Coverage for pygeodesy/triaxials.py: 95%
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« prev ^ index » next coverage.py v7.2.2, created at 2023-04-12 11:45 -0400
2# -*- coding: utf-8 -*-
4u'''Triaxal ellipsoid classes L{JacobiConformal}, Jacobi's conformal projection, trancoded
5from I{Charles Karney}'s C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/C++/
6doc/classGeographicLib_1_1JacobiConformal.html#details>} to pure Python, I{ordered} L{Triaxial}
7and I{unordered} L{Triaxial_} and miscellaneous classes L{BetaOmega2Tuple}, L{BetaOmega3Tuple},
8L{Jacobi2Tuple} and L{TriaxialError}.
10@see: U{Geodesics on a triaxial ellipsoid<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid#
11 Geodesics_on_a_triaxial_ellipsoid>} and U{Triaxial coordinate systems and their geometrical
12 interpretation<https://www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
14@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)
15@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)
16@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)
17@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)
18@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)
19@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)
20@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)
21@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)
22@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)
23@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)
24@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)
25@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)
26'''
27# make sure int/int division yields float quotient, see .basics
28from __future__ import division as _; del _ # PYCHOK semicolon
30# from pygeodesy.basics import isscalar, _zip # from .fsums, .namedTuples, .streprs
31from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, _EPS2e4, INT0, PI2, PI_3, PI4, \
32 _0_0, _0_5, _1_0, _N_2_0, isfinite, isnear1, \
33 _4_0 # PYCHOK used!
34from pygeodesy.datums import Datum, Ellipsoid, _spherical_datum, _WGS84
35# from pygeodesy.ellipsoids import Ellipsoid # from .datums
36# from pygeodesy.elliptic import Elliptic # ._MODS
37# from pygeodesy.errors import _ValueError # from .streprs
38from pygeodesy.fmath import Fdot, fdot, fmean_, hypot, hypot_, _hypot21_, norm2
39from pygeodesy.fsums import Fsum, fsum_, isscalar, Property_RO
40from pygeodesy.interns import NN, _a_, _b_, _beta_, _c_, _distant_, _height_, \
41 _inside_, _near_, _not_, _NL_, _NLATvar_, _NOTEQUAL_, \
42 _null_, _opposite_, _outside_, _SPACE_, _spherical_, \
43 _too_, _x_, _y_, _COMMA_ # PYCHOK used!
44# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .vector3d
45from pygeodesy.named import _NamedBase, _NamedEnum, _NamedEnumItem, \
46 _NamedTuple, _Pass, _lazyNamedEnumItem as _lazy
47from pygeodesy.namedTuples import LatLon3Tuple, Vector3Tuple, Vector4Tuple
48# from pygeodesy.props import Property_RO # from .fsums
49from pygeodesy.streprs import Fmt, _ValueError, _zip
50from pygeodesy.units import Degrees, Float, Height_, Meter, Meter2, Meter3, Radians, Radius
51from pygeodesy.utily import asin1, atan2d, km2m, m2km, sincos2, sincos2d, sincos2d_
52from pygeodesy.vector3d import _ALL_LAZY, _MODS, _otherV3d, Vector3d
54from math import atan2, fabs, sqrt
56__all__ = _ALL_LAZY.triaxials
57__version__ = '23.04.02'
59_E = _WGS84.ellipsoid
60_not_ordered_ = _not_('ordered')
61_omega_ = 'omega'
62_TRIPS = 537 # max 55, Eberly 1074?
63_WGS84_35abc = _E.a + 35, _E.a - 35, _E.b
64del _E
67class _ToNamedBase(_NamedBase):
68 '''(INTERNAL) C{-.toDegrees}, C{-.toRadians} base.
69 '''
70 def _toDegrees(self, a, b, *c, **toDMS_kwds):
71 if toDMS_kwds:
72 toDMS = _MODS.dms.toDMS
73 a = toDMS(a.toDegrees(), **toDMS_kwds)
74 b = toDMS(b.toDegrees(), **toDMS_kwds)
75 elif isinstance(a, Degrees) and \
76 isinstance(b, Degrees):
77 return self
78 else:
79 a, b = a.toDegrees(), b.toDegrees()
80 return self.classof(a, b, *c, name=self.name)
82 def _toRadians(self, a, b, *c):
83 return self if isinstance(a, Radians) and \
84 isinstance(b, Radians) else \
85 self.classof(a.toRadians(), b.toRadians(),
86 *c, name=self.name)
89class BetaOmega2Tuple(_NamedTuple, _ToNamedBase):
90 '''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and
91 longitude C{beta} and C{omega} both in C{Radians} (or
92 C{Degrees}).
93 '''
94 _Names_ = (_beta_, _omega_)
95 _Units_ = (_Pass, _Pass)
97 def toDegrees(self, **toDMS_kwds):
98 '''Convert this L{BetaOmega2Tuple} to C{Degrees} or C{toDMS}.
100 @return: L{BetaOmega2Tuple}C{(beta, omega)} with
101 C{beta} and C{omega} both in C{Degrees}
102 or as an L{toDMS} string provided some
103 B{C{toDMS_kwds}} are supplied.
104 '''
105 return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds)
107 def toRadians(self):
108 '''Convert this L{BetaOmega2Tuple} to C{Radians}.
110 @return: L{BetaOmega2Tuple}C{(beta, omega)} with
111 C{beta} and C{omega} both in C{Radians}.
112 '''
113 return _ToNamedBase._toRadians(self, *self)
116class BetaOmega3Tuple(_NamedTuple, _ToNamedBase):
117 '''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and
118 longitude C{beta} and C{omega} both in C{Radians} (or C{Degrees})
119 and the C{height}, rather the (signed) I{distance} to the triaxial's
120 surface (measured along the radial line to the triaxial's center)
121 in C{meter}, conventionally.
122 '''
123 _Names_ = BetaOmega2Tuple._Names_ + (_height_,)
124 _Units_ = BetaOmega2Tuple._Units_ + ( Meter,)
126 def toDegrees(self, **toDMS_kwds):
127 '''Convert this L{BetaOmega3Tuple} to C{Degrees} or C{toDMS}.
129 @return: L{BetaOmega3Tuple}C{(beta, omega, height)} with
130 C{beta} and C{omega} both in C{Degrees} or as an
131 L{toDMS} string provided some B{C{toDMS_kwds}}
132 are supplied.
133 '''
134 return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds)
136 def toRadians(self):
137 '''Convert this L{BetaOmega3Tuple} to C{Radians}.
