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:Charles@Karney.com>} (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 isscalar, map1, _zip, _ValueError 

34from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, _EPS2e4, INT0, PI2, PI_3, PI4, \ 

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

36 _4_0 # PYCHOK used! 

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

38# from pygeodesy.dms import toDMS # _MODS 

39# from pygeodesy.ellipsoids import Ellipsoid # 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 

43from pygeodesy.fsums import Fsum, fsumf_, fsum1f_ 

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

45 _height_, _inside_, _near_, _not_, _NOTEQUAL_, _null_, \ 

46 _opposite_, _outside_, _SPACE_, _spherical_, _too_, \ 

47 _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 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__ = '23.08.09' 

63 

64_not_ordered_ = _not_('ordered') 

65_omega_ = 'omega' 

66_TRIPS = 537 # 52..58, Eberly 1074? 

67 

68 

69class _NamedTupleTo(_NamedTuple): # in .testNamedTuples 

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

71 ''' 

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

73 if toDMS_kwds: 

74 toDMS = _MODS.dms.toDMS 

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

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

77 elif isinstance(a, Degrees) and \ 

78 isinstance(b, Degrees): 

79 return self 

80 else: 

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

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

83 

84 def _toRadians(self, a, b, *c): 

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

86 isinstance(b, Radians) else \ 

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

88 *c, name=self.name) 

89 

90 

91class BetaOmega2Tuple(_NamedTupleTo): 

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, **toDMS_kwds): 

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

101 

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

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

104 or as a L{toDMS} string provided some 

105 B{C{toDMS_kwds}} keyword arguments are 

106 specified. 

107 ''' 

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

109 

110 def toRadians(self): 

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

112 

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

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

115 ''' 

116 return _NamedTupleTo._toRadians(self, *self) 

117 

118 

119class BetaOmega3Tuple(_NamedTupleTo): 

120 '''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and 

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

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

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

124 in C{meter}, conventionally. 

125 ''' 

126 _Names_ = BetaOmega2Tuple._Names_ + (_height_,) 

127 _Units_ = BetaOmega2Tuple._Units_ + ( Meter,) 

128 

129 def toDegrees(self, **toDMS_kwds): 

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

131 

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

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

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

135 keyword arguments are specified. 

136 ''' 

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

138 

139 def toRadians(self): 

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

141 

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

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

144 ''' 

145 return _NamedTupleTo._toRadians(self, *self) 

146 

147 def to2Tuple(self): 

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

149 ''' 

150 return BetaOmega2Tuple(*self[:2]) 

151 

152 

153class Jacobi2Tuple(_NamedTupleTo): 

154 '''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y} 

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

156 ''' 

157 _Names_ = (_x_, _y_) 

158 _Units_ = (_Pass, _Pass) 

159 

160 def toDegrees(self, **toDMS_kwds): 

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

162 

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

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

165 provided some B{C{toDMS_kwds}} keyword 

166 arguments are specified. 

167 ''' 

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

169 

170 def toRadians(self): 

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

172 

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

174 and C{y} both in L{Radians}. 

175 ''' 

176 return _NamedTupleTo._toRadians(self, *self) 

177 

178 

179class Triaxial_(_NamedEnumItem): 

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

181 

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

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

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

185 I{omega}=0. 

186 

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

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

189 

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

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

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

193 L{Degrees} if converted). 

194 ''' 

195 _ijk = _kji = None 

196 _unordered = True 

197 

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

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

200 

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

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

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

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

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

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

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

208 

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

210 ''' 

211 try: 

212 a = a_triaxial 

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

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

215 except (TypeError, ValueError) as x: 

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

217 if name: 

218 self.name = name 

219 

220 a, b, c = self._abc3 = t 

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

222 s, _, t = sorted(t) 

223 if not (isfinite(t) and s > 0): 

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

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

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

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

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

229 

230 def __str__(self): 

231 return self.toStr() 

232 

233 @Property_RO 

234 def a(self): 

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

236 ''' 

237 a, _, _ = self._abc3 

238 return a 

239 

240 @Property_RO 

241 def _a2b2(self): 

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

243 ''' 

244 a, b, _ = self._abc3 

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

246 

247 @Property_RO 

248 def _a2_b2(self): 

249 '''(INTERNAL) Get C{(a/b)**2}. 

