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, _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 _lazyNamedEnumItem as _lazy, _name__, _NamedEnum, \

50 _NamedEnumItem, _NamedTuple, _Pass

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__ = '24.05.28'

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):

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 C{B{name}=NN} (C{str}).

198

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

200 '''

201 try:

202 try:

203 a = a_triaxial

204 t = a._abc3

205 except AttributeError:

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

207 except (TypeError, ValueError) as x:

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

209 if name:

210 self.name = name

211

212 a, b, c = self._abc3 = t

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

214 s, _, t = sorted(t)

215 if not (isfinite(t) and s > 0):

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

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

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

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

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

221

222 def __str__(self):

223 return self.toStr()

224

225 @Property_RO

226 def a(self):

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

228 '''

229 a, _, _ = self._abc3

230 return a

231

232 @Property_RO

233 def _a2b2(self):

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

235 '''

236 a, b, _ = self._abc3

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

238

239 @Property_RO

240 def _a2_b2(self):

241 '''(INTERNAL) Get C{(a/b)**2}.

242 '''

243 a, b, _ = self._abc3

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

245

246 @Property_RO

247 def _a2c2(self):

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

249 '''

250 a, _, c = self._abc3

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

252

253 @Property_RO

254 def area(self):

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

256 '''

257 c, b, a = sorted(self._abc3)

258 if a > c:

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

260 Ellipsoid(a, b=c).areax # a == b

261 else: # a == c == b

262 a = Meter2(area=a**2 * PI4)

263 return a

264

265 def area_p(self, p=1.6075):

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

267

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

269 for "near-flat" triaxials.

270

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

272 '''

273 a, b, c = self._abc3

274 if a == b == c:

275 a *= a

276 else:

277 _p = pow

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

279 return Meter2(area_p=a * PI4)

280

281 @Property_RO

282 def b(self):

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

284 '''

285 _, b, _ = self._abc3

286 return b

287

288 @Property_RO

289 def _b2c2(self):

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

291 '''

292 _, b, c = self._abc3

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

294

295 @Property_RO

296 def c(self):

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

298 '''

299 _, _, c = self._abc3

300 return c

301

302 @Property_RO

303 def _c2_b2(self):

304 '''(INTERNAL) Get C{(c/b)**2}.

305 '''

306 _, b, c = self._abc3

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

308

309 @Property_RO

310 def e2ab(self):

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

312 '''

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

314

315 @Property_RO

316 def _1e2ab(self):

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

318 '''

319 a, b, _ = self._abc3

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

321

322 @Property_RO

323 def e2ac(self):

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

325 '''

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

327

328 @Property_RO

329 def _1e2ac(self):

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

331 '''

332 a, _, c = self._abc3

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

334

335 @Property_RO

336 def e2bc(self):

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

338 '''

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

340

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

342

343 @property_RO

344 def _Elliptic(self):

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

346 '''

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

348 return E

349

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

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

352 from a Point-Of-View in space.

353

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

355 '''

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

357

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

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

360

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

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

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

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

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

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

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

368 value to skip validation.

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

370

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

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

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

374

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

376 root finding or validation failed.

377

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

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

380 '''

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

382

383 i, h = None, v.length

384 if h < EPS0: # EPS

385 x = y = z = _0_0

386 h -= min(self._abc3) # nearest

387 elif r: # .isSpherical

388 x, y, z = v.times(r / h).xyz

389 h -= r

390 else:

391 x, y, z = v.xyz

392 try:

393 if normal: # plumb to surface

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

395 else: # radial to center

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

397 h = v.minus_(x, y, z).length

398 except Exception as e:

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

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

401 h = -h # below the surface

402 return Vector4Tuple(x, y, z, h, iteration=i,

403 name=_name__(name, name__=self.height4))

404

405 @Property_RO

406 def isOrdered(self):

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

408 '''

409 a, b, c = self._abc3

410 return bool(a >= b > c) # b > c!

