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.dms import toDMS # _MODS 

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

40# from pygeodesy.elliptic import Elliptic # _MODS 

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

42from pygeodesy.fmath import Fdot, fdot, fmean_, hypot, hypot_, norm2, sqrt0 

43from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1f_ 

44from pygeodesy.interns import NN, _a_, _b_, _beta_, _c_, _distant_, _finite_, \ 

45 _height_, _inside_, _near_, _negative_, _not_, \ 

46 _NOTEQUAL_, _null_, _opposite_, _outside_, _SPACE_, \ 

47 _spherical_, _too_, _x_, _y_ 

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

49from pygeodesy.named import _NamedEnum, _NamedEnumItem, _NamedTuple, _Pass, \ 

50 _lazyNamedEnumItem as _lazy 

51from pygeodesy.namedTuples import LatLon3Tuple, Vector3Tuple, Vector4Tuple 

52from pygeodesy.props import Property_RO, property_RO 

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

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

55 Radius, Scalar_, _toDegrees, _toRadians 

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__ = '24.04.30' 

63 

64_not_ordered_ = _not_('ordered') 

65_omega_ = 'omega' 

66_TRIPS = 269 # 48-55, Eberly 1074? 

67 

68 

69class _NamedTupleTo(_NamedTuple): # in .testNamedTuples, like .cartesianBase.RadiusThetaPhi3Tuple 

70 '''(INTERNAL) Base for C{-.toDegrees}, C{-.toRadians}. 

71 ''' 

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

73 a, b, _ = _toDegrees(self, a, b, **toDMS_kwds) 

74 return _ or self.classof(a, b, *c, name=self.name) 

75 

76 def _toRadians(self, a, b, *c): 

77 a, b, _ = _toRadians(self, a, b) 

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

79 

80 

81class BetaOmega2Tuple(_NamedTupleTo): 

82 '''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and 

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

84 L{Degrees}). 

85 ''' 

86 _Names_ = (_beta_, _omega_) 

87 _Units_ = (_Pass, _Pass) 

88 

89 def toDegrees(self, **toDMS_kwds): 

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

91 

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

93 C{beta} and C{omega} both in L{Degrees} 

94 or as a L{toDMS} string provided some 

95 B{C{toDMS_kwds}} keyword arguments are 

96 specified. 

97 ''' 

98 return _NamedTupleTo._toDegrees(self, *self, **toDMS_kwds) 

99 

100 def toRadians(self): 

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

102 

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

104 C{beta} and C{omega} both in L{Radians}. 

105 ''' 

106 return _NamedTupleTo._toRadians(self, *self) 

107 

108 

109class BetaOmega3Tuple(_NamedTupleTo): 

110 '''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and 

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

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

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

114 in C{meter}, conventionally. 

115 ''' 

116 _Names_ = BetaOmega2Tuple._Names_ + (_height_,) 

117 _Units_ = BetaOmega2Tuple._Units_ + ( Meter,) 

118 

119 def toDegrees(self, **toDMS_kwds): 

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

121 

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

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

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

125 keyword arguments are specified. 

126 ''' 

127 return _NamedTupleTo._toDegrees(self, *self, **toDMS_kwds) 

128 

129 def toRadians(self): 

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

131 

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

133 C{beta} and C{omega} both in L{Radians}. 

134 ''' 

135 return _NamedTupleTo._toRadians(self, *self) 

136 

137 def to2Tuple(self): 

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

139 ''' 

140 return BetaOmega2Tuple(*self[:2]) 

141 

142 

143class Jacobi2Tuple(_NamedTupleTo): 

144 '''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y} 

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

146 ''' 

147 _Names_ = (_x_, _y_) 

148 _Units_ = (_Pass, _Pass) 

149 

150 def toDegrees(self, **toDMS_kwds): 

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

152 

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

154 both in L{Degrees} or as a L{toDMS} string 

155 provided some B{C{toDMS_kwds}} keyword 

156 arguments are specified. 

157 ''' 

158 return _NamedTupleTo._toDegrees(self, *self, **toDMS_kwds) 

159 

160 def toRadians(self): 

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

162 

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

164 and C{y} both in L{Radians}. 

165 ''' 

166 return _NamedTupleTo._toRadians(self, *self) 

167 

168 

169class Triaxial_(_NamedEnumItem): 

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

171 

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

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

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

175 I{omega}=0. 

176 

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

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

179 

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

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

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

183 L{Degrees} if converted). 

184 ''' 

185 _ijk = _kji = None 

186 _unordered = True 

187 

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

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

190 

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

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

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

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

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

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

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

198 

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

200 ''' 

201 try: 

202 a = a_triaxial 

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

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

205 except (TypeError, ValueError) as x: 

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

207 if name: 

208 self.name = name 

209 

210 a, b, c = self._abc3 = t 

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

212 s, _, t = sorted(t) 

213 if not (isfinite(t) and s > 0): 

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

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

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

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

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

219 

220 def __str__(self): 

221 return self.toStr() 

222 

223 @Property_RO 

224 def a(self): 

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

226 ''' 

227 a, _, _ = self._abc3 

228 return a 

229 

230 @Property_RO 

231 def _a2b2(self): 

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

233 ''' 

234 a, b, _ = self._abc3 

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

236 

237 @Property_RO 

238 def _a2_b2(self): 

239 '''(INTERNAL) Get C{(a/b)**2}. 