139 @return: L{BetaOmega3Tuple}C{(beta, omega, height)} with
140 C{beta} and C{omega} both in C{Radians}.
141 '''
142 return _ToNamedBase._toRadians(self, *self)
144 def to2Tuple(self):
145 '''Reduce this L{BetaOmega3Tuple} to a L{BetaOmega2Tuple}.
146 '''
147 return BetaOmega2Tuple(*self[:2])
150class Jacobi2Tuple(_NamedTuple, _ToNamedBase):
151 '''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y}
152 projection, both in C{Radians} (or C{Degrees}).
153 '''
154 _Names_ = (_x_, _y_)
155 _Units_ = (_Pass, _Pass)
157 def toDegrees(self, **toDMS_kwds):
158 '''Convert this L{Jacobi2Tuple} to C{Degrees} or C{toDMS}.
160 @return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y}
161 both in C{Degrees} or as an L{toDMS} string
162 provided some B{C{toDMS_kwds}} are supplied.
163 '''
164 return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds)
166 def toRadians(self):
167 '''Convert this L{Jacobi2Tuple} to C{Radians}.
169 @return: L{Jacobi2Tuple}C{(x, y)} with C{x}
170 and C{y} both in C{Radians}.
171 '''
172 return _ToNamedBase._toRadians(self, *self)
175class Triaxial_(_NamedEnumItem):
176 '''I{Unordered} triaxial ellipsoid and base class.
178 Triaxial ellipsoids with right-handed semi-axes C{a}, C{b} and C{c}, oriented
179 such that the large principal ellipse C{ab} is the equator I{Z}=0, I{beta}=0,
180 while the small principal ellipse C{ac} is the prime meridian, plane I{Y}=0,
181 I{omega}=0.
183 The four umbilic points, C{abs}(I{omega}) = C{abs}(I{beta}) = C{PI/2}, lie on
184 the middle principal ellipse C{bc} in plane I{X}=0, I{omega}=C{PI/2}.
186 @note: I{Geodetic} C{lat}- and C{lon}gitudes are in C{degrees}, I{geodetic}
187 C{phi} and C{lam}bda are in C{radians}, but I{ellipsoidal} lat- and
188 longitude C{beta} and C{omega} are in C{Radians} by default (or in
189 C{Degrees} if converted).
190 '''
191 _ijk = _kji = None
192 _unordered = True
194 def __init__(self, a_triaxial, b=None, c=None, name=NN):
195 '''New I{unordered} L{Triaxial_}.
197 @arg a_triaxial: C{X} semi-axis (C{scalar}, conventionally in C{meter})
198 or an other L{Triaxial} or L{Triaxial_} instance.
199 @kwarg b: C{Y} semi-axis (C{meter}, same units as B{C{a}}), required
200 if C{B{a_triaxial} is scalar}, ignored otherwise.
201 @kwarg c: C{Z} semi-axis (C{meter}, same units as B{C{a}}), required
202 if C{B{a_triaxial} is scalar}, ignored otherwise.
203 @kwarg name: Optional name (C{str}).
205 @raise TriaxialError: Invalid semi-axis or -axes.
206 '''
207 try:
208 a = a_triaxial
209 t = a._abc3 if isinstance(a, Triaxial_) else (
210 Radius(a=a), Radius(b=b), Radius(c=c))
211 except (TypeError, ValueError) as x:
212 raise TriaxialError(a=a, b=b, c=c, cause=x)
213 if name:
214 self.name = name
216 a, b, c = self._abc3 = t
217 if self._unordered: # == not isinstance(self, Triaxial)
218 s, _, t = sorted(t)
219 if not (isfinite(t) and s > 0):
220 raise TriaxialError(a=a, b=b, c=c) # txt=_invalid_
221 elif not (isfinite(a) and a >= b >= c > 0):
222 raise TriaxialError(a=a, b=b, c=c, txt=_not_ordered_)
223 elif not (a > c and self._a2c2 > 0 and self.e2ac > 0):
224 raise TriaxialError(a=a, c=c, e2ac=self.e2ac, txt=_spherical_)
226 def __str__(self):
227 return self.toStr()
229 @Property_RO
230 def a(self):
231 '''Get the (largest) C{x} semi-axis (C{meter}, conventionally).
232 '''
233 a, _, _ = self._abc3
234 return a
236 @Property_RO
237 def _a2b2(self):
238 '''(INTERNAL) Get C{a**2 - b**2} == E_sub_e**2.
239 '''
240 a, b, _ = self._abc3
241 return ((a - b) * (a + b)) if a != b else _0_0
243 @Property_RO
244 def _a2_b2(self):
245 '''(INTERNAL) Get C{(a/b)**2}.
246 '''
247 a, b, _ = self._abc3
248 return (a / b)**2 if a != b else _1_0
250 @Property_RO
251 def _a2c2(self):
252 '''(INTERNAL) Get C{a**2 - c**2} == E_sub_x**2.
253 '''
254 a, _, c = self._abc3
255 return ((a - c) * (a + c)) if a != c else _0_0
257 @Property_RO
258 def area(self):
259 '''Get the surface area (C{meter} I{squared}).
260 '''
261 c, b, a = sorted(self._abc3)
262 if a > c:
263 a = Triaxial(a, b, c).area if a > b else \
264 Ellipsoid(a, b=c).areax # a == b
265 else: # a == c == b
266 a = Meter2(area=a**2 * PI4)
267 return a
269 def area_p(self, p=1.6075):
270 '''I{Approximate} the surface area (C{meter} I{squared}).
272 @kwarg p: Exponent (C{scalar} > 0), 1.6 for near-spherical or 1.5849625007
273 for "near-flat" triaxials.
275 @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Approximate_formula>}.
276 '''
277 a, b, c = self._abc3
278 if a == b == c:
279 a *= a
280 else:
281 _p = pow
282 a = _p(fmean_(_p(a * b, p), _p(a * c, p), _p(b * c, p)), _1_0 / p)
283 return Meter2(area_p=a * PI4)
285 @Property_RO
286 def b(self):
287 '''Get the (middle) C{y} semi-axis (C{meter}, same units as B{C{a}}).
288 '''
289 _, b, _ = self._abc3
290 return b
292 @Property_RO
293 def _b2c2(self):
294 '''(INTERNAL) Get C{b**2 - c**2} == E_sub_y**2.
295 '''
296 _, b, c = self._abc3
297 return ((b - c) * (b + c)) if b != c else _0_0
299 @Property_RO
300 def c(self):
301 '''Get the (smallest) C{z} semi-axis (C{meter}, same units as B{C{a}}).
302 '''
303 _, _, c = self._abc3
304 return c
306 @Property_RO
307 def _c2_b2(self):
308 '''(INTERNAL) Get C{(c/b)**2}.