411

412 @Property_RO

413 def isSpherical(self):

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

415 '''

416 a, b, c = self._abc3

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

418

419 def _norm2(self, s, c, *a):

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

421 '''

422 if fabs(_hypot21(s, c)) > EPS02:

423 s, c = norm2(s, c)

424 if a:

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

426 return float0_(s, c)

427

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

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

430

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

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

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

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

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

436

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

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

439

440 @raise TriaxialError: Zero length cartesian or vector.

441

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

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

444 '''

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

446 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered

447 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2

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

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

450 if n.length < EPS0:

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

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

453

454 @Property_RO

455 def _normal3d(self):

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

457 '''

458 d = max(self._abc3)

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

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

461

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

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

464

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

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

467 '''

468 ijk = self._order_ijk(**reverse)

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

470

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

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

473

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

475 '''

476 ijk = self._order_ijk(**reverse)

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

478

479 @Property_RO

480 def _ordered4(self):

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

482 '''

483 def _order2(reverse, a, b, c):

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

485

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

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

488 '''

489 i, j, k = 0, 1, 2 # range(3)

490 if a < b:

491 a, b, i, j = b, a, j, i

492 if a < c:

493 a, c, i, k = c, a, k, i

494 if b < c:

495 b, c, j, k = c, b, k, j

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

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

498 return (a, b, c), ijk

499

500 abc, T = self._abc3, self

501 if not self.isOrdered:

502 abc, ijk = _order2(False, *abc)

503 if ijk:

504 _, kji = _order2(True, *ijk)

505 T = Triaxial_(*abc)

506 T._ijk, T._kji = ijk, kji

507 return abc + (T,)

508

509 def _order_ijk(self, reverse=False):

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

511 '''

512 return self._kji if reverse else self._ijk

513

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

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

516 '''

517 def _rphi(a, b, sphi, cphi):

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

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

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

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

522

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

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

525

526 a, b, c = self._abc3

527 if a != b:

528 a = _rphi(a, b, sb, cb)

529 if a != c:

530 c = _rphi(a, c, sa, ca)

531 z, r = c * sa, c * ca

532 x, y = r * cb, r * sb

533 return x, y, z

534

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

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

537

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

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

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

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

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

543

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

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

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

547 respectively outside this triaxial.

548

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

550 '''

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

552

553 def toEllipsoid(self, **name):

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

555

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

557

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

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

560

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

562

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

564 '''

565 a, b, c = self._abc3

566 if a == b:

567 b = c # N = c-Z

568 elif b == c: # N = a-X

569 a, b = b, a

570 elif a != c: # N = b-Y

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

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

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

574

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

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

577

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

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

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

581

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

583 '''

584 T = Triaxial_

585 t = T.a,

586 J = JacobiConformalSpherical

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

588 t += T.e2ab, T.e2bc, T.e2ac

589 J = JacobiConformal

590 if isinstance(self, J):

591 t += J.xyQ2,

592 t += T.volume, T.area

593 return self._instr(name, prec, props=t, area_p=self.area_p()) # __name__

594

595 @Property_RO

596 def volume(self):

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

598 '''

599 a, b, c = self._abc3

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

601

602

603 class Triaxial(Triaxial_):

604 '''I{Ordered} triaxial ellipsoid.

605

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

607 '''

608 _unordered = False

609

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

611 '''New I{ordered} L{Triaxial}.

612

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

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

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

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

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

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

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

620

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

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

623

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

625 '''

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

627

628 @Property_RO

629 def _a2b2_a2c2(self):

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

631 '''

632 return self._a2b2 / self._a2c2

633

634 @Property_RO

635 def area(self):

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

637

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

639 '''

640 a, b, c = self._abc3

641 if a != b:

642 kp2, k2 = self._k2_kp2 # swapped!

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

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

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

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

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

648 aE.fF(r) * c2 / s)

649 a = Meter2(area=a * b * PI2)

650 else: # a == b > c

651 a = Ellipsoid(a, b=c).areax

652 return a

653

654 def _exyz3(self, u):

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

656 '''

657 if u > 0:

658 u2 = u**2

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

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

661 else:

662 x = y = u = _0_0

663 return x, y, u

664

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

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

667 and height to cartesian.

