1

2# -*- coding: utf-8 -*-

3

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

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

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

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

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

9

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

11 see 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

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

32

33 from pygeodesy.basics import isLatLon, isscalar, map2, _zip, _ValueError

34 from 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!

37 from 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

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

43 from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1f_

44 from 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

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

50 _lazyNamedEnumItem as _lazy

51 from pygeodesy.namedTuples import LatLon3Tuple, Vector3Tuple, Vector4Tuple

52 from pygeodesy.props import Property_RO, property_RO

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

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

55 Radius, Scalar_, _toDegrees, _toRadians

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

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

58

59 from math import atan2, fabs, sqrt

60

61__all__ = _ALL_LAZY.triaxials

62__version__ = '24.04.14'

63

64_not_ordered_ = _not_('ordered')

65_omega_ = 'omega'

66_TRIPS = 269 # 48-55, Eberly 1074?

67

68

69 class _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

81 class 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

109 class 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

143 class 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

169 class 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

597 class 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

907 class 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

1021 class 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

1109 class TriaxialError(_ValueError):

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

1111 '''

1112 pass # ...

1113

1114

1115 class 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

1125 Triaxials = 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_EWGS84_35 = _EWGS84.a + 35, _EWGS84.a - 35, _EWGS84.b

1131 Triaxials._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', *map2(m2km, _EWGS84_35)))

1144 del _EWGS84_35

1145

1146

1147 def _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

1157 def _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

1213 def 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

1257 def _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

1264 def _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

1317 def _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

1404 def _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

1411 def _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

1446 def _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

1455 if __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,

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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

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