309 '''
310 _, b, c = self._abc3
311 return (c / b)**2 if b != c else _1_0
313 @Property_RO
314 def e2ab(self):
315 '''Get the C{ab} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (b/a)**2}.
316 '''
317 return Float(e2ab=(_1_0 - self._1e2ab) or _0_0)
319 @Property_RO
320 def _1e2ab(self):
321 '''(INTERNAL) Get C{1 - e2ab} == C{(b/a)**2}.
322 '''
323 a, b, _ = self._abc3
324 return (b / a)**2 if a != b else _1_0
326 @Property_RO
327 def e2bc(self):
328 '''Get the C{bc} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/b)**2}.
329 '''
330 return Float(e2bc=(_1_0 - self._1e2bc) or _0_0)
332 _1e2bc = _c2_b2 # C{1 - e2bc} == C{(c/b)**2}
334 @Property_RO
335 def e2ac(self):
336 '''Get the C{ac} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/a)**2}.
337 '''
338 return Float(e2ac=(_1_0 - self._1e2ac) or _0_0)
340 @Property_RO
341 def _1e2ac(self):
342 '''(INTERNAL) Get C{1 - e2ac} == C{(c/a)**2}.
343 '''
344 a, _, c = self._abc3
345 return (c / a)**2 if a != c else _1_0
347 @Property_RO
348 def _Elliptic(self):
349 '''(INTERNAL) Get class L{Elliptic} once.
350 '''
351 return _MODS.elliptic.Elliptic
353 def hartzell4(self, pov, los=None, name=NN):
354 '''Compute the intersection of this triaxial's surface with a Line-Of-Sight
355 from a Point-Of-View in space.
357 @see: Function L{pygeodesy.hartzell4} for further details.
358 '''
359 return hartzell4(pov, los=los, tri_biax=self, name=name)
361 def height4(self, x_xyz, y=None, z=None, normal=True, eps=EPS):
362 '''Compute the projection on and the height of a cartesian above or below
363 this triaxial's surface.
365 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
366 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
367 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}.
368 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}.
369 @kwarg normal: If C{True} the projection is perpendicular to (the nearest
370 point on) this triaxial's surface, otherwise the C{radial}
371 line to this triaxial's center (C{bool}).
372 @kwarg eps: Tolerance for root finding and validation (C{scalar}), use a
373 negative value to skip validation.
375 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates
376 C{x}, C{y} and C{z} of the projection on or the intersection
377 with and with the height C{h} above or below the triaxial's
378 surface in C{meter}, conventionally.
380 @raise TriaxialError: Non-cartesian B{C{xyz}}, invalid B{C{eps}}, no
381 convergence in root finding or validation failed.
383 @see: Method L{Ellipsoid.height4} and I{Eberly}'s U{Distance from a Point
384 to ... an Ellipsoid ...<https://www.GeometricTools.com/Documentation/
385 DistancePointEllipseEllipsoid.pdf>}.
386 '''
387 v, r = _otherV3d_(x_xyz, y, z), self.isSpherical
389 i, h = None, v.length
390 if h < EPS0: # EPS
391 x = y = z = _0_0
392 h -= min(self._abc3) # nearest
393 elif r: # .isSpherical
394 x, y, z = v.times(r / h).xyz
395 h -= r
396 else:
397 x, y, z = v.xyz
398 try:
399 if normal: # perpendicular to triaxial
400 x, y, z, h, i = _normalTo5(x, y, z, self, eps=eps)
401 else: # radially to triaxial's center
402 x, y, z = self._radialTo3(z, hypot(x, y), y, x)
403 h = v.minus_(x, y, z).length
404 except Exception as e:
405 raise TriaxialError(x=x, y=y, z=z, cause=e)
406 if h > 0 and self.sideOf(v, eps=EPS0) < 0:
407 h = -h # below the surface
408 return Vector4Tuple(x, y, z, h, iteration=i, name=self.height4.__name__)
410 @Property_RO
411 def isOrdered(self):
412 '''Is this triaxial I{ordered} and not I{spherical} (C{bool})?
413 '''
414 a, b, c = self._abc3
415 return bool(a >= b > c) # b > c!
417 @Property_RO
418 def isSpherical(self):
419 '''Is this triaxial I{spherical} (C{Radius} or INT0)?
420 '''
421 a, b, c = self._abc3
422 return a if a == b == c else INT0
424 def normal3d(self, x_xyz, y=None, z=None, length=_1_0):
425 '''Get a 3-D vector at a cartesian on, perpendicular to this triaxial's surface.
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}).
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}}.
436 @note: Cartesian location C{(B{x}, B{y}, B{z})} must be on this triaxial's
437 surface, use method L{Triaxial.sideOf} to validate.
438 '''
439 # n = 2 * (x / a2, y / b2, z / c2)
440 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered
441 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2
442 # == unit(x, y / _1e2ab, z / _1e2ac).times(length)
443 n = self._normal3d.times_(*_otherV3d_(x_xyz, y, z).xyz)
444 if n.length < EPS0:
445 raise TriaxialError(x=x_xyz, y=y, z=z, txt=_null_)
446 return n.times(length / n.length)
448 @Property_RO
449 def _normal3d(self):
450 '''(INTERNAL) Get M{Vector3d((d/a)**2, (d/b)**2, (d/c)**2)}, M{d = max(a, b, c)}.
451 '''
452 d = max(self._abc3)
453 t = tuple(((d / x)**2 if x != d else _1_0) for x in self._abc3)
454 return Vector3d(*t, name=self.normal3d.__name__)
456 def _norm2(self, s, c, *a):
457 '''(INTERNAL) Normalize C{s} and C{c} iff not already.
458 '''
459 if fabs(s) > _1_0 or fabs(c) > _1_0 or \
460 fabs(_hypot21_(s, c)) > EPS0:
461 s, c = norm2(s, c)
462 if a:
463 s, c = norm2(s * self.b, c * a[0])
464 return (s or _0_0), (c or _0_0)
466 def _order3(self, *abc, **reverse): # reverse=False
467 '''(INTERNAL) Un-/Order C{a}, C{b} and C{c}.
469 @return: 3-Tuple C{(a, b, c)} ordered by or un-ordered
470 (reverse-ordered) C{ijk} if C{B{reverse}=True}.
471 '''
472 ijk = self._order_ijk(**reverse)
473 return _getitems(abc, *ijk) if ijk else abc
475 def _order3d(self, v, **reverse): # reverse=False
476 '''(INTERNAL) Un-/Order a C{Vector3d}.
478 @return: Vector3d(x, y, z) un-/ordered.