668

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

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

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

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

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

674

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

676

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

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

679 '''

680 if height:

681 h = self._Height(height)

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

683 else:

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

685 if z: # and x and y:

686 sa, ca = SinCos2(beta)

687 sb, cb = SinCos2(omega)

688

689 r = self._a2b2_a2c2

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

691 y *= ca * sb

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

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

694

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

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

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

698

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

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

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

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

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

704

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

706

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

708

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

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

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

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

713 '''

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

715 return Vector3Tuple(*t, **name)

716

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

718 '''Project a cartesian on this triaxial.

719

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

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

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

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

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

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

726 see method L{Triaxial.height4}.

727

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

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

730 '''

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

732

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

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

735

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

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

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

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

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

741

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

743

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

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

746 '''

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

748

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

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

751

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

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

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

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

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

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

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

759

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

761

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

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

764 '''

765 sa, ca = self._norm2(slat, clat)

766 sb, cb = self._norm2(slon, clon)

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

768

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

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

771 '''

772 ca_x_sb = ca * sb

773 h = self._Height(height)

774 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2

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

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

777 x = (h + n) * ca * cb

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

779 z = (h + n * self._1e2ac) * sa

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

781

782 def _Height(self, height):

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

784 '''

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

786

787 @Property_RO

788 def _k2_kp2(self):

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

790 '''

791 # k2 = a2b2 / a2c2 * c2_b2

792 # kp2 = b2c2 / a2c2 * a2_b2

793 # b2 = b**2

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

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

796 # aE = Elliptic(kp2, 0, k2, 1)

797 return (self._a2b2_a2c2 * self._c2_b2,

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

799

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

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

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

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

804 '''

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

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

807

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

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

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

811

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

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

814

815 x, y, z = self._abc3

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

817 y *= ca * sb

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

819 return x, y, z

820

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

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

823 and height.

824

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

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

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

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

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

830

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

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

833 units as this triaxial's axes.

834

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

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

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

838 '''

839 v = _otherV3d_(x_xyz, y, z)

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

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

842

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

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

845

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

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

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

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

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

851 as the axes).

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

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

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

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

856

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

858

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

860

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

862 '''

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

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

865 if s: # PYCHOK no cover

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

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

868

869 if h:

870 if normal:

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

872 elif v.length > EPS0:

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

874 return v.xyz # Vector3Tuple

875

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

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

878

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

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

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

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

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

884

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

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

887 as this triaxial's axes.

888

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

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

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

892 '''

893 v = _otherV3d_(x_xyz, y, z)

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

895 self._1e2bc, # == 1 - e_sub_y**2

896 _1_0)

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

898 return LatLon3Tuple(*t, **name)

899

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

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

902 '''

903 x, y, z = s.xyz

904 d = hypot( x, y)

905 a = atan2_(z, d)

906 b = atan2_(y, x)

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

908 return a, b, h

909

910

911 class JacobiConformal(Triaxial):

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

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

914

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

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

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

918 returned in the case of an ellipsoid of revolution.

919

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

921 licensed under the MIT/X11 License.

922

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

924 L{JacobiConformalSpherical}.

925

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

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

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

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

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

931 '''

932

933 @Property_RO

934 def _xE(self):

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

936 '''

937 k2, kp2 = self._k2_kp2

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

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

940

941 def xR(self, omega):

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

943

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

945

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

947 '''

948 return self.xR_(*SinCos2(omega))

949

950 def xR_(self, somega, comega):

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

952

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

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

955

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

957 '''

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

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

960

961 @Property_RO

962 def xyQ2(self):

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

964 '''

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

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

967 name=JacobiConformal.xyQ2.name)

968

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

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

971

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

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

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

975

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

977 '''

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

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

980

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

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

983

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

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

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

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

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

989

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

991 '''

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

993 self.yR_(sbeta, cbeta),

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

995

996 @Property_RO

997 def _yE(self):

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

999 '''

1000 kp2, k2 = self._k2_kp2 # swapped!