250 ''' 

251 a, b, _ = self._abc3 

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

253 

254 @Property_RO 

255 def _a2c2(self): 

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

257 ''' 

258 a, _, c = self._abc3 

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

260 

261 @Property_RO 

262 def area(self): 

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

264 ''' 

265 c, b, a = sorted(self._abc3) 

266 if a > c: 

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

268 Ellipsoid(a, b=c).areax # a == b 

269 else: # a == c == b 

270 a = Meter2(area=a**2 * PI4) 

271 return a 

272 

273 def area_p(self, p=1.6075): 

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

275 

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

277 for "near-flat" triaxials. 

278 

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

280 ''' 

281 a, b, c = self._abc3 

282 if a == b == c: 

283 a *= a 

284 else: 

285 _p = pow 

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

287 return Meter2(area_p=a * PI4) 

288 

289 @Property_RO 

290 def b(self): 

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

292 ''' 

293 _, b, _ = self._abc3 

294 return b 

295 

296 @Property_RO 

297 def _b2c2(self): 

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

299 ''' 

300 _, b, c = self._abc3 

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

302 

303 @Property_RO 

304 def c(self): 

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

306 ''' 

307 _, _, c = self._abc3 

308 return c 

309 

310 @Property_RO 

311 def _c2_b2(self): 

312 '''(INTERNAL) Get C{(c/b)**2}. 

313 ''' 

314 _, b, c = self._abc3 

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

316 

317 @Property_RO 

318 def e2ab(self): 

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

320 ''' 

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

322 

323 @Property_RO 

324 def _1e2ab(self): 

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

326 ''' 

327 a, b, _ = self._abc3 

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

329 

330 @Property_RO 

331 def e2ac(self): 

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

333 ''' 

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

335 

336 @Property_RO 

337 def _1e2ac(self): 

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

339 ''' 

340 a, _, c = self._abc3 

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

342 

343 @Property_RO 

344 def e2bc(self): 

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

346 ''' 

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

348 

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

350 

351 @property_RO 

352 def _Elliptic(self): 

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

354 ''' 

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

356 return E 

357 

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

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

360 from a Point-Of-View in space. 

361 

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

363 ''' 

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

365 

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

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

368 this triaxial's surface. 

369 

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

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

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

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

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

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

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

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

378 negative value to skip validation. 

379 

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

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

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

383 surface in C{meter}, conventionally. 

384 

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

386 convergence in root finding or validation failed. 

387 

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

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

390 DistancePointEllipseEllipsoid.pdf>}. 

391 ''' 

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

393 

394 i, h = None, v.length 

395 if h < EPS0: # EPS 

396 x = y = z = _0_0 

397 h -= min(self._abc3) # nearest 

398 elif r: # .isSpherical 

399 x, y, z = v.times(r / h).xyz 

400 h -= r 

401 else: 

402 x, y, z = v.xyz 

403 try: 

404 if normal: # perpendicular to triaxial 

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

406 else: # radially to triaxial's center 

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

408 h = v.minus_(x, y, z).length 

409 except Exception as e: 

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

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

412 h = -h # below the surface 

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

414 

415 @Property_RO 

416 def isOrdered(self): 

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

418 ''' 

419 a, b, c = self._abc3 

420 return bool(a >= b > c) # b > c! 

421 

422 @Property_RO 

423 def isSpherical(self): 

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

425 ''' 

426 a, b, c = self._abc3 

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

428 

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

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

431 

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

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

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

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

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

437 

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

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

440 

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

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

443 ''' 

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

445 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered 

446 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2 

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

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

449 if n.length < EPS0: 

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

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

452 

453 @Property_RO 

454 def _normal3d(self): 

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

456 ''' 

457 d = max(self._abc3) 

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

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

460 

461 def _norm2(self, s, c, *a): 

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

463 ''' 

464 if fabs(_hypot21(s, c)) > EPS02: 

465 s, c = norm2(s, c) 

466 if a: 

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

468 return float0_(s, c) 

469 

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

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

472 

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

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

475 ''' 

476 ijk = self._order_ijk(**reverse) 

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

478 

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

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

481 

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

483 ''' 

484 ijk = self._order_ijk(**reverse) 

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

486 

487 @Property_RO 

488 def _ordered4(self): 

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

490 ''' 

491 def _order2(reverse, a, b, c): 

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

493 

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

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

496 ''' 

497 i, j, k = 0, 1, 2 # range(3) 

498 if a < b: 

499 a, b, i, j = b, a, j, i 

500 if a < c: 

501 a, c, i, k = c, a, k, i 

502 if b < c: 

503 b, c, j, k = c, b, k, j 

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

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

506 return (a, b, c), ijk 

507 

508 abc, T = self._abc3, self 

509 if not self.isOrdered: 