240 ''' 

241 a, b, _ = self._abc3 

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

243 

244 @Property_RO 

245 def _a2c2(self): 

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

247 ''' 

248 a, _, c = self._abc3 

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

250 

251 @Property_RO 

252 def area(self): 

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

254 ''' 

255 c, b, a = sorted(self._abc3) 

256 if a > c: 

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

258 Ellipsoid(a, b=c).areax # a == b 

259 else: # a == c == b 

260 a = Meter2(area=a**2 * PI4) 

261 return a 

262 

263 def area_p(self, p=1.6075): 

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

265 

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

267 for "near-flat" triaxials. 

268 

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

270 ''' 

271 a, b, c = self._abc3 

272 if a == b == c: 

273 a *= a 

274 else: 

275 _p = pow 

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

277 return Meter2(area_p=a * PI4) 

278 

279 @Property_RO 

280 def b(self): 

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

282 ''' 

283 _, b, _ = self._abc3 

284 return b 

285 

286 @Property_RO 

287 def _b2c2(self): 

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

289 ''' 

290 _, b, c = self._abc3 

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

292 

293 @Property_RO 

294 def c(self): 

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

296 ''' 

297 _, _, c = self._abc3 

298 return c 

299 

300 @Property_RO 

301 def _c2_b2(self): 

302 '''(INTERNAL) Get C{(c/b)**2}. 

303 ''' 

304 _, b, c = self._abc3 

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

306 

307 @Property_RO 

308 def e2ab(self): 

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

310 ''' 

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

312 

313 @Property_RO 

314 def _1e2ab(self): 

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

316 ''' 

317 a, b, _ = self._abc3 

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

319 

320 @Property_RO 

321 def e2ac(self): 

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

323 ''' 

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

325 

326 @Property_RO 

327 def _1e2ac(self): 

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

329 ''' 

330 a, _, c = self._abc3 

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

332 

333 @Property_RO 

334 def e2bc(self): 

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

336 ''' 

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

338 

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

340 

341 @property_RO 

342 def _Elliptic(self): 

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

344 ''' 

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

346 return E 

347 

348 def hartzell4(self, pov, los=False, name=NN): 

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

350 from a Point-Of-View in space. 

351 

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

353 ''' 

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

355 

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

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

358 

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

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

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

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

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

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

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

366 value to skip validation. 

367 

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

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

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

371 

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

373 root finding or validation failed. 

374 

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

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

377 ''' 

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

379 

380 i, h = None, v.length 

381 if h < EPS0: # EPS 

382 x = y = z = _0_0 

383 h -= min(self._abc3) # nearest 

384 elif r: # .isSpherical 

385 x, y, z = v.times(r / h).xyz 

386 h -= r 

387 else: 

388 x, y, z = v.xyz 

389 try: 

390 if normal: # plumb to surface 

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

392 else: # radial to center 

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

394 h = v.minus_(x, y, z).length 

395 except Exception as e: 

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

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

398 h = -h # below the surface 

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

400 

401 @Property_RO 

402 def isOrdered(self): 

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

404 ''' 

405 a, b, c = self._abc3 

406 return bool(a >= b > c) # b > c! 

407 

408 @Property_RO 

409 def isSpherical(self): 

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

411 ''' 

412 a, b, c = self._abc3 

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

414 

415 def _norm2(self, s, c, *a): 

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

417 ''' 

418 if fabs(_hypot21(s, c)) > EPS02: 

419 s, c = norm2(s, c) 

420 if a: 

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

422 return float0_(s, c) 

423 

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

425 '''Get a 3-D vector at a cartesian on and 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 @raise TriaxialError: Zero length cartesian or vector. 

437 

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

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

440 ''' 

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

442 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered 

443 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2 

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

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

446 if n.length < EPS0: 

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

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

449 

450 @Property_RO 

451 def _normal3d(self): 

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

453 ''' 

454 d = max(self._abc3) 

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

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

457 

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

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

460 

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

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

463 ''' 

464 ijk = self._order_ijk(**reverse) 

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

466 

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

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

469 

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

471 ''' 

472 ijk = self._order_ijk(**reverse) 

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

474 

475 @Property_RO 

476 def _ordered4(self): 

477 '''(INTERNAL) Helper for C{_hartzell3} and C{_normalTo5}. 

