding C{B{name}="hartzell4"} (C{str}). 

1257 

1258 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h} 

1259 the distance from B{C{pov}} to C{(x, y, z)} I{along the} B{C{los}}, all 

1260 in C{meter}, conventionally. 

1261 

1262 @raise TriaxialError: Invalid B{C{pov}} or B{C{pov}} inside the tri-/biaxial or 

1263 invalid B{C{los}} or B{C{los}} points outside or away from 

1264 the tri-/biaxial. 

1265 

1266 @raise TypeError: Invalid B{C{tri_biax}}, C{ellipsoid} or C{datum}. 

1267 

1268 @see: Class L{pygeodesy3.Los}, functions L{pygeodesy.tyr3d} and L{pygeodesy.hartzell} 

1269 and U{lookAtSpheroid<https://PyPI.org/project/pymap3d>} and U{"Satellite 

1270 Line-of-Sight Intersection with Earth"<https://StephenHartzell.Medium.com/ 

1271 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

1272 ''' 

1273 if isinstance(tri_biax, Triaxial_): 

1274 T = tri_biax 

1275 else: 

1276 D = tri_biax if isinstance(tri_biax, Datum) else \ 

1277 _spherical_datum(tri_biax, name__=hartzell4) # _dunder_nameof 

1278 T = D.ellipsoid._triaxial 

1279 try: 

1280 v, h, i = _hartzell3(pov, los, T) 

1281 except Exception as x: 

1282 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x) 

1283 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, # _dunder_nameof 

1284 name=_name__(name, name__=hartzell4)) 

1285 

1286 

1287def _hypot21(x, y, z=0): 

1288 '''(INTERNAL) Compute M{x**2 + y**2 + z**2 - 1} with C{max(fabs(x), 

1289 fabs(y), fabs(z))} rarely greater than 1.0. 

1290 ''' 

1291 return fsumf_(_1_0, x**2, y**2, (z**2 if z else _0_0), _N_2_0) 

1292 

1293 

1294def _otherV3d_(x_xyz, y, z, **name): 

1295 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}. 

1296 ''' 

1297 return Vector3d(x_xyz, y, z, **name) if isscalar(x_xyz) else \ 

1298 _otherV3d(x_xyz=x_xyz, **name) 

1299 

1300 

1301def _plumbTo4(x, y, a, b, eps=EPS): 

1302 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}. 

1303 

1304 @see: Function C{pygeodesy.ellipsoids._plumbTo3} and I{Eberly}'s U{Distance 

1305 from a Point to ... an Ellipse ...<https://www.GeometricTools.com/ 

1306 Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1307 ''' 

1308 if b > a: 

1309 b, a, d, i = _plumbTo4(y, x, b, a, eps=eps) 

1310 return a, b, d, i 

1311 

1312 if not (b > 0 and isfinite(a)): 

1313 raise _ValueError(a=a, b=b) 

1314 

1315 i, _a = None, fabs 

1316 if y: 

1317 if x: 

1318 u = _a(x / a) 

1319 v = _a(y / b) 

1320 g = _hypot21(u, v) 

1321 if _a(g) < EPS02: # on the ellipse 

1322 a, b, d = x, y, _0_0 

1323 else: 

1324 r = (b / a)**2 

1325 t, i = _rootXd(_1_0 / r, 0, u, 0, v, g, eps) 

1326 a = x / (t * r + _1_0) 

1327 b = y / (t + _1_0) 

1328 d = hypot(x - a, y - b) 

1329 else: # x == 0 

1330 if y < 0: 

1331 b = -b 

1332 a, d = x, _a(y - b) 

1333 

1334 else: # y == 0 

1335 n = a * x 

1336 d = (a + b) * (a - b) 

1337 if d > _a(n): # PYCHOK no cover 

1338 r = n / d 

1339 a *= r 

1340 r = _1_0 - r**2 

1341 if r > EPS02: 

1342 b *= sqrt(r) 

1343 d = hypot(x - a, b) 

1344 else: 

1345 b = _0_0 

1346 d = _a(x - a) 

1347 else: 

1348 if x < 0: 

1349 a = -a 

1350 b, d = y, _a(x - a) 

1351 return a, b, d, i 

1352 

1353 

1354def _plumbTo5(x, y, z, Tun, eps=EPS): # MCCABE 19 in .testTriaxials 

1355 '''(INTERNAL) Nearest point on and distance to an I{un-/ordered} triaxial. 

1356 

1357 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https:// 

1358 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1359 ''' 

1360 a, b, c, T = Tun._ordered4 

1361 if Tun is not T: # T is ordered, Tun isn't 

1362 t = T._order3(x, y, z) + (T,) 

1363 a, b, c, d, i = _plumbTo5(*t, eps=eps) 

1364 return T._order3(a, b, c, reverse=True) + (d, i) 

1365 

1366 if not (c > 0 and isfinite(a)): 

1367 raise _ValueError(a=a, b=b, c=c) 

1368 

1369 if eps > 0: 

1370 val = max(eps * 1e8, EPS) 

1371 else: # no validation 

1372 val, eps = 0, max(EPS0, -eps) 

1373 

1374 i, _a = None, fabs 

1375 if z: 

1376 if y: 

1377 if x: 

1378 u = _a(x / a) 

1379 v = _a(y / b) 

1380 w = _a(z / c) 