510 abc, ijk = _order2(False, *abc) 

511 if ijk: 

512 _, kji = _order2(True, *ijk) 

513 T = Triaxial_(*abc) 

514 T._ijk, T._kji = ijk, kji 

515 return abc + (T,) 

516 

517 def _order_ijk(self, reverse=False): 

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

519 ''' 

520 return self._kji if reverse else self._ijk 

521 

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

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

524 ''' 

525 def _rphi(a, b, sphi, cphi): 

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

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

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

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

530 

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

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

533 

534 a, b, c = self._abc3 

535 if a != b: 

536 a = _rphi(a, b, sb, cb) 

537 if a != c: 

538 c = _rphi(a, c, sa, ca) 

539 z, r = c * sa, c * ca 

540 x, y = r * cb, r * sb 

541 return x, y, z 

542 

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

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

545 

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

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

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

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

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

551 

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

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

554 if in- respectively outside this triaxial. 

555 

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

557 ''' 

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

559 

560 def _sqrt(self, x): 

561 '''(INTERNAL) Helper, see L{pygeodesy.sqrt0}. 

562 ''' 

563 if x < 0: 

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

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

566 

567 def toEllipsoid(self, name=NN): 

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

569 

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

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

572 

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

574 

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

576 ''' 

577 a, b, c = self._abc3 

578 if a == b: 

579 b = c # N = c-Z 

580 elif b == c: # N = a-X 

581 a, b = b, a 

582 elif a != c: # N = b-Y 

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

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

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

586 

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

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

589 

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

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

592 this triaxial's name. 

593 

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

595 ''' 

596 T = Triaxial_ 

597 t = T.a, 

598 J = JacobiConformalSpherical 

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

600 t += T.e2ab, T.e2bc, T.e2ac 

601 J = JacobiConformal 

602 if isinstance(self, J): 

603 t += J.xyQ2, 

604 t += T.volume, T.area 

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

606 

607 @Property_RO 

608 def volume(self): 

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

610 ''' 

611 a, b, c = self._abc3 

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

613 

614 

615class Triaxial(Triaxial_): 

616 '''I{Ordered} triaxial ellipsoid. 

617 

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

619 ''' 

620 _unordered = False 

621 

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

623 '''New I{ordered} L{Triaxial}. 

624 

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

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

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

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

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

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

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

632 

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

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

635 

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

637 ''' 

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

639 

640 @Property_RO 

641 def _a2b2_a2c2(self): 

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

643 ''' 

644 return self._a2b2 / self._a2c2 

645 

646 @Property_RO 

647 def area(self): 

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

649 

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

651 ''' 

652 a, b, c = self._abc3 

653 if a != b: 

654 kp2, k2 = self._k2_kp2 # swapped! 

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

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

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

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

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

660 aE.fF(r) * c2 / s) 

661 a = Meter2(area=a * b * PI2) 

662 else: # a == b > c 

663 a = Ellipsoid(a, b=c).areax 

664 return a 

665 

666 def _exyz3(self, u): 

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

668 ''' 

669 if u > 0: 

670 u2 = u**2 

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

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

673 else: 

674 x = y = u = _0_0 

675 return x, y, u 

676 

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

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

679 and height to cartesian. 

680 

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

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

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

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

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

686 

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

688 

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

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

691 ''' 

692 if height: 

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

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

695 else: 

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

697 if z: # and x and y: 

698 sa, ca = SinCos2(beta) 

699 sb, cb = SinCos2(omega) 

700 

701 r = self._a2b2_a2c2 

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

703 y *= ca * sb 

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

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

706 

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

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

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

710 

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

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

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

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

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

716 

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

718 

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

720 

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

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

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

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

725 ''' 

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

727 return Vector3Tuple(*t, name=name) 

728 

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

730 '''Project a cartesian on this triaxial. 

731 

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

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

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

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

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

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

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

739 

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

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

742 ''' 

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

744 _ = t.rename(name) 

745 return t 

746 

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

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

749 

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

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

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

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

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

762 

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

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

765 

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

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

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

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

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

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

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

773 

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

775 

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

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

778 ''' 

779 sa, ca = self._norm2(slat, clat) 

780 sb, cb = self._norm2(slon, clon) 

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

782 

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

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

785 ''' 

786 ca_x_sb = ca * sb 

787 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2 

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

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

790 x = (h + n) * ca * cb 

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

792 z = (h + n * self._1e2ac) * sa 

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

794 

795 @Property_RO 

796 def _k2_kp2(self): 