478 ''' 

479 def _order2(reverse, a, b, c): 

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

481 

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

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

484 ''' 

485 i, j, k = 0, 1, 2 # range(3) 

486 if a < b: 

487 a, b, i, j = b, a, j, i 

488 if a < c: 

489 a, c, i, k = c, a, k, i 

490 if b < c: 

491 b, c, j, k = c, b, k, j 

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

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

494 return (a, b, c), ijk 

495 

496 abc, T = self._abc3, self 

497 if not self.isOrdered: 

498 abc, ijk = _order2(False, *abc) 

499 if ijk: 

500 _, kji = _order2(True, *ijk) 

501 T = Triaxial_(*abc) 

502 T._ijk, T._kji = ijk, kji 

503 return abc + (T,) 

504 

505 def _order_ijk(self, reverse=False): 

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

507 ''' 

508 return self._kji if reverse else self._ijk 

509 

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

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

512 ''' 

513 def _rphi(a, b, sphi, cphi): 

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

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

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

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

518 

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

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

521 

522 a, b, c = self._abc3 

523 if a != b: 

524 a = _rphi(a, b, sb, cb) 

525 if a != c: 

526 c = _rphi(a, c, sa, ca) 

527 z, r = c * sa, c * ca 

528 x, y = r * cb, r * sb 

529 return x, y, z 

530 

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

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

533 

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

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

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

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

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

539 

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

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

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

543 respectively outside this triaxial. 

544 

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

546 ''' 

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

548 

549 def toEllipsoid(self, name=NN): 

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

551 

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

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

554 

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

556 

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

558 ''' 

559 a, b, c = self._abc3 

560 if a == b: 

561 b = c # N = c-Z 

562 elif b == c: # N = a-X 

563 a, b = b, a 

564 elif a != c: # N = b-Y 

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

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

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

568 

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

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

571 

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

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

574 this triaxial's name. 

575 

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

577 ''' 

578 T = Triaxial_ 

579 t = T.a, 

580 J = JacobiConformalSpherical 

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

582 t += T.e2ab, T.e2bc, T.e2ac 

583 J = JacobiConformal 

584 if isinstance(self, J): 

585 t += J.xyQ2, 

586 t += T.volume, T.area 

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

588 

589 @Property_RO 

590 def volume(self): 

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

592 ''' 

593 a, b, c = self._abc3 

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

595 

596 

597class Triaxial(Triaxial_): 

598 '''I{Ordered} triaxial ellipsoid. 

599 

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

601 ''' 

602 _unordered = False 

603 

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

605 '''New I{ordered} L{Triaxial}. 

606 

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

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

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

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

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

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

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

614 

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

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

617 

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

619 ''' 

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

621 

622 @Property_RO 

623 def _a2b2_a2c2(self): 

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

625 ''' 

626 return self._a2b2 / self._a2c2 

627 

628 @Property_RO 

629 def area(self): 

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

631 

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

633 ''' 

634 a, b, c = self._abc3 

635 if a != b: 

636 kp2, k2 = self._k2_kp2 # swapped! 

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

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

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

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

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

642 aE.fF(r) * c2 / s) 

643 a = Meter2(area=a * b * PI2) 

644 else: # a == b > c 

645 a = Ellipsoid(a, b=c).areax 

646 return a 

647 

648 def _exyz3(self, u): 

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

650 ''' 

651 if u > 0: 

652 u2 = u**2 

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

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

655 else: 

656 x = y = u = _0_0 

657 return x, y, u 

658 

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

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

661 and height to cartesian. 

662 

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

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

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

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

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

668 

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

670 

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

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

673 ''' 

674 if height: 

675 h = self._Height(height) 

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

677 else: 

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

679 if z: # and x and y: 

680 sa, ca = SinCos2(beta) 

681 sb, cb = SinCos2(omega) 

682 

683 r = self._a2b2_a2c2 

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

685 y *= ca * sb 

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

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

688 

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

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

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

692 

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

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

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

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

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

698 

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

700 

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

702 

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

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

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

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

707 ''' 

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

709 return Vector3Tuple(*t, name=name) 

710 

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

712 '''Project a cartesian on this triaxial. 