1381 g = _hypot21(u, v, w) 

1382 if _a(g) < EPS02: # on the ellipsoid 

1383 a, b, c, d = x, y, z, _0_0 

1384 else: 

1385 r = T._1e2ac # (c / a)**2 

1386 s = T._1e2bc # (c / b)**2 

1387 t, i = _rootXd(_1_0 / r, _1_0 / s, u, v, w, g, eps) 

1388 a = x / (t * r + _1_0) 

1389 b = y / (t * s + _1_0) 

1390 c = z / (t + _1_0) 

1391 d = hypot_(x - a, y - b, z - c) 

1392 else: # x == 0 

1393 a = x # 0 

1394 b, c, d, i = _plumbTo4(y, z, b, c, eps=eps) 

1395 elif x: # y == 0 

1396 b = y # 0 

1397 a, c, d, i = _plumbTo4(x, z, a, c, eps=eps) 

1398 else: # x == y == 0 

1399 if z < 0: 

1400 c = -c 

1401 a, b, d = x, y, _a(z - c) 

1402 

1403 else: # z == 0 

1404 t = True 

1405 d = T._a2c2 # (a + c) * (a - c) 

1406 n = a * x 

1407 if d > _a(n): 

1408 u = n / d 

1409 d = T._b2c2 # (b + c) * (b - c) 

1410 n = b * y 

1411 if d > _a(n): 

1412 v = n / d 

1413 n = _hypot21(u, v) 

1414 if n < 0: 

1415 a *= u 

1416 b *= v 

1417 c *= sqrt(-n) 

1418 d = hypot_(x - a, y - b, c) 

1419 t = False 

1420 if t: 

1421 c = z # signed-0 

1422 a, b, d, i = _plumbTo4(x, y, a, b, eps=eps) 

1423 

1424 if val > 0: # validate 

1425 e = T.sideOf(a, b, c, eps=val) 

1426 if e: # not near the ellipsoid's surface 

1427 raise _ValueError(a=a, b=b, c=c, d=d, 

1428 sideOf=e, eps=val) 

1429 if d: # angle of delta and normal vector 

1430 m = Vector3d(x, y, z).minus_(a, b, c) 

1431 if m.euclid > val: 

1432 m = m.unit() 

1433 n = T.normal3d(a, b, c) 

1434 e = n.dot(m) # n.negate().dot(m) 

1435 if not isnear1(_a(e), eps1=val): 

1436 raise _ValueError(n=n, m=m, 

1437 dot=e, eps=val) 

1438 return a, b, c, d, i 

1439 

1440 

1441def _rootXd(r, s, u, v, w, g, eps): 

1442 '''(INTERNAL) Robust 2d- or 3d-root finder: 2d- if C{s == v == 0} else 3d-root. 

1443 

1444 @see: I{Eberly}'s U{Robust Root Finders ...<https://www.GeometricTools.com/ 

1445 Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1446 ''' 

1447 _1, __2 = _1_0, _0_5 

1448 _a, _h21 = fabs, _hypot21 

1449 

1450 u *= r 

1451 v *= s # 0 for 2d-root 

1452 t0 = w - _1 

1453 t1 = _0_0 if g < 0 else (hypot_(u, w, v) - _1) 

1454 # assert t0 <= t1 

1455 for i in range(1, _TRIPS): # 48-55 

1456 e = _a(t0 - t1) 

1457 if e < eps: 

1458 break 

1459 t = (t0 + t1) * __2 

1460 if t in (t0, t1): 

1461 break 

1462 g = _h21(u / (t + r), w / (t + _1), 

1463 (v / (t + s)) if v else 0) 

1464 if g > 0: 

1465 t0 = t 

1466 elif g < 0: 

1467 t1 = t 

1468 else: 

1469 break 

1470 else: # PYCHOK no cover 

1471 t = Fmt.no_convergence(e, eps) 

1472 raise _ValueError(t, txt__=_rootXd) 

1473 return t, i 

1474 

1475 

1476def _sideOf(xyz, abc, eps=EPS): 

1477 '''(INTERNAL) Helper for C{_hartzell3}, M{.sideOf} and M{.reverseCartesian}. 

1478 

1479 @return: M{sum((x / a)**2 for x, a in zip(xyz, abc)) - 1} or C{INT0}. 

1480 ''' 

1481 s = fsumf_(_N_1_0, *((x / a)**2 for x, a in _zip(xyz, abc) if a)) # strict=True 

1482 return INT0 if fabs(s) < eps else s 

1483 

1484 

1485if __name__ == '__main__': 

1486 

1487 from pygeodesy import printf 

1488 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_ 

1489 

1490 # __doc__ of this file, force all into registery 

1491 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_) 

1492 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t)) 

1493 

1494# **) MIT License 

1495# 

1496# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1497# 

1498# Permission is hereby granted, free of charge, to any person obtaining a 

1499# copy of this software and associated documentation files (the "Software"), 

1500# to deal in the Software without restriction, including without limitation 

1501# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1502# and/or sell copies of the Software, and to permit persons to whom the 

1503# Software is furnished to do so, subject to the following conditions: 

1504# 

1505# The above copyright notice and this permission notice shall be included 

1506# in all copies or substantial portions of the Software. 

1507# 

1508# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1509# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1510# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1511# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1512# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1513# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1514# OTHER DEALINGS IN THE SOFTWARE.