479 '''
480 ijk = self._order_ijk(**reverse)
481 return v.classof(*_getitems(v.xyz, *ijk)) if ijk else v
483 @Property_RO
484 def _ordered4(self):
485 '''(INTERNAL) Helper for C{_hartzell3d2} and C{_normalTo5}.
486 '''
487 def _order2(reverse, a, b, c):
488 '''(INTERNAL) Un-Order C{a}, C{b} and C{c}.
490 @return: 2-Tuple C{((a, b, c), ijk)} with C{a} >= C{b} >= C{c}
491 and C{ijk} a 3-tuple with the initial indices.
492 '''
493 i, j, k = 0, 1, 2
494 if a < b:
495 a, b, i, j = b, a, j, i
496 if a < c:
497 a, c, i, k = c, a, k, i
498 if b < c:
499 b, c, j, k = c, b, k, j
500 # reverse (k, j, i) since (a, b, c) is reversed-sorted
501 ijk = (k, j, i) if reverse else (None if i < j < k else (i, j, k))
502 return (a, b, c), ijk
504 abc, T = self._abc3, self
505 if not self.isOrdered:
506 abc, ijk = _order2(False, *abc)
507 if ijk:
508 _, kji = _order2(True, *ijk)
509 T = Triaxial_(*abc)
510 T._ijk, T._kji = ijk, kji
511 return abc + (T,)
513 def _order_ijk(self, reverse=False):
514 '''(INTERNAL) Get the un-/order indices.
515 '''
516 return self._kji if reverse else self._ijk
518 def _radialTo3(self, sbeta, cbeta, somega, comega):
519 '''(INTERNAL) I{Unordered} helper for C{.height4}.
520 '''
521 def _rphi(a, b, sphi, cphi):
522 # <https://WikiPedia.org/wiki/Ellipse#Polar_form_relative_to_focus>
523 # polar form: radius(phi) = a * b / hypot(a * sphi, b * cphi)
524 return (b / hypot(sphi, b / a * cphi)) if a > b else (
525 (a / hypot(cphi, a / b * sphi)) if a < b else a)
527 sa, ca = self._norm2(sbeta, cbeta)
528 sb, cb = self._norm2(somega, comega)
530 a, b, c = self._abc3
531 if a != b:
532 a = _rphi(a, b, sb, cb)
533 if a != c:
534 c = _rphi(a, c, sa, ca)
535 z, r = c * sa, c * ca
536 x, y = r * cb, r * sb
537 return x, y, z
539 def sideOf(self, x_xyz, y=None, z=None, eps=EPS4):
540 '''Is a cartesian above, below or on the surface of this triaxial?
542 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
543 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
544 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}.
545 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}.
546 @kwarg eps: Near surface tolerance(C{scalar}).
548 @return: C{INT0} if C{(B{x}, B{y}, B{z})} is near this triaxial's surface
549 within tolerance B{C{eps}}, otherwise a neg- or positive C{float}
550 if in- respectively outside this triaxial.
552 @see: Methods L{Triaxial.height4} and L{Triaxial.normal3d}.
553 '''
554 return _sideOf(_otherV3d_(x_xyz, y, z).xyz, self._abc3, eps=eps)
556 def _sqrt(self, x):
557 '''(INTERNAL) Helper.
558 '''
559 if x < 0:
560 raise TriaxialError(Fmt.PAREN(sqrt=x))
561 return _0_0 if x < EPS02 else sqrt(x)
563 def toEllipsoid(self, name=NN):
564 '''Convert this triaxial to an L{Ellipsoid}, provided C{a == b} or C{b == c}.
566 @return: An L{Ellipsoid} with north along this C{Z} axis if C{a == b},
567 this C{Y} axis if C{a == c} or this C{X} axis if C{b == c}.
569 @raise TriaxialError: This C{a != b}, C{b != c} and C{c != a}.
571 @see: Method L{Ellipsoid.toTriaxial}.
572 '''
573 a, b, c = self._abc3
574 if a == b: # N = Z
575 b = c
576 elif b == c: # N = X
577 a, b = b, a
578 elif a != c:
579 t = _SPACE_(_a_, _NOTEQUAL_, _b_, _NOTEQUAL_, _c_)
580 raise TriaxialError(a=a, b=b, c=c, txt=t)
581 return Ellipsoid(a, b=b, name=name or self.name)
583 def toStr(self, prec=9, name=NN, **unused): # PYCHOK signature
584 '''Return this C{Triaxial} as a string.
586 @kwarg prec: Precision, number of decimal digits (0..9).
587 @kwarg name: Override name (C{str}) or C{None} to exclude
588 this triaxial's name.
590 @return: This C{Triaxial}'s attributes (C{str}).
591 '''
592 T = Triaxial_
593 t = T.a, T.b, T.c, T.e2ab, T.e2bc, T.e2ac
594 if isinstance(self, JacobiConformal):
595 t += JacobiConformal.xyQ2,
596 t += T.volume, T.area
597 return self._instr(name, prec, props=t, area_p=self.area_p())
599 @Property_RO
600 def volume(self):
601 '''Get the volume (C{meter**3}), M{4 / 3 * PI * a * b * c}.
602 '''
603 a, b, c = self._abc3
604 return Meter3(volume=a * b * c * PI_3 * _4_0)
607class Triaxial(Triaxial_):
608 '''I{Ordered} triaxial ellipsoid.
610 @see: L{Triaxial_} for more information.
611 '''
612 _unordered = False
614 def __init__(self, a_triaxial, b=None, c=None, name=NN):
615 '''New I{ordered} L{Triaxial}.
617 @arg a_triaxial: Largest semi-axis (C{scalar}, conventionally in C{meter})
618 or an other L{Triaxial} or L{Triaxial_} instance.
619 @kwarg b: Middle semi-axis (C{meter}, same units as B{C{a}}), required
620 if C{B{a_triaxial} is scalar}, ignored otherwise.
621 @kwarg c: Smallest semi-axis (C{meter}, same units as B{C{a}}), required
622 if C{B{a_triaxial} is scalar}, ignored otherwise.
623 @kwarg name: Optional name (C{str}).
625 @note: The semi-axes must be ordered as C{B{a} >= B{b} >= B{c} > 0} and
626 must be ellipsoidal, C{B{a} > B{c}}.
628 @raise TriaxialError: Semi-axes not ordered, spherical or invalid.
629 '''
630 Triaxial_.__init__(self, a_triaxial, b=b, c=c, name=name)
632 @Property_RO
633 def _a2b2_a2c2(self):
634 ''' @see: Method C{forwardBetaOmega}.