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

798 ''' 

799 # k2 = a2b2 / a2c2 * c2_b2 

800 # kp2 = b2c2 / a2c2 * a2_b2 

801 # b2 = b**2 

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

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

804 # aE = Elliptic(kp2, 0, k2, 1) 

805 return (self._a2b2_a2c2 * self._c2_b2, 

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

807 

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

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

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

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

812 ''' 

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

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

815 

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

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

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

819 

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

821 z2 = Fsum(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2) 

822 

823 x, y, z = self._abc3 

824 x *= cb * self._sqrt(x2) 

825 y *= ca * sb 

826 z *= sa * self._sqrt(z2) 

827 return x, y, z 

828 

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

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

831 and height. 

832 

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

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

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

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

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

838 

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

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

841 units as this triaxial's axes. 

842 

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

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

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

846 ''' 

847 v = _otherV3d_(x_xyz, y, z) 

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

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

850 

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

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

853 

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

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

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

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

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

859 as the axes). 

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

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

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

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

864 

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

866 

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

868 

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

870 ''' 

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

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

873 if s: # PYCHOK no cover 

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

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

876 

877 if h: 

878 if normal: 

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

880 elif v.length > EPS0: 

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

882 return v.xyz # Vector3Tuple 

883 

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

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

886 

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

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

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

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

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

892 

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

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

895 as this triaxial's axes. 

896 

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

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

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

900 ''' 

901 v = _otherV3d_(x_xyz, y, z) 

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

903 self._1e2bc, # == 1 - e_sub_y**2 

904 _1_0) 

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

906 return LatLon3Tuple(*t, name=name) 

907 

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

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

910 ''' 

911 x, y, z = s.xyz 

912 d = hypot( x, y) 

913 a = atan2_(z, d) 

914 b = atan2_(y, x) 

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

916 return a, b, h 

917 

918 

919class JacobiConformal(Triaxial): 

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

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

922 

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

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

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

926 returned in the case of an ellipsoid of revolution. 

927 

928 Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2023) and 

929 licensed under the MIT/X11 License. 

930 

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

932 L{JacobiConformalSpherical}. 

933 

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

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

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

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

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

939 ''' 

940 

941 @Property_RO 

942 def _xE(self): 

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

944 ''' 

945 k2, kp2 = self._k2_kp2 

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

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

948 

949 def xR(self, omega): 

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

951 

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

953 

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

955 ''' 

956 return self.xR_(*SinCos2(omega)) 

957 

958 def xR_(self, somega, comega): 

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

960 

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

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

963 

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

965 ''' 

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

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

968 

969 @Property_RO 

970 def xyQ2(self): 

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

972 ''' 

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

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

975 name=JacobiConformal.xyQ2.name) 

976 

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

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

979 

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

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

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

983 

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

985 ''' 

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

987 name=name or self.xyR2.__name__) 

988 

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

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

991 

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

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

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

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

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

997 

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

999 ''' 

1000 return Jacobi2Tuple(self.xR_(somega, comega), 

1001 self.yR_(sbeta, cbeta), 

1002 name=name or self.xyR2_.__name__) 

1003 

1004 @Property_RO 

1005 def _yE(self): 

1006 '''(INTERNAL) Get the x-elliptic function. 

1007 ''' 

1008 kp2, k2 = self._k2_kp2 # swapped! 

1009 # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc 

1010 return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2) 

1011 

1012 def yR(self, beta): 

1013 '''Compute a Jacobi Conformal C{y} projection. 

1014 

1015 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

1016 

1017 @return: The C{y} projection (L{Radians}). 

1018 ''' 

1019 return self.yR_(*SinCos2(beta)) 

1020 

1021 def yR_(self, sbeta, cbeta): 

1022 '''Compute a Jacobi Conformal C{y} projection. 

1023 

1024 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

1025 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

1026 

1027 @return: The C{y} projection (L{Radians}). 

1028 ''' 

1029 s, c = self._norm2(sbeta, cbeta, self.c) 

1030 return Radians(y=self._yE.fPi(s, c) * self._c2_b2) 

1031 

1032 

1033class JacobiConformalSpherical(JacobiConformal): 

1034 '''An alternate, I{spherical} L{JacobiConformal} projection. 

1035 

1036 @see: L{JacobiConformal} for other and more details. 

1037 ''' 

1038 _ab = _bc = 0 

1039 

1040 def __init__(self, radius_triaxial, ab=0, bc=0, name=NN): 

1041 '''New L{JacobiConformalSpherical}. 