713 

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

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

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

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

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

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

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

721 

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

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

724 ''' 

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

726 _ = t.rename(name) 

727 return t 

728 

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

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

731 

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

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

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

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

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

737 

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

739 

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

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

742 ''' 

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

744 

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

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

747 

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

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

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

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

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

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

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

755 

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

757 

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

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

760 ''' 

761 sa, ca = self._norm2(slat, clat) 

762 sb, cb = self._norm2(slon, clon) 

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

764 

765 def _forwardLatLon3(self, height, name, sa, ca, sb, cb): 

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

767 ''' 

768 ca_x_sb = ca * sb 

769 h = self._Height(height) 

770 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2 

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

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

773 x = (h + n) * ca * cb 

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

775 z = (h + n * self._1e2ac) * sa 

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

777 

778 def _Height(self, height): 

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

780 ''' 

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

782 

783 @Property_RO 

784 def _k2_kp2(self): 

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

786 ''' 

787 # k2 = a2b2 / a2c2 * c2_b2 

788 # kp2 = b2c2 / a2c2 * a2_b2 

789 # b2 = b**2 

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

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

792 # aE = Elliptic(kp2, 0, k2, 1) 

793 return (self._a2b2_a2c2 * self._c2_b2, 

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

795 

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

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

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

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

800 ''' 

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

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

803 

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

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

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

807 

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

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

810 

811 x, y, z = self._abc3 

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

813 y *= ca * sb 

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

815 return x, y, z 

816 

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

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

819 and height. 

820 

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

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

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

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

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

826 

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

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

829 units as this triaxial's axes. 

830 

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

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

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

834 ''' 

835 v = _otherV3d_(x_xyz, y, z) 

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

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

838 

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

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

841 

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

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

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

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

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

847 as the axes). 

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

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

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

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

852 

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

854 

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

856 

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

858 ''' 

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

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

861 if s: # PYCHOK no cover 

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

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

864 

865 if h: 

866 if normal: 

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

868 elif v.length > EPS0: 

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

870 return v.xyz # Vector3Tuple 

871 

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

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

874 

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

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

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

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

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

880 

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

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

883 as this triaxial's axes. 

884 

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

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

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

888 ''' 

889 v = _otherV3d_(x_xyz, y, z) 

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

891 self._1e2bc, # == 1 - e_sub_y**2 

892 _1_0) 

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

894 return LatLon3Tuple(*t, name=name) 

895 

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

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

898 ''' 

899 x, y, z = s.xyz 

900 d = hypot( x, y) 

901 a = atan2_(z, d) 

902 b = atan2_(y, x) 

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

904 return a, b, h 

905 

906 

907class JacobiConformal(Triaxial): 

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

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

910 

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

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

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

914 returned in the case of an ellipsoid of revolution. 

915 

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

917 licensed under the MIT/X11 License. 

918 

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

920 L{JacobiConformalSpherical}. 

921 

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

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

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

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

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

927 ''' 

928 

929 @Property_RO 

930 def _xE(self): 

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

932 ''' 

933 k2, kp2 = self._k2_kp2 

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

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

936 

937 def xR(self, omega): 

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

939 

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

941 

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

943 ''' 

944 return self.xR_(*SinCos2(omega)) 

945 

946 def xR_(self, somega, comega): 

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

948 

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

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

951 

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

953 ''' 

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

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

956 

957 @Property_RO 

958 def xyQ2(self): 

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

960 ''' 

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

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

963 name=JacobiConformal.xyQ2.name) 

964 

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

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

967 

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

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

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

971 

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

973 ''' 

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

975 name=name or self.xyR2.__name__) 

976 

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

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

979 

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

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

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

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

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

985 

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

987 ''' 

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

989 self.yR_(sbeta, cbeta), 

990 name=name or self.xyR2_.__name__) 

991 

992 @Property_RO 

993 def _yE(self): 

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

995 ''' 

996 kp2, k2 = self._k2_kp2 # swapped! 

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

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

999 

1000 def yR(self, beta): 

1001 '''Compute a Jacobi Conformal C{y} projection. 

1002 

1003 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

1004 

1005 @return: The C{y} projection (L{Radians}). 

1006 ''' 

1007 return self.yR_(*SinCos2(beta)) 

1008 

1009 def yR_(self, sbeta, cbeta): 

1010 '''Compute a Jacobi Conformal C{y} projection. 

1011 

1012 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

1013 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

1014 

1015 @return: The C{y} projection (L{Radians}). 

1016 ''' 

1017 s, c = self._norm2(sbeta, cbeta, self.c) 

1018 return Radians(y=self._yE.fPi(s, c) * self._c2_b2) 

1019 

1020 

1021class JacobiConformalSpherical(JacobiConformal): 

1022 '''An alternate, I{spherical} L{JacobiConformal} projection. 

1023 

1024 @see: L{JacobiConformal} for other and more details. 

1025 ''' 

1026 _ab = _bc = 0 

1027 

1028 def __init__(self, radius_triaxial, ab=0, bc=0, name=NN): 

1029 '''New L{JacobiConformalSpherical}. 

1030 

1031 @arg radius_triaxial: Radius (C{scalar}, conventionally in 

1032 C{meter}) or an other L{JacobiConformalSpherical}, 

1033 L{JacobiConformal} or ordered L{Triaxial}. 

1034 @kwarg ab: Relative magnitude of C{B{a} - B{b}} (C{meter}, 

1035 same units as C{scalar B{radius}}. 