1001 # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc

1002 return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2)

1003

1004 def yR(self, beta):

1005 '''Compute a Jacobi Conformal C{y} projection.

1006

1007 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).

1008

1009 @return: The C{y} projection (L{Radians}).

1010 '''

1011 return self.yR_(*SinCos2(beta))

1012

1013 def yR_(self, sbeta, cbeta):

1014 '''Compute a Jacobi Conformal C{y} projection.

1015

1016 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).

1017 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).

1018

1019 @return: The C{y} projection (L{Radians}).

1020 '''

1021 s, c = self._norm2(sbeta, cbeta, self.c)

1022 return Radians(y=self._yE.fPi(s, c) * self._c2_b2)

1023

1024

1025 class JacobiConformalSpherical(JacobiConformal):

1026 '''An alternate, I{spherical} L{JacobiConformal} projection.

1027

1028 @see: L{JacobiConformal} for other and more details.

1029 '''

1030 _ab = _bc = 0

1031

1032 def __init__(self, radius_triaxial, ab=0, bc=0, **name):

1033 '''New L{JacobiConformalSpherical}.

1034

1035 @arg radius_triaxial: Radius (C{scalar}, conventionally in

1036 C{meter}) or an other L{JacobiConformalSpherical},

1037 L{JacobiConformal} or ordered L{Triaxial}.

1038 @kwarg ab: Relative magnitude of C{B{a} - B{b}} (C{meter},

1039 same units as C{scalar B{radius}}.

1040 @kwarg bc: Relative magnitude of C{B{b} - B{c}} (C{meter},

1041 same units as C{scalar B{radius}}.

1042 @kwarg name: Optional C{B{name}=NN} (C{str}).

1043

1044 @raise TriaxialError: Invalid B{C{radius_triaxial}}, negative

1045 B{C{ab}}, negative B{C{bc}} or C{(B{ab}

1046 + B{bc})} not positive.

1047

1048 @note: If B{C{radius_triaxial}} is a L{JacobiConformalSpherical}

1049 and if B{C{ab}} and B{C{bc}} are both zero or C{None},

1050 the B{C{radius_triaxial}}'s C{ab}, C{bc}, C{a}, C{b}

1051 and C{c} are copied.

1052 '''

1053 try:

1054 r = radius_triaxial

1055 if isinstance(r, Triaxial): # ordered only

1056 t = r._abc3

1057 j = isinstance(r, JacobiConformalSpherical) and not bool(ab or bc)

1058 else:

1059 t = (Radius(radius=r),) * 3

1060 j = False

1061 self._ab = r.ab if j else Scalar_(ab=ab) # low=0

1062 self._bc = r.bc if j else Scalar_(bc=bc) # low=0

1063 if (self.ab + self.bc) <= 0:

1064 raise ValueError(_negative_)

1065 a, _, c = self._abc3 = t

1066 if not (a >= c and isfinite(self._a2b2)

1067 and isfinite(self._a2c2)):

1068 raise ValueError(_not_(_finite_))

1069 except (TypeError, ValueError) as x:

1070 raise TriaxialError(radius_triaxial=r, ab=ab, bc=bc, cause=x)

1071 if name:

1072 self.name = name

1073

1074 @Property_RO

1075 def ab(self):

1076 '''Get relative magnitude C{ab} (C{meter}, same units as B{C{a}}).

1077 '''

1078 return self._ab

1079

1080 @Property_RO

1081 def _a2b2(self):

1082 '''(INTERNAL) Get C{a**2 - b**2} == ab * (a + b).

1083 '''

1084 a, b, _ = self._abc3

1085 return self.ab * (a + b)

1086

1087 @Property_RO

1088 def _a2c2(self):

1089 '''(INTERNAL) Get C{a**2 - c**2} == a2b2 + b2c2.