1042 

1043 @arg radius_triaxial: Radius (C{scalar}, conventionally in 

1044 C{meter}) or an other L{JacobiConformalSpherical}, 

1045 L{JacobiConformal} or ordered L{Triaxial}. 

1046 @kwarg ab: Relative magnitude of C{B{a} - B{b}} (C{meter}, 

1047 same units as C{scalar B{radius}}. 

1048 @kwarg bc: Relative magnitude of C{B{b} - B{c}} (C{meter}, 

1049 same units as C{scalar B{radius}}. 

1050 @kwarg name: Optional name (C{str}). 

1051 

1052 @raise TriaxialError: Invalid B{C{radius_triaxial}}, negative 

1053 B{C{ab}}, negative B{C{bc}} or C{(B{ab} 

1054 + B{bc})} not positive. 

1055 

1056 @note: If B{C{radius_triaxial}} is a L{JacobiConformalSpherical} 

1057 and if B{C{ab}} and B{C{bc}} are both zero or C{None}, 

1058 the B{C{radius_triaxial}}'s C{ab}, C{bc}, C{a}, C{b} 

1059 and C{c} are copied. 

1060 ''' 

1061 try: 

1062 r, j = radius_triaxial, False 

1063 if isinstance(r, Triaxial): # ordered only 

1064 if (not (ab or bc)) and isinstance(r, JacobiConformalSpherical): 

1065 j = True 

1066 t = r._abc3 

1067 else: 

1068 t = (Radius(radius=r),) * 3 

1069 self._ab = r.ab if j else Scalar_(ab=ab) # low=0 

1070 self._bc = r.bc if j else Scalar_(bc=bc) # low=0 

1071 if (self.ab + self.bc) <= 0: 

1072 raise ValueError('(ab + bc)') 

1073 a, _, c = self._abc3 = t 

1074 if not (a >= c and isfinite(self._a2b2) 

1075 and isfinite(self._a2c2)): 

1076 raise ValueError(_not_(_finite_)) 

1077 except (TypeError, ValueError) as x: 

1078 raise TriaxialError(radius_triaxial=r, ab=ab, bc=bc, cause=x) 

1079 if name: 

1080 self.name = name 

1081 

1082 @Property_RO 

1083 def ab(self): 

1084 '''Get relative magnitude C{ab} (C{meter}, same units as B{C{a}}). 

1085 ''' 

1086 return self._ab 

1087 

1088 @Property_RO 

1089 def _a2b2(self): 

1090 '''(INTERNAL) Get C{a**2 - b**2} == ab * (a + b). 

1091 ''' 

1092 a, b, _ = self._abc3 

1093 return self.ab * (a + b) 

1094 

1095 @Property_RO 

1096 def _a2c2(self): 

1097 '''(INTERNAL) Get C{a**2 - c**2} == a2b2 + b2c2. 

1098 ''' 

1099 return self._a2b2 + self._b2c2 

1100 

1101 @Property_RO 

1102 def bc(self): 

1103 '''Get relative magnitude C{bc} (C{meter}, same units as B{C{a}}). 

1104 ''' 

1105 return self._bc 

1106 

1107 @Property_RO 

1108 def _b2c2(self): 

1109 '''(INTERNAL) Get C{b**2 - c**2} == bc * (b + c). 

1110 ''' 

1111 _, b, c = self._abc3 

1112 return self.bc * (b + c) 

1113 

1114 @Property_RO 

1115 def radius(self): 

1116 '''Get radius (C{meter}, conventionally). 

1117 ''' 

1118 return self.a 

1119 

1120 

1121class TriaxialError(_ValueError): 

1122 '''Raised for L{Triaxial} issues. 

1123 ''' 

1124 pass # ... 

1125 

1126 

1127class Triaxials(_NamedEnum): 

1128 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class 

1129 to accommodate the L{_LazyNamedEnumItem} properties. 

1130 ''' 

1131 def _Lazy(self, *abc, **name): 

1132 '''(INTERNAL) Instantiate the C{Triaxial}. 

1133 ''' 

1134 a, b, c = map(km2m, abc) 

1135 return Triaxial(a, b, c, **name) 

1136 

1137Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton 

1138'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.''' 