1036 @kwarg bc: Relative magnitude of C{B{b} - B{c}} (C{meter}, 

1037 same units as C{scalar B{radius}}. 

1038 @kwarg name: Optional name (C{str}). 

1039 

1040 @raise TriaxialError: Invalid B{C{radius_triaxial}}, negative 

1041 B{C{ab}}, negative B{C{bc}} or C{(B{ab} 

1042 + B{bc})} not positive. 

1043 

1044 @note: If B{C{radius_triaxial}} is a L{JacobiConformalSpherical} 

1045 and if B{C{ab}} and B{C{bc}} are both zero or C{None}, 

1046 the B{C{radius_triaxial}}'s C{ab}, C{bc}, C{a}, C{b} 

1047 and C{c} are copied. 

1048 ''' 

1049 try: 

1050 r = radius_triaxial 

1051 if isinstance(r, Triaxial): # ordered only 

1052 t = r._abc3 

1053 j = isinstance(r, JacobiConformalSpherical) and not bool(ab or bc) 

1054 else: 

1055 t = (Radius(radius=r),) * 3 

1056 j = False 

1057 self._ab = r.ab if j else Scalar_(ab=ab) # low=0 

1058 self._bc = r.bc if j else Scalar_(bc=bc) # low=0 

1059 if (self.ab + self.bc) <= 0: 

1060 raise ValueError(_negative_) 

1061 a, _, c = self._abc3 = t 

1062 if not (a >= c and isfinite(self._a2b2) 

1063 and isfinite(self._a2c2)): 

1064 raise ValueError(_not_(_finite_)) 

1065 except (TypeError, ValueError) as x: 

1066 raise TriaxialError(radius_triaxial=r, ab=ab, bc=bc, cause=x) 

1067 if name: 

1068 self.name = name 

1069 

1070 @Property_RO 

1071 def ab(self): 

1072 '''Get relative magnitude C{ab} (C{meter}, same units as B{C{a}}). 

1073 ''' 

1074 return self._ab 

1075 

1076 @Property_RO 

1077 def _a2b2(self): 

1078 '''(INTERNAL) Get C{a**2 - b**2} == ab * (a + b). 

1079 ''' 

1080 a, b, _ = self._abc3 

1081 return self.ab * (a + b) 

1082 

1083 @Property_RO 

1084 def _a2c2(self): 

1085 '''(INTERNAL) Get C{a**2 - c**2} == a2b2 + b2c2. 

1086 ''' 

1087 return self._a2b2 + self._b2c2 

1088 

1089 @Property_RO 

1090 def bc(self): 

1091 '''Get relative magnitude C{bc} (C{meter}, same units as B{C{a}}). 

1092 ''' 

1093 return self._bc 

1094 

1095 @Property_RO 

1096 def _b2c2(self): 

1097 '''(INTERNAL) Get C{b**2 - c**2} == bc * (b + c). 

1098 ''' 

1099 _, b, c = self._abc3 

1100 return self.bc * (b + c) 

1101 

1102 @Property_RO 

1103 def radius(self): 

1104 '''Get radius (C{meter}, conventionally). 

1105 ''' 

1106 return self.a 

1107 

1108 

1109class TriaxialError(_ValueError): 

1110 '''Raised for L{Triaxial} issues. 

1111 ''' 

1112 pass # ... 

1113 

1114 

1115class Triaxials(_NamedEnum): 

1116 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class 

1117 to accommodate the L{_LazyNamedEnumItem} properties. 

1118 ''' 

1119 def _Lazy(self, *abc, **name): 

1120 '''(INTERNAL) Instantiate the C{Triaxial}. 

1121 ''' 

1122 a, b, c = map(km2m, abc) 

1123 return Triaxial(a, b, c, **name) 

1124 

1125Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton 

1126'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.''' 

1127# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in Km 

1128# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf> 

1129# <https://link.Springer.com/article/10.1007/s00190-022-01650-9> 

1130_abc84_35 = (_EWGS84.a + 35), (_EWGS84.a - 35), _EWGS84.b 

1131Triaxials._assert( # a (Km) b (Km) c (Km) planet 

1132 Amalthea = _lazy('Amalthea', 125.0, 73.0, 64), # Jupiter 

1133 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus 

1134 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544), 

1135 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn 

1136 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter 

1137 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter 

1138 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3), 

1139 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn 

1140 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus 

1141 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth 

1142 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn 

1143 WGS84_35 = _lazy('WGS84_35', *map(m2km, _abc84_35))) 

1144del _abc84_35, _EWGS84 

1145 

1146 

1147def _getitems(items, *indices): 

1148 '''(INTERNAL) Get the C{items} at the given I{indices}. 