635 '''
636 return self._a2b2 / self._a2c2
638 @Property_RO
639 def area(self):
640 '''Get the surface area (C{meter} I{squared}).
642 @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Surface_area>}.
643 '''
644 a, b, c = self._abc3
645 if a != b:
646 kp2, k2 = self._k2_kp2 # swapped!
647 aE = self._Elliptic(k2, _0_0, kp2, _1_0)
648 c2 = self._1e2ac # cos(phi)**2 == (c/a)**2
649 s2 = self. e2ac # sin(phi)**2 == 1 - c2
650 s = sqrt(s2)
651 r = asin1(s) # phi == atan2(sqrt(c2), s)
652 b *= fsum_(aE.fE(r) * s, c / a * c / b,
653 aE.fF(r) * c2 / s, floats=True)
654 a = Meter2(area=a * b * PI2)
655 else: # a == b > c
656 a = Ellipsoid(a, b=c).areax
657 return a
659 def _exyz3(self, u):
660 '''(INTERNAL) Helper for C{.forwardBetOmg}.
661 '''
662 if u > 0:
663 u2 = u**2
664 x = self._sqrt(_1_0 + self._a2c2 / u2) * u
665 y = self._sqrt(_1_0 + self._b2c2 / u2) * u
666 else:
667 x = y = u = _0_0
668 return x, y, u
670 def forwardBetaOmega(self, beta, omega, height=0, name=NN):
671 '''Convert I{ellipsoidal} lat- and longitude C{beta}, C{omega}
672 and height to cartesian.
674 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
675 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
676 @arg height: Height above or below the ellipsoid's surface (C{meter}, same
677 units as this triaxial's C{a}, C{b} and C{c} semi-axes).
678 @kwarg name: Optional name (C{str}).
680 @return: A L{Vector3Tuple}C{(x, y, z)}.
682 @see: Method L{Triaxial.reverseBetaOmega} and U{Expressions (23-25)<https://
683 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
684 '''
685 if height:
686 h = Height_(height=height, low=-self.c, Error=TriaxialError)
687 x, y, z = self._exyz3(h + self.c)
688 else:
689 x, y, z = self._abc3 # == self._exyz3(self.c)
690 if z: # and x and y:
691 sa, ca = _SinCos2(beta)
692 sb, cb = _SinCos2(omega)
694 r = self._a2b2_a2c2
695 x *= cb * self._sqrt(ca**2 + r * sa**2)
696 y *= ca * sb
697 z *= sa * self._sqrt(_1_0 - r * cb**2)
698 return Vector3Tuple(x, y, z, name=name)
700 def forwardBetaOmega_(self, sbeta, cbeta, somega, comega, name=NN):
701 '''Convert I{ellipsoidal} lat- and longitude C{beta} and C{omega}
702 to cartesian coordinates I{on the triaxial's surface}.
704 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
705 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
706 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
707 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
708 @kwarg name: Optional name (C{str}).
710 @return: A L{Vector3Tuple}C{(x, y, z)} on the surface.
712 @raise TriaxialError: This triaxial is near-spherical.
714 @see: Method L{Triaxial.reverseBetaOmega}, U{Triaxial ellipsoid coordinate
715 system<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid#
716 Triaxial_ellipsoid_coordinate_system>} and U{expressions (23-25)<https://
717 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
718 '''
719 t = self._radialTo3(sbeta, cbeta, somega, comega)
720 return Vector3Tuple(*t, name=name)
722 def forwardCartesian(self, x_xyz, y=None, z=None, name=NN, **normal_eps):
723 '''Project a cartesian on this triaxial.
725 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
726 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
727 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}.
728 @kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}.
729 @kwarg name: Optional name (C{str}).
730 @kwarg normal_eps: Optional keyword arguments C{B{normal}=True} and
731 C{B{eps}=EPS}, see method L{Triaxial.height4}.
733 @see: Method L{Triaxial.height4} for further information and method
734 L{Triaxial.reverseCartesian} to reverse the projection.
735 '''
736 t = self.height4(x_xyz, y, z, **normal_eps)
737 _ = t.rename(name)
738 return t
740 def forwardLatLon(self, lat, lon, height=0, name=NN):
741 '''Convert I{geodetic} lat-, longitude and heigth to cartesian.
743 @arg lat: Geodetic latitude (C{degrees}).
744 @arg lon: Geodetic longitude (C{degrees}).
745 @arg height: Height above the ellipsoid (C{meter}, same units
746 as this triaxial's C{a}, C{b} and C{c} axes).
747 @kwarg name: Optional name (C{str}).
749 @return: A L{Vector3Tuple}C{(x, y, z)}.
751 @see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https://
752 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
753 '''
754 return self._forwardLatLon3(height, name, *sincos2d_(lat, lon))
756 def forwardLatLon_(self, slat, clat, slon, clon, height=0, name=NN):
757 '''Convert I{geodetic} lat-, longitude and heigth to cartesian.
759 @arg slat: Geodetic latitude C{sin(lat)} (C{scalar}).
760 @arg clat: Geodetic latitude C{cos(lat)} (C{scalar}).
761 @arg slon: Geodetic longitude C{sin(lon)} (C{scalar}).
762 @arg clon: Geodetic longitude C{cos(lon)} (C{scalar}).
763 @arg height: Height above the ellipsoid (C{meter}, same units
764 as this triaxial's axes C{a}, C{b} and C{c}).
765 @kwarg name: Optional name (C{str}).
767 @return: A L{Vector3Tuple}C{(x, y, z)}.
769 @see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https://
770 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
771 '''
772 sa, ca = self._norm2(slat, clat)
773 sb, cb = self._norm2(slon, clon)
774 return self._forwardLatLon3(height, name, sa, ca, sb, cb)
776 def _forwardLatLon3(self, h, name, sa, ca, sb, cb):
777 '''(INTERNAL) Helper for C{.forwardLatLon} and C{.forwardLatLon_}.
778 '''
779 ca_x_sb = ca * sb
780 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2
781 t = fsum_(_1_0, -self.e2ac * sa**2, -self.e2ab * ca_x_sb**2, floats=True)
782 N = self.a / self._sqrt(t) # prime vertical
783 x = (h + N) * ca * cb
784 y = (h + N * self._1e2ab) * ca_x_sb
785 z = (h + N * self._1e2ac) * sa
786 return Vector3Tuple(x, y, z, name=name)
788 @Property_RO
789 def _k2_kp2(self):
790 '''(INTERNAL) Get C{k2} and C{kp2} for C{._xE}, C{._yE} and C{.area}.