1139# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in Km 

1140# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf> 

1141# <https://link.Springer.com/article/10.1007/s00190-022-01650-9> 

1142_E = _WGS84.ellipsoid 

1143Triaxials._assert( # a (Km) b (Km) c (Km) planet 

1144 Amalthea = _lazy('Amalthea', 125.0, 73.0, 64), # Jupiter 

1145 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus 

1146 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544), 

1147 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn 

1148 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter 

1149 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter 

1150 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3), 

1151 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn 

1152 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus 

1153 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth 

1154 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn 

1155 WGS84_35 = _lazy('WGS84_35', *map1(m2km, _E.a + 35, _E.a - 35, _E.b))) 

1156del _E 

1157 

1158 

1159def _getitems(items, *indices): 

1160 '''(INTERNAL) Get the C{items} at the given I{indices}. 

1161 

1162 @return: C{Type(items[i] for i in indices)} with 

1163 C{Type = type(items)}, any C{type} having 

1164 the special method C{__getitem__}. 

1165 ''' 

1166 return type(items)(map(items.__getitem__, indices)) 

1167 

1168 

1169def _hartzell3d2(pov, los, Tun): # MCCABE 13 in .ellipsoidal.hartzell4, .formy.hartzell 

1170 '''(INTERNAL) Hartzell's "Satellite Line-of-Sight Intersection ...", 

1171 formula for I{un-/ordered} triaxials. 

1172 ''' 

1173 a, b, c, T = Tun._ordered4 

1174 

1175 a2 = a**2 # largest, factored out 

1176 b2, p2 = (b**2, T._1e2ab) if b != a else (a2, _1_0) 

1177 c2, q2 = (c**2, T._1e2ac) if c != a else (a2, _1_0) 

1178 

1179 p3 = T._order3d(_otherV3d(pov=pov)) 

1180 u3 = T._order3d(_otherV3d(los=los)) if los else p3.negate() 

1181 u3 = u3.unit() # unit vector, opposing signs 

1182 

1183 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz 

1184 ux, vy, wz = u3.times_(p3).xyz 

1185 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

1186 

1187 t = (p2 * c2), c2, b2 

1188 m = fdot(t, u2, v2, w2) # a2 factored out 

1189 if m < EPS0: # zero or near-null LOS vector 

1190 raise _ValueError(_near_(_null_)) 

1191 

1192 r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

1193 -w2 * y2, b2 * u2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2, 

1194 -w2 * x2 * p2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2) 

1195 if r > 0: # a2 factored out 

1196 r = sqrt(r) * b * c # == a * a * b * c / a2 

1197 elif r < 0: # LOS pointing away from or missing the triaxial 

1198 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

1199 

1200 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

1201 if d > 0: # POV inside or LOS missing, outside the triaxial 

1202 s = fsumf_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0) # like _sideOf 

1203 raise _ValueError(_outside_ if s > 0 else _inside_) 

1204 elif fsum1f_(x2, y2, z2) < d**2: # d past triaxial's center 

1205 raise _ValueError(_too_(_distant_)) 

1206 

1207 v = p3.minus(u3.times(d)) # Vector3d 

1208 h = p3.minus(v).length # distance to triaxial 

1209 return T._order3d(v, reverse=True), h 

1210 

1211 

1212def hartzell4(pov, los=None, tri_biax=_WGS84, name=NN): 

1213 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight 

1214 from a Point-Of-View outside. 

1215 

1216 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple} 

1217 or L{Vector3d}). 

1218 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Vector3d}) or 

1219 C{None} to point to the tri-/biaxial's center. 

1220 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal} or 

1221 L{JacobiConformalSpherical}) or biaxial ellipsoid (L{Datum}, 

1222 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} radius, 

1223 conventionally in C{meter}). 

1224 @kwarg name: Optional name (C{str}). 

1225 

1226 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with 

1227 C{h} the distance from B{C{pov}} to C{(x, y, z)} along the B{C{los}}, 

1228 all in C{meter}, conventionally. 

1229 

1230 @raise TriaxialError: Null B{C{pov}} or B{C{los}}, or B{C{pov}} is inside the 

1231 tri-/biaxial or B{C{los}} points outside the tri-/biaxial 

1232 or points in an opposite direction. 

1233 

1234 @raise TypeError: Invalid B{C{pov}} or B{C{los}}. 

1235 

1236 @see: Function L{pygeodesy.hartzell}, L{pygeodesy.tyr3d} for B{C{los}} and 

1237 U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell. 

1238 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

1239 ''' 

1240 if isinstance(tri_biax, Triaxial_): 

1241 T = tri_biax 

1242 else: 

1243 D = tri_biax if isinstance(tri_biax, Datum) else \ 

1244 _spherical_datum(tri_biax, name=hartzell4.__name__) 

1245 T = D.ellipsoid._triaxial 

1246 

1247 try: 

1248 v, h = _hartzell3d2(pov, los, T) 

1249 except Exception as x: 

1250 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x) 

1251 return Vector4Tuple(v.x, v.y, v.z, h, name=name or hartzell4.__name__) 

1252 

1253 

1254def _hypot21(x, y, z=0): 

1255 '''(INTERNAL) Compute M{x**2 + y**2 + z**2 - 1} with C{max(fabs(x), 

1256 fabs(y), fabs(z))} rarely greater than 1.0. 