1149 

1150 @return: C{Type(items[i] for i in indices)} with 

1151 C{Type = type(items)}, any C{type} having 

1152 the special method C{__getitem__}. 

1153 ''' 

1154 return type(items)(map(items.__getitem__, indices)) 

1155 

1156 

1157def _hartzell3(pov, los, Tun): # in .ellipsoids.hartzell4, .formy.hartzell 

1158 '''(INTERNAL) Hartzell's "Satellite Line-of-Sight Intersection ...", 

1159 formula from a Point-Of-View to an I{un-/ordered} Triaxial. 

1160 ''' 

1161 def _toUvwV3d(los, pov): 

1162 try: # pov must be LatLon or Cartesian if los is a Los 

1163 v = los.toUvw(pov) 

1164 except (AttributeError, TypeError): 

1165 v = _otherV3d(los=los) 

1166 return v 

1167 

1168 p3 = _otherV3d(pov=pov.toCartesian() if isLatLon(pov) else pov) 

1169 if los is True: # normal 

1170 a, b, c, d, i = _normalTo5(p3.x, p3.y, p3.z, Tun) 

1171 return type(p3)(a, b, c), d, i 

1172 

1173 u3 = p3.negate() if los is False or los is None else _toUvwV3d(los, pov) 

1174 

1175 a, b, c, T = Tun._ordered4 

1176 

1177 a2 = a**2 # largest, factored out 

1178 b2, p2 = (b**2, T._1e2ab) if b != a else (a2, _1_0) 

1179 c2, q2 = (c**2, T._1e2ac) if c != a else (a2, _1_0) 

1180 

1181 p3 = T._order3d(p3) 

1182 u3 = T._order3d(u3).unit() # unit vector, opposing signs 

1183 

1184 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz 

1185 ux, vy, wz = u3.times_(p3).xyz 

1186 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

1187 

1188 t = (p2 * c2), c2, b2 

1189 m = fdot(t, u2, v2, w2) # a2 factored out 

1190 if m < EPS0: # zero or near-null LOS vector 

1191 raise _ValueError(_near_(_null_)) 

1192 

1193 r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

1194 -w2 * y2, -u2 * y2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2, 

1195 -w2 * x2 * p2, b2 * u2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2) 

1196 if r > 0: # a2 factored out 

1197 r = sqrt(r) * b * c # == a * a * b * c / a2 

1198 elif r < 0: # LOS pointing away from or missing the triaxial 

1199 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

1200 

1201 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

1202 if d > 0: # POV inside or LOS outside or missing the triaxial 

1203 s = fsumf_(_N_1_0, x2 / a2, y2 / b2, z2 / c2) # like _sideOf 

1204 raise _ValueError(_outside_ if s > 0 else _inside_) 

1205 elif fsum1f_(x2, y2, z2) < d**2: # d past triaxial's center 

1206 raise _ValueError(_too_(_distant_)) 

1207 

1208 v = p3.minus(u3.times(d)) # cartesian type(pov) or Vector3d 

1209 h = p3.minus(v).length # distance to pov == -d 

1210 return T._order3d(v, reverse=True), h, None 

1211 

1212 

1213def hartzell4(pov, los=False, tri_biax=_WGS84, name=NN): 

1214 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight 

1215 from a Point-Of-View outside. 

1216 

1217 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple} 

1218 C{LatLon} or L{Vector3d}). 

1219 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Los}, L{Vector3d}), 

1220 C{True} for the I{normal, plumb} onto the surface or C{False} or 

1221 C{None} to point to the center of the tri-/biaxial. 

1222 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal} or 

1223 L{JacobiConformalSpherical}) or biaxial ellipsoid (L{Datum}, 

1224 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} radius, 

1225 conventionally in C{meter}). 

1226 @kwarg name: Optional name (C{str}). 

1227 

1228 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h} 

1229 the distance from B{C{pov}} to C{(x, y, z)} I{along the} B{C{los}}, all 

1230 in C{meter}, conventionally. 

1231 

1232 @raise TriaxialError: Invalid B{C{pov}} or B{C{pov}} inside the tri-/biaxial or 

1233 invalid B{C{los}} or B{C{los}} points outside or away from 

1234 the tri-/biaxial. 

1235 

1236 @raise TypeError: Invalid B{C{tri_biax}}, C{ellipsoid} or C{datum}. 