1090 '''

1091 return self._a2b2 + self._b2c2

1092

1093 @Property_RO

1094 def bc(self):

1095 '''Get relative magnitude C{bc} (C{meter}, same units as B{C{a}}).

1096 '''

1097 return self._bc

1098

1099 @Property_RO

1100 def _b2c2(self):

1101 '''(INTERNAL) Get C{b**2 - c**2} == bc * (b + c).

1102 '''

1103 _, b, c = self._abc3

1104 return self.bc * (b + c)

1105

1106 @Property_RO

1107 def radius(self):

1108 '''Get radius (C{meter}, conventionally).

1109 '''

1110 return self.a

1111

1112

1113 class TriaxialError(_ValueError):

1114 '''Raised for L{Triaxial} issues.

1115 '''

1116 pass # ...

1117

1118

1119 class Triaxials(_NamedEnum):

1120 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class

1121 to accommodate the L{_LazyNamedEnumItem} properties.

1122 '''

1123 def _Lazy(self, *abc, **name):

1124 '''(INTERNAL) Instantiate the C{Triaxial}.

1125 '''

1126 a, b, c = map(km2m, abc)

1127 return Triaxial(a, b, c, **name)

1128

1129 Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton

1130'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.'''

1131# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in Km

1132# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf>

1133# <https://link.Springer.com/article/10.1007/s00190-022-01650-9>

1134_abc84_35 = (_EWGS84.a + 35), (_EWGS84.a - 35), _EWGS84.b

1135 Triaxials._assert( # a (Km) b (Km) c (Km) planet

1136 Amalthea = _lazy('Amalthea', 125.0, 73.0, 64), # Jupiter

1137 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus

1138 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544),

1139 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn

1140 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter

1141 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter

1142 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3),

1143 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn

1144 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus

1145 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth

1146 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn

1147 WGS84_35 = _lazy('WGS84_35', *map(m2km, _abc84_35)))

1148 del _abc84_35, _EWGS84

1149

1150

1151 def _getitems(items, *indices):

1152 '''(INTERNAL) Get the C{items} at the given I{indices}.

1153

1154 @return: C{Type(items[i] for i in indices)} with

1155 C{Type = type(items)}, any C{type} having

1156 the special method C{__getitem__}.

1157 '''

1158 return type(items)(map(items.__getitem__, indices))

1159

1160

1161 def _hartzell3(pov, los, Tun): # in .Ellipsoid.hartzell4, .formy.hartzell

1162 '''(INTERNAL) Hartzell's "Satellite Line-of-Sight Intersection ...",

1163 formula from a Point-Of-View to an I{un-/ordered} Triaxial.

1164 '''

1165 def _toUvwV3d(los, pov):

1166 try: # pov must be LatLon or Cartesian if los is a Los

1167 v = los.toUvw(pov)

1168 except (AttributeError, TypeError):

1169 v = _otherV3d(los=los)

1170 return v

1171

1172 p3 = _otherV3d(pov=pov.toCartesian() if isLatLon(pov) else pov)

1173 if los is True: # normal

1174 a, b, c, d, i = _normalTo5(p3.x, p3.y, p3.z, Tun)

1175 return type(p3)(a, b, c), d, i

1176

1177 u3 = p3.negate() if los is False or los is None else _toUvwV3d(los, pov)

1178

1179 a, b, c, T = Tun._ordered4

1180

1181 a2 = a**2 # largest, factored out

1182 b2, p2 = (b**2, T._1e2ab) if b != a else (a2, _1_0)

1183 c2, q2 = (c**2, T._1e2ac) if c != a else (a2, _1_0)

1184

1185 p3 = T._order3d(p3)

1186 u3 = T._order3d(u3).unit() # unit vector, opposing signs

1187

1188 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz

1189 ux, vy, wz = u3.times_(p3).xyz

1190 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz

1191

1192 t = (p2 * c2), c2, b2

1193 m = fdot(t, u2, v2, w2) # a2 factored out

1194 if m < EPS0: # zero or near-null LOS vector

1195 raise _ValueError(_near_(_null_))