1257 ''' 

1258 return fsumf_(_1_0, x**2, y**2, z**2, _N_2_0) if z else \ 

1259 fsumf_(_1_0, x**2, y**2, _N_2_0) 

1260 

1261 

1262def _normalTo4(x, y, a, b, eps=EPS): 

1263 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}. 

1264 

1265 @see: Function C{pygeodesy.ellipsoids._normalTo3} and I{Eberly}'s U{Distance 

1266 from a Point to ... an Ellipsoid ...<https://www.GeometricTools.com/ 

1267 Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1268 ''' 

1269 if a < b: 

1270 b, a, d, i = _normalTo4(y, x, b, a, eps=eps) 

1271 return a, b, d, i 

1272 

1273 if not (b > 0 and isfinite(a)): 

1274 raise _ValueError(a=a, b=b) 

1275 

1276 i = None 

1277 if y: 

1278 if x: 

1279 u = fabs(x / a) 

1280 v = fabs(y / b) 

1281 g = _hypot21(u, v) 

1282 if g: 

1283 r = (a / b)**2 

1284 t, i = _rootXd(r, 0, u, 0, v, g, eps) 

1285 a = x / (t / r + _1_0) 

1286 b = y / (t + _1_0) 

1287 d = hypot(x - a, y - b) 

1288 else: # on the ellipse 

1289 a, b, d = x, y, _0_0 

1290 else: # x == 0 

1291 if y < 0: 

1292 b = -b 

1293 a, d = x, fabs(y - b) 

1294 

1295 else: # y == 0 

1296 n = a * x 

1297 d = (a + b) * (a - b) 

1298 if d > fabs(n): # PYCHOK no cover 

1299 r = n / d 

1300 a *= r 

1301 b *= sqrt(_1_0 - r**2) 

1302 d = hypot(x - a, b) 

1303 else: 

1304 if x < 0: 

1305 a = -a 

1306 b, d = y, fabs(x - a) 

1307 return a, b, d, i 

1308 

1309 

1310def _normalTo5(x, y, z, Tun, eps=EPS): # MCCABE 19 

1311 '''(INTERNAL) Nearest point on and distance to an I{un-/ordered} triaxial. 

1312 

1313 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https:// 

1314 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

1315 ''' 

1316 a, b, c, T = Tun._ordered4 

1317 if Tun is not T: # T is ordered, Tun isn't 

1318 t = T._order3(x, y, z) + (T,) 

1319 a, b, c, d, i = _normalTo5(*t, eps=eps) 

1320 return T._order3(a, b, c, reverse=True) + (d, i) 

1321 

1322 if not (isfinite(a) and c > 0): 

1323 raise _ValueError(a=a, b=b, c=c) 

1324 

1325 if eps > 0: 

1326 val = max(eps * 1e8, EPS) 

1327 else: # no validation 

1328 val, eps = 0, -eps 

1329 

1330 i = None 

1331 if z: 

1332 if y: 

1333 if x: 

1334 u = fabs(x / a) 

1335 v = fabs(y / b) 

1336 w = fabs(z / c) 

1337 g = _hypot21(u, v, w) 

1338 if g: 

1339 r = T._1e2ac # (c / a)**2 

1340 s = T._1e2bc # (c / b)**2 

1341 t, i = _rootXd(_1_0 / r, _1_0 / s, u, v, w, g, eps) 

1342 a = x / (t * r + _1_0) 

1343 b = y / (t * s + _1_0) 

1344 c = z / (t + _1_0) 

1345 d = hypot_(x - a, y - b, z - c) 

1346 else: # on the ellipsoid 

1347 a, b, c, d = x, y, z, _0_0 

1348 else: # x == 0 

1349 a = x # 0 

1350 b, c, d, i = _normalTo4(y, z, b, c, eps=eps) 

1351 elif x: # y == 0 

1352 b = y # 0 

1353 a, c, d, i = _normalTo4(x, z, a, c, eps=eps) 

1354 else: # x == y == 0 

1355 if z < 0: 