791 '''
792 # k2 = a2b2 / a2c2 * c2_b2
793 # kp2 = b2c2 / a2c2 * a2_b2
794 # b2 = b**2
795 # xE = Elliptic(k2, -a2b2 / b2, kp2, a2_b2)
796 # yE = Elliptic(kp2, +b2c2 / b2, k2, c2_b2)
797 # aE = Elliptic(kp2, 0, k2, 1)
798 return (self._a2b2 / self._a2c2 * self._c2_b2,
799 self._b2c2 / self._a2c2 * self._a2_b2)
801 def _radialTo3(self, sbeta, cbeta, somega, comega):
802 '''(INTERNAL) Convert I{ellipsoidal} lat- and longitude C{beta} and
803 C{omega} to cartesian coordinates I{on the triaxial's surface},
804 also I{ordered} helper for C{.height4}.
805 '''
806 sa, ca = self._norm2(sbeta, cbeta)
807 sb, cb = self._norm2(somega, comega)
809 b2_a2 = self._1e2ab # == (b/a)**2
810 c2_a2 = -self._1e2ac # == -(c/a)**2
811 a2c2_a2 = self. e2ac # (a**2 - c**2) / a**2 == 1 - (c/a)**2
813 x = Fsum(_1_0, -b2_a2 * sa**2, c2_a2 * ca**2).fover(a2c2_a2)
814 z = Fsum(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2)
816 x = self.a * cb * self._sqrt(x)
817 y = self.b * ca * sb
818 z = self.c * sa * self._sqrt(z)
819 return x, y, z
821 def reverseBetaOmega(self, x_xyz, y=None, z=None, name=NN):
822 '''Convert cartesian to I{ellipsoidal} lat- and longitude, C{beta}, C{omega}
823 and height.
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 name (C{str}).
831 @return: A L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and
832 C{omega} in C{Radians} and (radial) C{height} in C{meter}, same
833 units as this triaxial's axes.
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=name)
843 def reverseCartesian(self, x_xyz, y=None, z=None, h=0, normal=True, eps=_EPS2e4, name=NN):
844 '''"Unproject" a cartesian on to a cartesion I{off} this triaxial's surface.
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 name (C{str}).
857 @return: A L{Vector3Tuple}C{(x, y, z)}.
859 @raise TrialError: Cartesian C{(x, y, z)} not on this triaxial's surface.
861 @see: Methods L{Triaxial.forwardCartesian} and L{Triaxial.height4}.
862 '''
863 v = _otherV3d_(x_xyz, y, z, name=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)
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
876 def reverseLatLon(self, x_xyz, y=None, z=None, name=NN):
877 '''Convert cartesian to I{geodetic} lat-, longitude and height.
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 name (C{str}).
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.
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=name)
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
911class 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.
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 (or a sphere, I{currently
919 not supported}).
921 Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2020) and
922 licensed under the MIT/X11 License.
924 @note: This constructor can not be used to specify a sphere.
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# @Property_RO
933# def ab(self):
934# '''Get relative magnitude C{ab} (C{None} or C{meter}, same units as B{C{a}}).
935# '''
936# return self._ab
938# @Property_RO
939# def bc(self):
940# '''Get relative magnitude C{bc} (C{None} or C{meter}, same units as B{C{a}}).
941# '''
942# return self._bc
944 @Property_RO
945 def _xE(self):
946 '''(INTERNAL) Get the x-elliptic function.
947 '''
948 k2, kp2 = self._k2_kp2
949 # -a2b2 / b2 == (b2 - a2) / b2 == 1 - a2 / b2 == 1 - a2_b2
950 return self._Elliptic(k2, _1_0 - self._a2_b2, kp2, self._a2_b2)
952 def xR(self, omega):
953 '''Compute a Jacobi Conformal C{x} projection.
955 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
957 @return: The C{x} projection (C{Radians}).
958 '''
959 return self.xR_(*_SinCos2(omega))
961 def xR_(self, somega, comega):
962 '''Compute a Jacobi Conformal C{x} projection.
964 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
965 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
967 @return: The C{x} projection (C{Radians}).
968 '''
969 s, c = self._norm2(somega, comega, self.a)
970 return Radians(x=self._xE.fPi(s, c) * self._a2_b2)
972 @Property_RO
973 def xyQ2(self):
974 '''Get the Jacobi Conformal quadrant size (L{Jacobi2Tuple}C{(x, y)}).
975 '''
976 return Jacobi2Tuple(Radians(x=self._a2_b2 * self._xE.cPi),
977 Radians(y=self._c2_b2 * self._yE.cPi),
978 name=JacobiConformal.xyQ2.name)
980 def xyR2(self, beta, omega, name=NN):
981 '''Compute a Jacobi Conformal C{x} and C{y} projection.
983 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
984 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
985 @kwarg name: Optional name (C{str}).
987 @return: A L{Jacobi2Tuple}C{(x, y)}.
988 '''
989 return self.xyR2_(*(_SinCos2(beta) + _SinCos2(omega)),
990 name=name or self.xyR2.__name__)
992 def xyR2_(self, sbeta, cbeta, somega, comega, name=NN):
993 '''Compute a Jacobi Conformal C{x} and C{y} projection.
995 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
996 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
997 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
998 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
999 @kwarg name: Optional name (C{str}).
1001 @return: A L{Jacobi2Tuple}C{(x, y)}.
1002 '''
1003 return Jacobi2Tuple(self.xR_(somega, comega),
1004 self.yR_(sbeta, cbeta),
1005 name=name or self.xyR2_.__name__)
1007 @Property_RO
1008 def _yE(self):
1009 '''(INTERNAL) Get the x-elliptic function.
1010 '''
1011 kp2, k2 = self._k2_kp2 # swapped!
1012 # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc
1013 return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2)
1015 def yR(self, beta):
1016 '''Compute a Jacobi Conformal C{y} projection.
1018 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
1020 @return: The C{y} projection (C{Radians}).
1021 '''
1022 return self.yR_(*_SinCos2(beta))
1024 def yR_(self, sbeta, cbeta):
1025 '''Compute a Jacobi Conformal C{y} projection.
1027 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
1028 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
1030 @return: The C{y} projection (C{Radians}).
1031 '''
1032 s, c = self._norm2(sbeta, cbeta, self.c)
1033 return Radians(y=self._yE.fPi(s, c) * self._c2_b2)
1036class TriaxialError(_ValueError):
1037 '''Raised for L{Triaxial} issues.
1038 '''
1039 pass # ...
1042class Triaxials(_NamedEnum):
1043 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class
1044 to accommodate the L{_LazyNamedEnumItem} properties.
1045 '''
1046 def _Lazy(self, *abc, **name):
1047 '''(INTERNAL) Instantiate the C{Triaxial}.