1237 

1238 @see: Class L{pygeodesy3.Los}, functions L{pygeodesy.tyr3d} and L{pygeodesy.hartzell} 

1239 and U{lookAtSpheroid<https://PyPI.org/project/pymap3d>} and U{"Satellite 

1240 Line-of-Sight Intersection with Earth"<https://StephenHartzell.Medium.com/ 

1241 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

1242 ''' 

1243 n = hartzell4.__name__ 

1244 if isinstance(tri_biax, Triaxial_): 

1245 T = tri_biax 

1246 else: 

1247 D = tri_biax if isinstance(tri_biax, Datum) else \ 

1248 _spherical_datum(tri_biax, name=n) 

1249 T = D.ellipsoid._triaxial 

1250 try: 

1251 v, h, i = _hartzell3(pov, los, T) 

1252 except Exception as x: 

1253 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x) 

1254 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name=name or n) 

1255 

1256 

1257def _hypot21(x, y, z=0): 

1258 '''(INTERNAL) Compute M{x**2 + y**2 + z**2 - 1} with C{max(fabs(x), 

1259 fabs(y), fabs(z))} rarely greater than 1.0. 

1260 ''' 

1261 return fsumf_(_1_0, x**2, y**2, (z**2 if z else _0_0), _N_2_0) 

1262 

1263 

1264def _normalTo4(x, y, a, b, eps=EPS): 

1265 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}. 

1266 

1267 @see: Function C{pygeodesy.ellipsoids._normalTo3} and I{Eberly}'s U{Distance 

1268 from a Point to ... an Ellipse ...<https://www.GeometricTools.com/ 

1269 Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1270 ''' 

1271 if b > a: 

1272 b, a, d, i = _normalTo4(y, x, b, a, eps=eps) 

1273 return a, b, d, i 

1274 

1275 if not (b > 0 and isfinite(a)): 

1276 raise _ValueError(a=a, b=b) 

1277 

1278 i, _a = None, fabs 

1279 if y: 

1280 if x: 

1281 u = _a(x / a) 

1282 v = _a(y / b) 

1283 g = _hypot21(u, v) 

1284 if _a(g) < EPS02: # on the ellipse 

1285 a, b, d = x, y, _0_0 

1286 else: 

1287 r = (b / a)**2 

1288 t, i = _rootXd(_1_0 / r, 0, u, 0, v, g, eps) 

1289 a = x / (t * r + _1_0) 

1290 b = y / (t + _1_0) 

1291 d = hypot(x - a, y - b) 

1292 else: # x == 0 

1293 if y < 0: 

1294 b = -b 

1295 a, d = x, _a(y - b) 

1296 

1297 else: # y == 0 

1298 n = a * x 

1299 d = (a + b) * (a - b) 

1300 if d > _a(n): # PYCHOK no cover 

1301 r = n / d 

1302 a *= r 

1303 r = _1_0 - r**2 

1304 if r > EPS02: 

1305 b *= sqrt(r) 

1306 d = hypot(x - a, b) 

1307 else: 

1308 b = _0_0 

1309 d = _a(x - a) 

1310 else: 

1311 if x < 0: 

1312 a = -a 

1313 b, d = y, _a(x - a) 

1314 return a, b, d, i 

1315 

1316 

1317def _normalTo5(x, y, z, Tun, eps=EPS): # MCCABE 19 

1318 '''(INTERNAL) Nearest point on and distance to an I{un-/ordered} triaxial. 

1319 

1320 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https:// 

1321 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1322 ''' 

1323 a, b, c, T = Tun._ordered4 

1324 if Tun is not T: # T is ordered, Tun isn't 

1325 t = T._order3(x, y, z) + (T,) 

1326 a, b, c, d, i = _normalTo5(*t, eps=eps) 

1327 return T._order3(a, b, c, reverse=True) + (d, i) 

1328 

1329 if not (c > 0 and isfinite(a)): 

1330 raise _ValueError(a=a, b=b, c=c) 

1331 

1332 if eps > 0: 

1333 val = max(eps * 1e8, EPS) 

1334 else: # no validation 

1335 val, eps = 0, max(EPS0, -eps) 

1336 

1337 i, _a = None, fabs 

1338 if z: 

1339 if y: 

1340 if x: 

1341 u = _a(x / a) 

1342 v = _a(y / b) 

1343 w = _a(z / c) 

1344 g = _hypot21(u, v, w) 

1345 if _a(g) < EPS02: # on the ellipsoid 

1346 a, b, c, d = x, y, z, _0_0 

1347 else: 

1348 r = T._1e2ac # (c / a)**2 

1349 s = T._1e2bc # (c / b)**2 

1350 t, i = _rootXd(_1_0 / r, _1_0 / s, u, v, w, g, eps) 

1351 a = x / (t * r + _1_0) 

1352 b = y / (t * s + _1_0) 

1353 c = z / (t + _1_0) 

1354 d = hypot_(x - a, y - b, z - c) 

1355 else: # x == 0 

1356 a = x # 0 

1357 b, c, d, i = _normalTo4(y, z, b, c, eps=eps) 

1358 elif x: # y == 0 

1359 b = y # 0 

1360 a, c, d, i = _normalTo4(x, z, a, c, eps=eps) 