1196

1197 r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,

1198 -w2 * y2, -u2 * y2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2,

1199 -w2 * x2 * p2, b2 * u2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2)

1200 if r > 0: # a2 factored out

1201 r = sqrt(r) * b * c # == a * a * b * c / a2

1202 elif r < 0: # LOS pointing away from or missing the triaxial

1203 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_)

1204

1205 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out

1206 if d > 0: # POV inside or LOS outside or missing the triaxial

1207 s = fsumf_(_N_1_0, x2 / a2, y2 / b2, z2 / c2) # like _sideOf

1208 raise _ValueError(_outside_ if s > 0 else _inside_)

1209 elif fsum1f_(x2, y2, z2) < d**2: # d past triaxial's center

1210 raise _ValueError(_too_(_distant_))

1211

1212 v = p3.minus(u3.times(d)) # cartesian type(pov) or Vector3d

1213 h = p3.minus(v).length # distance to pov == -d

1214 return T._order3d(v, reverse=True), h, None

1215

1216

1217 def hartzell4(pov, los=False, tri_biax=_WGS84, **name):

1218 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight

1219 from a Point-Of-View outside.

1220

1221 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple}

1222 C{LatLon} or L{Vector3d}).

1223 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Los}, L{Vector3d}),

1224 C{True} for the I{normal, plumb} onto the surface or C{False} or

1225 C{None} to point to the center of the tri-/biaxial.

1226 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal} or

1227 L{JacobiConformalSpherical}) or biaxial ellipsoid (L{Datum},

1228 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} radius,

1229 conventionally in C{meter}).

1230 @kwarg name: Optional, overriding C{B{name}="hartzell4"} (C{str}).

1231

1232 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h}

1233 the distance from B{C{pov}} to C{(x, y, z)} I{along the} B{C{los}}, all

1234 in C{meter}, conventionally.

1235

1236 @raise TriaxialError: Invalid B{C{pov}} or B{C{pov}} inside the tri-/biaxial or

1237 invalid B{C{los}} or B{C{los}} points outside or away from

1238 the tri-/biaxial.

1239

1240 @raise TypeError: Invalid B{C{tri_biax}}, C{ellipsoid} or C{datum}.

1241

1242 @see: Class L{pygeodesy3.Los}, functions L{pygeodesy.tyr3d} and L{pygeodesy.hartzell}

1243 and U{lookAtSpheroid<https://PyPI.org/project/pymap3d>} and U{"Satellite

1244 Line-of-Sight Intersection with Earth"<https://StephenHartzell.Medium.com/

1245 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.

1246 '''

1247 if isinstance(tri_biax, Triaxial_):

1248 T = tri_biax

1249 else:

1250 D = tri_biax if isinstance(tri_biax, Datum) else \

1251 _spherical_datum(tri_biax, name__=hartzell4) # _dunder_nameof

1252 T = D.ellipsoid._triaxial

1253 try:

1254 v, h, i = _hartzell3(pov, los, T)

1255 except Exception as x:

1256 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x)

1257 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, # _dunder_nameof

1258 name=_name__(name, name__=hartzell4))

1259

1260

1261 def _hypot21(x, y, z=0):

1262 '''(INTERNAL) Compute M{x**2 + y**2 + z**2 - 1} with C{max(fabs(x),

1263 fabs(y), fabs(z))} rarely greater than 1.0.

1264 '''

1265 return fsumf_(_1_0, x**2, y**2, (z**2 if z else _0_0), _N_2_0)

1266

1267

1268 def _normalTo4(x, y, a, b, eps=EPS):

1269 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}.