1048 '''
1049 a, b, c = map(km2m, abc)
1050 return Triaxial(a, b, c, **name)
1052Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton
1053'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.'''
1054# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in km
1055# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf>
1056# <https://link.Springer.com/article/10.1007/s00190-022-01650-9>
1057Triaxials._assert( # a (km) b (km) c (km) planet
1058 Amalthea = _lazy('Amalthea', 125.0, 73.0, 64), # Jupiter
1059 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus
1060 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544),
1061 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn
1062 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter
1063 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter
1064 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3),
1065 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn
1066 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus
1067 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth
1068 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn
1069 WGS84_35 = _lazy('WGS84_35', *map(m2km, _WGS84_35abc)))
1071del _WGS84_35abc
1074def _getitems(items, *indices):
1075 '''(INTERNAL) Get the C{items} at the given I{indices}.
1077 @return: C{Type(items[i] for i in indices)} with
1078 C{Type = type(items)}, any C{type} having
1079 the special method C{__getitem__}.
1080 '''
1081 return type(items)(map(items.__getitem__, indices))
1084def _hartzell3d2(pov, los, Tun): # MCCABE 13 in .formy.hartzell
1085 '''(INTERNAL) Hartzell's "Satellite Lin-of-Sight Intersection ...",
1086 formula for I{un-/ordered} triaxials.
1087 '''
1088 a, b, c, T = Tun._ordered4
1090 a2 = a**2 # largest, factored out
1091 b2, p2 = (b**2, (b / a)**2) if b != a else (a2, _1_0)
1092 c2, q2 = c**2, (c / a)**2
1094 p3 = T._order3d(_otherV3d(pov=pov))
1095 u3 = T._order3d(_otherV3d(los=los)) if los else p3.negate()
1096 u3 = u3.unit() # unit vector, opposing signs
1098 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz
1099 ux, vy, wz = u3.times_(p3).xyz
1100 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz
1102 t = (p2 * c2), c2, b2
1103 m = fdot(t, u2, v2, w2) # a2 factored out
1104 if m < EPS0: # zero or near-null LOS vector
1105 raise _ValueError(_near_(_null_))
1107 r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,
1108 -w2 * y2, b2 * u2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2,
1109 -w2 * x2 * p2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True)
1110 if r > 0: # a2 factored out
1111 r = sqrt(r) * b * c # == a * a * b * c / a2
1112 elif r < 0: # LOS pointing away from or missing the triaxial
1113 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_)
1115 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out
1116 if d > 0: # POV inside or LOS missing, outside the triaxial
1117 s = fsum_(_1_0, x2 / a2, y2 / b2, z2 / c2, _N_2_0, floats=True) # like _sideOf
1118 raise _ValueError(_outside_ if s > 0 else _inside_)
1119 elif fsum_(x2, y2, z2, floats=True) < d**2: # d past triaxial's center
1120 raise _ValueError(_too_(_distant_))
1122 v = p3.minus(u3.times(d)) # Vector3d
1123 h = p3.minus(v).length # distance to triaxial
1124 return T._order3d(v, reverse=True), h
1127def hartzell4(pov, los=None, tri_biax=_WGS84, name=NN):
1128 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight
1129 from a Point-Of-View outside.
1131 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple}
1132 or L{Vector3d}).
1133 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Vector3d}) or
1134 C{None} to point to the tri-/biaxial's center.
1135 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal})
1136 or biaxial ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2},
1137 L{a_f2Tuple} or C{scalar} radius in C{meter}).
1138 @kwarg name: Optional name (C{str}).
1140 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h}
1141 the distance from B{C{pov}} to C{(x, y, z)} along B{C{los}}.
1143 @raise TriaxialError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}} is inside
1144 the tri-/biaxial or B{C{los}} points outside the
1145 tri-/biaxial or points in an opposite direction.
1147 @raise TypeError: Invalid B{C{pov}} or B{C{los}}.
1149 @see: Function L{pygeodesy.hartzell}, L{pygeodesy.tyr3d} for B{C{los}} and
1150 U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.
1151 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
1152 '''
1153 if isinstance(tri_biax, Triaxial_):
1154 T = tri_biax
1155 else:
1156 D = tri_biax if isinstance(tri_biax, Datum) else \
1157 _spherical_datum(tri_biax, name=hartzell4.__name__)
1158 E = D.ellipsoid
1159 T = Triaxial_(E.a, E.a, E.b, name=E.name)
1161 try:
1162 v, h = _hartzell3d2(pov, los, T)
1163 except Exception as x:
1164 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x)
1165 return Vector4Tuple(v.x, v.y, v.z, h, name=name or hartzell4.__name__)
1168def _normalTo4(x, y, a, b, eps=EPS): # MCCABE 14
1169 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}.
1171 @see: Function C{pygeodesy.ellipsoids._normalTo3} and I{Eberly}'s U{Distance
1172 from a Point to ... an Ellipsoid ...<https://www.GeometricTools.com/
1173 Documentation/DistancePointEllipseEllipsoid.pdf>}.
1174 '''
1175 def _root2d(r, u, v, g, eps):
1176 # robust root finder
1177 _1, __2 = _1_0, _0_5
1178 _a, _h2 = fabs, _hypot21_
1179 u *= r
1180 t0 = v - _1
1181 t1 = _0_0 if g < 0 else _h2(u, v)
1182 for i in range(1, _TRIPS):
1183 t = (t0 + t1) * __2
1184 if t in (t0, t1) or _a(t0 - t1) < eps:
1185 break
1186 g = _h2(u / (t + r), v / (t + _1))
1187 if g > 0:
1188 t0 = t
1189 elif g < 0:
1190 t1 = t
1191 else:
1192 break
1193 else: # PYCHOK no cover
1194 e = _a(t0 - t1)
1195 t = _root2d.__name__
1196 raise _ValueError(Fmt.no_convergence(e, eps), txt=t)
1197 return t, i
1199 if a < b:
1200 b, a, d, i = _normalTo4(y, x, b, a, eps=eps)
1201 return a, b, d, i
1203 if not (isfinite(a) and b > 0):
1204 raise _ValueError(a=a, b=b)
1206 i = None
1207 if y:
1208 if x:
1209 u = fabs(x / a)
1210 v = fabs(y / b)
1211 g = _hypot21_(u, v)
1212 if g:
1213 r = (a / b)**2
1214 t, i = _root2d(r, u, v, g, eps)
1215 a = x / (t / r + _1_0)
1216 b = y / (t + _1_0)
1217 d = hypot(x - a, y - b)
1218 else: # on the ellipse
1219 a, b, d = x, y, _0_0
1220 else: # x == 0
1221 if y < 0:
1222 b = -b
1223 a, d = x, fabs(y - b)
1225 else: # y == 0
1226 n = a * x
1227 d = (a + b) * (a - b)
1228 if d > fabs(n): # PYCHOK no cover
1229 r = n / d
1230 a *= r
1231 b *= sqrt(_1_0 - r**2)
1232 d = hypot(x - a, b)
1233 else:
1234 if x < 0:
1235 a = -a
1236 b, d = y, fabs(x - a)
1237 return a, b, d, i
1240def _normalTo5(x, y, z, Tun, eps=EPS): # MCCABE 24
1241 '''(INTERNAL) Nearest point on and distance to an I{un- or ordered} triaxial.