1356 c = -c 

1357 a, b, d = x, y, fabs(z - c) 

1358 

1359 else: # z == 0 

1360 t = False 

1361 n = a * x 

1362 d = T._a2c2 # (a + c) * (a - c) 

1363 if d > fabs(n): 

1364 u = n / d 

1365 n = b * y 

1366 d = T._b2c2 # (b + c) * (b - c) 

1367 if d > fabs(n): 

1368 v = n / d 

1369 n = _hypot21(u, v) 

1370 if n < 0: 

1371 a *= u 

1372 b *= v 

1373 c *= sqrt(-n) 

1374 d = hypot_(x - a, y - b, c) 

1375 t = True 

1376 if not t: 

1377 c = z # 0 

1378 a, b, d, i = _normalTo4(x, y, a, b, eps=eps) 

1379 

1380 if val > 0: # validate 

1381 e = T.sideOf(a, b, c, eps=val) 

1382 if e: # not near the ellipsoid's surface 

1383 raise _ValueError(a=a, b=b, c=c, d=d, 

1384 sideOf=e, eps=val) 

1385 if d: # angle of delta and normal vector 

1386 m = Vector3d(x, y, z).minus_(a, b, c) 

1387 if m.euclid > val: 

1388 m = m.unit() 

1389 n = T.normal3d(a, b, c) 

1390 e = n.dot(m) # n.negate().dot(m) 

1391 if not isnear1(fabs(e), eps1=val): 

1392 raise _ValueError(n=n, m=m, 

1393 dot=e, eps=val) 

1394 return a, b, c, d, i 

1395 

1396 

1397def _otherV3d_(x_xyz, y, z, **name): 

1398 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}. 

1399 ''' 

1400 return Vector3d(x_xyz, y, z, **name) if isscalar(x_xyz) else \ 

1401 _otherV3d(x_xyz=x_xyz) 

1402 

1403 

1404def _rootXd(r, s, u, v, w, g, eps): 

1405 '''(INTERNAL) Robust 2d- or 3d-root finder: 

1406 2d- if C{s == v == 0} otherwise 3d-root. 

1407 ''' 

1408 _1, __2 = _1_0, _0_5 

1409 _a, _h2 = fabs, _hypot21 

1410 

1411 u *= r 

1412 v *= s # 0 for 2d-root 

1413 t0 = w - _1 

1414 t1 = _0_0 if g < 0 else _h2(u, w, v) 

1415 for i in range(1, _TRIPS): 

1416 e = _a(t0 - t1) 

1417 if e < eps: 

1418 break 

1419 t = (t0 + t1) * __2 

1420 if t in (t0, t1): 

1421 break 

1422 g = _h2(u / (t + r), w / (t + _1), 

1423 (v / (t + s)) if v else 0) 

1424 if g > 0: 

1425 t0 = t 

1426 elif g < 0: 

1427 t1 = t 

1428 else: 

1429 break 

1430 else: # PYCHOK no cover 

1431 t = Fmt.no_convergence(e, eps) 

1432 raise _ValueError(t, txt=_rootXd.__name__) 

1433 return t, i 

1434 

1435 

1436def _sideOf(xyz, abc, eps=EPS): # in .formy 

1437 '''(INTERNAL) Helper for C{_hartzell3d2}, M{.sideOf} and M{.reverseCartesian}. 

1438 

1439 @return: M{sum((x / a)**2 for x, a in zip(xyz, abc)) - 1} or C{INT0}, 

1440 ''' 

1441 s = _hypot21(*((x / a) for x, a in _zip(xyz, abc) if a)) # strict=True 

1442 return s if fabs(s) > eps else INT0 

1443 

1444 

1445if __name__ == '__main__': 

1446 

1447 from pygeodesy import printf 

1448 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_ 

1449 

1450 # __doc__ of this file, force all into registery 

1451 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_) 

1452 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t)) 

1453 

1454# **) MIT License 

1455# 

1456# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved. 

1457# 

1458# Permission is hereby granted, free of charge, to any person obtaining a 

1459# copy of this software and associated documentation files (the "Software"), 

1460# to deal in the Software without restriction, including without limitation 

1461# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

1462# and/or sell copies of the Software, and to permit persons to whom the 

1463# Software is furnished to do so, subject to the following conditions: 

1464# 

1465# The above copyright notice and this permission notice shall be included 

1466# in all copies or substantial portions of the Software. 

1467# 

1468# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

1469# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

1470# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

1471# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

1472# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

1473# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

1474# OTHER DEALINGS IN THE SOFTWARE.