1270

1271 @see: Function C{pygeodesy.ellipsoids._normalTo3} and I{Eberly}'s U{Distance

1272 from a Point to ... an Ellipse ...<https://www.GeometricTools.com/

1273 Documentation/DistancePointEllipseEllipsoid.pdf>}.

1274 '''

1275 if b > a:

1276 b, a, d, i = _normalTo4(y, x, b, a, eps=eps)

1277 return a, b, d, i

1278

1279 if not (b > 0 and isfinite(a)):

1280 raise _ValueError(a=a, b=b)

1281

1282 i, _a = None, fabs

1283 if y:

1284 if x:

1285 u = _a(x / a)

1286 v = _a(y / b)

1287 g = _hypot21(u, v)

1288 if _a(g) < EPS02: # on the ellipse

1289 a, b, d = x, y, _0_0

1290 else:

1291 r = (b / a)**2

1292 t, i = _rootXd(_1_0 / r, 0, u, 0, v, g, eps)

1293 a = x / (t * r + _1_0)

1294 b = y / (t + _1_0)

1295 d = hypot(x - a, y - b)

1296 else: # x == 0

1297 if y < 0:

1298 b = -b

1299 a, d = x, _a(y - b)

1300

1301 else: # y == 0

1302 n = a * x

1303 d = (a + b) * (a - b)

1304 if d > _a(n): # PYCHOK no cover

1305 r = n / d

1306 a *= r

1307 r = _1_0 - r**2

1308 if r > EPS02:

1309 b *= sqrt(r)

1310 d = hypot(x - a, b)

1311 else:

1312 b = _0_0

1313 d = _a(x - a)

1314 else:

1315 if x < 0:

1316 a = -a

1317 b, d = y, _a(x - a)

1318 return a, b, d, i

1319

1320

1321 def _normalTo5(x, y, z, Tun, eps=EPS): # MCCABE 19

1322 '''(INTERNAL) Nearest point on and distance to an I{un-/ordered} triaxial.

1323

1324 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https://

1325 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}.

1326 '''

1327 a, b, c, T = Tun._ordered4

1328 if Tun is not T: # T is ordered, Tun isn't

1329 t = T._order3(x, y, z) + (T,)

1330 a, b, c, d, i = _normalTo5(*t, eps=eps)

1331 return T._order3(a, b, c, reverse=True) + (d, i)

1332

1333 if not (c > 0 and isfinite(a)):

1334 raise _ValueError(a=a, b=b, c=c)

1335

1336 if eps > 0:

1337 val = max(eps * 1e8, EPS)

1338 else: # no validation

1339 val, eps = 0, max(EPS0, -eps)

1340

1341 i, _a = None, fabs

1342 if z:

1343 if y:

1344 if x:

1345 u = _a(x / a)

1346 v = _a(y / b)

1347 w = _a(z / c)

1348 g = _hypot21(u, v, w)

1349 if _a(g) < EPS02: # on the ellipsoid

1350 a, b, c, d = x, y, z, _0_0

1351 else:

1352 r = T._1e2ac # (c / a)**2

1353 s = T._1e2bc # (c / b)**2

1354 t, i = _rootXd(_1_0 / r, _1_0 / s, u, v, w, g, eps)

1355 a = x / (t * r + _1_0)

1356 b = y / (t * s + _1_0)

1357 c = z / (t + _1_0)

1358 d = hypot_(x - a, y - b, z - c)

1359 else: # x == 0

1360 a = x # 0

1361 b, c, d, i = _normalTo4(y, z, b, c, eps=eps)

1362 elif x: # y == 0

1363 b = y # 0

1364 a, c, d, i = _normalTo4(x, z, a, c, eps=eps)

1365 else: # x == y == 0

1366 if z < 0:

1367 c = -c

1368 a, b, d = x, y, _a(z - c)

1369

1370 else: # z == 0

1371 t = True

1372 d = T._a2c2 # (a + c) * (a - c)

1373 n = a * x

1374 if d > _a(n):

1375 u = n / d

1376 d = T._b2c2 # (b + c) * (b - c)

1377 n = b * y

1378 if d > _a(n):

1379 v = n / d

1380 n = _hypot21(u, v)

1381 if n < 0:

1382 a *= u

1383 b *= v

1384 c *= sqrt(-n)

1385 d = hypot_(x - a, y - b, c)

1386 t = False

1387 if t:

1388 c = z # signed-0

1389 a, b, d, i = _normalTo4(x, y, a, b, eps=eps)

1390

1391 if val > 0: # validate

1392 e = T.sideOf(a, b, c, eps=val)

1393 if e: # not near the ellipsoid's surface

1394 raise _ValueError(a=a, b=b, c=c, d=d,

1395 sideOf=e, eps=val)

1396 if d: # angle of delta and normal vector

1397 m = Vector3d(x, y, z).minus_(a, b, c)

1398 if m.euclid > val:

1399 m = m.unit()

1400 n = T.normal3d(a, b, c)

1401 e = n.dot(m) # n.negate().dot(m)

1402 if not isnear1(_a(e), eps1=val):

1403 raise _ValueError(n=n, m=m,

1404 dot=e, eps=val)

1405 return a, b, c, d, i

1406

1407

1408 def _otherV3d_(x_xyz, y, z, **name):

1409 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}.

1410 '''

1411 return Vector3d(x_xyz, y, z, **name) if isscalar(x_xyz) else \

1412 _otherV3d(x_xyz=x_xyz)

1413

1414

1415 def _rootXd(r, s, u, v, w, g, eps):

1416 '''(INTERNAL) Robust 2d- or 3d-root finder: 2d- if C{s == v == 0} else 3d-root.

1417

1418 @see: I{Eberly}'s U{Robust Root Finders ...<https://www.GeometricTools.com/

1419 Documentation/DistancePointEllipseEllipsoid.pdf>}.

1420 '''

1421 _1, __2 = _1_0, _0_5

1422 _a, _h21 = fabs, _hypot21

1423

1424 u *= r

1425 v *= s # 0 for 2d-root

1426 t0 = w - _1

1427 t1 = _0_0 if g < 0 else (hypot_(u, w, v) - _1)

1428 # assert t0 <= t1

1429 for i in range(1, _TRIPS): # 48-55

1430 e = _a(t0 - t1)

1431 if e < eps:

1432 break

1433 t = (t0 + t1) * __2

1434 if t in (t0, t1):

1435 break

1436 g = _h21(u / (t + r), w / (t + _1),

1437 (v / (t + s)) if v else 0)

1438 if g > 0:

1439 t0 = t

1440 elif g < 0:

1441 t1 = t

1442 else:

1443 break

1444 else: # PYCHOK no cover

1445 t = Fmt.no_convergence(e, eps)

1446 raise _ValueError(t, txt__=_rootXd)

1447 return t, i

1448

1449

1450 def _sideOf(xyz, abc, eps=EPS):

1451 '''(INTERNAL) Helper for C{_hartzell3}, M{.sideOf} and M{.reverseCartesian}.

1452

1453 @return: M{sum((x / a)**2 for x, a in zip(xyz, abc)) - 1} or C{INT0}.

1454 '''

1455 s = fsumf_(_N_1_0, *((x / a)**2 for x, a in _zip(xyz, abc) if a)) # strict=True

1456 return INT0 if fabs(s) < eps else s

1457

1458

1459 if __name__ == '__main__':

1460

1461 from pygeodesy import printf

1462 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_

1463

1464 # __doc__ of this file, force all into registery

1465 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_)

1466 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))

1467

1468# **) MIT License

1469#

1470# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1471#

1472# Permission is hereby granted, free of charge, to any person obtaining a

1473# copy of this software and associated documentation files (the "Software"),

1474# to deal in the Software without restriction, including without limitation

1475# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1476# and/or sell copies of the Software, and to permit persons to whom the

1477# Software is furnished to do so, subject to the following conditions:

1478#

1479# The above copyright notice and this permission notice shall be included

1480# in all copies or substantial portions of the Software.

1481#

1482# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1483# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1484# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1485# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1486# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1487# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1488# OTHER DEALINGS IN THE SOFTWARE.