1243 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https://
1244 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}.
1245 '''
1246 def _root3d(r, s, u, v, w, g, eps):
1247 # robust root finder
1248 _1, __2 = _1_0, _0_5
1249 _a, _h2 = fabs, _hypot21_
1250 u *= r
1251 v *= s
1252 t0 = w - _1
1253 t1 = _0_0 if g < 0 else _h2(u, v, w)
1254 for i in range(1, _TRIPS):
1255 t = (t0 + t1) * __2
1256 if t in (t0, t1) or _a(t0 - t1) < eps:
1257 break
1258 g = _h2(u / (t + r), v / (t + s), w / (t + _1))
1259 if g > 0:
1260 t0 = t
1261 elif g < 0:
1262 t1 = t
1263 else:
1264 break
1265 else: # PYCHOK no cover
1266 e = _a(t0 - t1)
1267 t = _root3d.__name__
1268 raise _ValueError(Fmt.no_convergence(e, eps), txt=t)
1269 return t, i
1271 a, b, c, T = Tun._ordered4
1272 if Tun is not T: # T is ordered, Tun isn't
1273 t = T._order3(x, y, z) + (T,)
1274 a, b, c, d, i = _normalTo5(*t, eps=eps)
1275 return T._order3(a, b, c, reverse=True) + (d, i)
1277 if not (isfinite(a) and c > 0):
1278 raise _ValueError(a=a, b=b, c=c)
1280 if eps > 0:
1281 val = max(eps * 1e8, EPS)
1282 else: # no validation
1283 val, eps = 0, -eps
1285 i = None
1286 if z:
1287 if y:
1288 if x:
1289 u = fabs(x / a)
1290 v = fabs(y / b)
1291 w = fabs(z / c)
1292 g = _hypot21_(u, v, w)
1293 if g:
1294 r = T._1e2ac # (c / a)**2
1295 s = T._1e2bc # (c / b)**2
1296 t, i = _root3d(_1_0 / r, _1_0 / s, u, v, w, g, eps)
1297 a = x / (t * r + _1_0)
1298 b = y / (t * s + _1_0)
1299 c = z / (t + _1_0)
1300 d = hypot_(x - a, y - b, z - c)
1301 else: # on the ellipsoid
1302 a, b, c, d = x, y, z, _0_0
1303 else: # x == 0
1304 a = x # 0
1305 b, c, d, i = _normalTo4(y, z, b, c, eps=eps)
1306 elif x: # y == 0
1307 b = y # 0
1308 a, c, d, i = _normalTo4(x, z, a, c, eps=eps)
1309 else: # x == y == 0
1310 if z < 0:
1311 c = -c
1312 a, b, d = x, y, fabs(z - c)
1314 else: # z == 0
1315 t = False
1316 n = a * x
1317 d = T._a2c2 # (a + c) * (a - c)
1318 if d > fabs(n):
1319 u = n / d
1320 n = b * y
1321 d = T._b2c2 # (b + c) * (b - c)
1322 if d > fabs(n):
1323 v = n / d
1324 n = _hypot21_(u, v)
1325 if n < 0:
1326 a *= u
1327 b *= v
1328 c *= sqrt(-n)
1329 d = hypot_(x - a, y - b, c)
1330 t = True
1331 if not t:
1332 c = z # 0
1333 a, b, d, i = _normalTo4(x, y, a, b, eps=eps)
1335 if val > 0: # validate
1336 e = T.sideOf(a, b, c, eps=val)
1337 if e: # not near the ellipsoid's surface
1338 raise _ValueError(a=a, b=b, c=c, d=d,
1339 sideOf=e, eps=val)
1340 if d: # angle of delta and normal vector
1341 m = Vector3d(x, y, z).minus_(a, b, c)
1342 if m.euclid > val:
1343 m = m.unit()
1344 n = T.normal3d(a, b, c)
1345 e = n.dot(m) # n.negate().dot(m)
1346 if not isnear1(fabs(e), eps1=val):
1347 raise _ValueError(n=n, m=m,
1348 dot=e, eps=val)
1349 return a, b, c, d, i
1352def _otherV3d_(x_xyz, y, z, name=NN):
1353 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}.
1354 '''
1355 return Vector3d(x_xyz, y, z, name=name) if isscalar(x_xyz) else \
1356 _otherV3d(x_xyz=x_xyz)
1359def _sideOf(xyz, abc, eps=EPS): # in .formy
1360 '''(INTERNAL) Helper for C{_hartzell3d2}, M{.sideOf} and M{.reverseCartesian}.
1362 @return: M{sum((x / a)**2 for x, a in zip(xyz, abc)) - 1} or C{INT0},
1363 '''
1364 s = _hypot21_(*((x / a) for x, a in _zip(xyz, abc) if a)) # strict=True
1365 return s if fabs(s) > eps else INT0
1368def _SinCos2(x):
1369 '''Get C{sin} and C{cos} of C{x} in C{Degrees}, C{Radians} or {radians}.
1370 '''
1371 return sincos2d(x) if isinstance(x, Degrees) else (
1372 sincos2(x) if isinstance(x, Radians) else
1373 sincos2(float(x))) # assume C{radians}
1376if __name__ == '__main__':
1378 from pygeodesy import printf
1380 # __doc__ of this file, force all into registery
1381 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_)
1382 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))
1384# **) MIT License
1385#
1386# Copyright (C) 2022-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1387#
1388# Permission is hereby granted, free of charge, to any person obtaining a
1389# copy of this software and associated documentation files (the "Software"),
1390# to deal in the Software without restriction, including without limitation
1391# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1392# and/or sell copies of the Software, and to permit persons to whom the
1393# Software is furnished to do so, subject to the following conditions:
1394#
1395# The above copyright notice and this permission notice shall be included
1396# in all copies or substantial portions of the Software.
1397#
1398# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1399# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1400# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1401# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1402# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1403# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1404# OTHER DEALINGS IN THE SOFTWARE.