1361 else: # x == y == 0 

1362 if z < 0: 

1363 c = -c 

1364 a, b, d = x, y, _a(z - c) 

1365 

1366 else: # z == 0 

1367 t = True 

1368 d = T._a2c2 # (a + c) * (a - c) 

1369 n = a * x 

1370 if d > _a(n): 

1371 u = n / d 

1372 d = T._b2c2 # (b + c) * (b - c) 

1373 n = b * y 

1374 if d > _a(n): 

1375 v = n / d 

1376 n = _hypot21(u, v) 

1377 if n < 0: 

1378 a *= u 

1379 b *= v 

1380 c *= sqrt(-n) 

1381 d = hypot_(x - a, y - b, c) 

1382 t = False 

1383 if t: 

1384 c = z # signed-0 

1385 a, b, d, i = _normalTo4(x, y, a, b, eps=eps) 

1386 

1387 if val > 0: # validate 

1388 e = T.sideOf(a, b, c, eps=val) 

1389 if e: # not near the ellipsoid's surface 

1390 raise _ValueError(a=a, b=b, c=c, d=d, 

1391 sideOf=e, eps=val) 

1392 if d: # angle of delta and normal vector 

1393 m = Vector3d(x, y, z).minus_(a, b, c) 

1394 if m.euclid > val: 

1395 m = m.unit() 

1396 n = T.normal3d(a, b, c) 

1397 e = n.dot(m) # n.negate().dot(m) 

1398 if not isnear1(_a(e), eps1=val): 

1399 raise _ValueError(n=n, m=m, 

1400 dot=e, eps=val) 

1401 return a, b, c, d, i 

1402 

1403 

1404def _otherV3d_(x_xyz, y, z, **name): 

1405 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}. 

1406 ''' 

1407 return Vector3d(x_xyz, y, z, **name) if isscalar(x_xyz) else \ 

1408 _otherV3d(x_xyz=x_xyz) 

1409 

1410 

1411def _rootXd(r, s, u, v, w, g, eps): 

1412 '''(INTERNAL) Robust 2d- or 3d-root finder: 2d- if C{s == v == 0} else 3d-root. 

1413 

1414 @see: I{Eberly}'s U{Robust Root Finders ...<https://www.GeometricTools.com/ 

1415 Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1416 ''' 

1417 _1, __2 = _1_0, _0_5 

1418 _a, _h21 = fabs, _hypot21 

1419 

1420 u *= r 

1421 v *= s # 0 for 2d-root 

1422 t0 = w - _1 

1423 t1 = _0_0 if g < 0 else (hypot_(u, w, v) - _1) 

1424 # assert t0 <= t1 

1425 for i in range(1, _TRIPS): # 48-55 

1426 e = _a(t0 - t1) 

1427 if e < eps: 

1428 break 

1429 t = (t0 + t1) * __2 

1430 if t in (t0, t1): 

1431 break 

1432 g = _h21(u / (t + r), w / (t + _1), 

1433 (v / (t + s)) if v else 0) 

1434 if g > 0: 

1435 t0 = t 

1436 elif g < 0: 

1437 t1 = t 

1438 else: 

1439 break 

1440 else: # PYCHOK no cover 

1441 t = Fmt.no_convergence(e, eps) 

1442 raise _ValueError(t, txt=_rootXd.__name__) 

1443 return t, i 

1444 

1445 

1446def _sideOf(xyz, abc, eps=EPS): 

1447 '''(INTERNAL) Helper for C{_hartzell3}, M{.sideOf} and M{.reverseCartesian}. 

1448 

1449 @return: M{sum((x / a)**2 for x, a in zip(xyz, abc)) - 1} or C{INT0}. 

1450 ''' 

1451 s = fsumf_(_N_1_0, *((x / a)**2 for x, a in _zip(xyz, abc) if a)) # strict=True 

1452 return INT0 if fabs(s) < eps else s 

1453 

1454 

1455if __name__ == '__main__': 

1456 

1457 from pygeodesy import printf 

1458 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_ 

1459 

1460 # __doc__ of this file, force all into registery 

1461 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_) 

1462 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t)) 

1463 

1464# **) MIT License 

1465# 

1466# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

1467# 

1468# Permission is hereby granted, free of charge, to any person obtaining a 

1469# copy of this software and associated documentation files (the "Software"), 

1470# to deal in the Software without restriction, including without limitation 

1471# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1472# and/or sell copies of the Software, and to permit persons to whom the 

1473# Software is furnished to do so, subject to the following conditions: 

1474# 

1475# The above copyright notice and this permission notice shall be included 

1476# in all copies or substantial portions of the Software. 

1477# 

1478# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1479# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1480# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1481# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1482# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1483# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1484# OTHER DEALINGS IN THE SOFTWARE.