Coverage for pygeodesy/triaxials.py: 94%
615 statements
« prev ^ index » next coverage.py v7.6.1, created at 2025-04-25 13:15 -0400
« prev ^ index » next coverage.py v7.6.1, created at 2025-04-25 13:15 -0400
2# -*- coding: utf-8 -*-
4u'''Triaxal ellipsoid classes I{ordered} L{Triaxial} and I{unordered} L{Triaxial_} and Jacobi
5conformal projections L{JacobiConformal} and L{JacobiConformalSpherical}, transcoded from
6I{Charles Karney}'s C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/C++/doc/
7classGeographicLib_1_1JacobiConformal.html#details>} to pure Python and miscellaneous classes
8L{BetaOmega2Tuple}, L{BetaOmega3Tuple}, L{Jacobi2Tuple} and L{TriaxialError}.
10Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2024). For more information,
11see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
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>}.
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=798619018175.109985352)
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=493855762247.691833496, 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=698880863325.757080078, 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=37838824733332.21875)
27@var Triaxials.Tethys: Triaxial(name='Tethys', a=535600, b=528200, c=525800, e2ab=0.027441672, e2bc=0.009066821, e2ac=0.036259685, volume=623086233855821440, area=3528073490771.394042969, area_p=3528074261832.738769531)
28@var Triaxials.WGS84_35: Triaxial(name='WGS84_35', a=6378172, b=6378102, c=6356752.314245179, e2ab=0.00002195, e2bc=0.006683478, e2ac=0.006705281, volume=1083207319768789942272, area=510065621722018.125, area_p=510065626587483.3125)
29'''
30# make sure int/int division yields float quotient, see .basics
31from __future__ import division as _; del _ # PYCHOK semicolon
33from pygeodesy.basics import _isin, isLatLon, isscalar
34from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INT0, PI2, PI_3, PI4, \
35 _EPS2e4, _SQRT2_2, float0_, isfinite, isnear1, _over, \
36 _0_0, _0_5, _1_0, _N_1_0, _64_0, _4_0 # PYCHOK used!
37from pygeodesy.datums import Datum, _spherical_datum, _WGS84, Ellipsoid, _EWGS84, Fmt
38# from pygeodesy.ellipsoids import Ellipsoid, _EWGS84 # from .datums
39# from pygeodesy.elliptic import Elliptic # _MODS
40from pygeodesy.errors import _AssertionError, _ValueError
41from pygeodesy.fmath import Fdot, fdot, fmean_, hypot, hypot_, norm2, sqrt0
42from pygeodesy.fsums import _Fsumf_, fsumf_, fsum1f_
43from pygeodesy.interns import NN, _a_, _b_, _beta_, _c_, _distant_, _DMAIN_, \
44 _finite_, _height_, _inside_, _near_, _negative_, \
45 _not_, _NOTEQUAL_, _null_, _opposite_, _outside_, \
46 _SPACE_, _spherical_, _too_, _x_, _y_
47# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .vector3d
48from pygeodesy.named import _lazyNamedEnumItem as _lazy, _name__, _NamedEnum, \
49 _NamedEnumItem, _Pass
50from pygeodesy.namedTuples import LatLon3Tuple, _NamedTupleTo, Vector3Tuple, \
51 Vector4Tuple
52from pygeodesy.props import Property_RO, property_ROver
53# from pygeodesy.streprs import Fmt # from .datums
54from pygeodesy.units import Degrees, Float, Height_, Meter, Meter2, Meter3, \
55 Radians, Radius, Scalar_
56from pygeodesy.utily import asin1, atan2, atan2d, km2m, m2km, SinCos2, sincos2d_
57from pygeodesy.vector3d import _otherV3d, Vector3d, _ALL_LAZY, _MODS
59from math import fabs, sqrt
61__all__ = _ALL_LAZY.triaxials
62__version__ = '25.04.14'
64_not_ordered_ = _not_('ordered')
65_omega_ = 'omega'
66_TRIPS = 359 # Eberly 1074?
69class _NamedTupleToX(_NamedTupleTo): # in .testNamedTuples
70 '''(INTERNAL) Base class for L{BetaOmega2Tuple},
71 L{BetaOmega3Tuple} and L{Jacobi2Tuple}.
72 '''
73 def _toDegrees(self, name, **toDMS_kwds):
74 '''(INTERNAL) Convert C{self[0:2]} to L{Degrees} or C{toDMS}.
75 '''
76 return self._toX3U(_NamedTupleTo._Degrees3, Degrees, name, *self, **toDMS_kwds)
78 def _toRadians(self, name):
79 '''(INTERNAL) Convert C{self[0:2]} to L{Radians}.
80 '''
81 return self._toX3U(_NamedTupleTo._Radians3, Radians, name, *self)
83 def _toX3U(self, _X3, U, name, a, b, *c, **kwds):
84 a, b, s = _X3(self, a, b, **kwds)
85 if s is None or name:
86 n = self._name__(name)
87 s = self.classof(a, b, *c, name=n).reUnit(U, U).toUnits()
88 return s
91class BetaOmega2Tuple(_NamedTupleToX):
92 '''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and
93 longitude C{beta} and C{omega} both in L{Radians} (or
94 L{Degrees}).
95 '''
96 _Names_ = (_beta_, _omega_)
97 _Units_ = (_Pass, _Pass)
99 def toDegrees(self, name=NN, **toDMS_kwds):
100 '''Convert this L{BetaOmega2Tuple} to L{Degrees} or C{toDMS}.
102 @kwarg name: Optional C{B{name}=NN} (C{str}).
104 @return: L{BetaOmega2Tuple}C{(beta, omega)} with C{beta} and
105 C{omega} both in L{Degrees} or as L{toDMS} strings
106 provided some B{C{toDMS_kwds}} keyword arguments are
107 specified.
108 '''
109 return self._toDegrees(name, **toDMS_kwds)
111 def toRadians(self, **name):
112 '''Convert this L{BetaOmega2Tuple} to L{Radians}.
114 @kwarg name: Optional C{B{name}=NN} (C{str}).
116 @return: L{BetaOmega2Tuple}C{(beta, omega)} with C{beta} and C{omega}
117 both in L{Radians}.
118 '''
119 return self._toRadians(name)
122class BetaOmega3Tuple(_NamedTupleToX):
123 '''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and
124 longitude C{beta} and C{omega} both in L{Radians} (or L{Degrees})
125 and the C{height}, rather the (signed) I{distance} to the triaxial's
126 surface (measured along the radial line to the triaxial's center)
127 in C{meter}, conventionally.
128 '''
129 _Names_ = BetaOmega2Tuple._Names_ + (_height_,)
130 _Units_ = BetaOmega2Tuple._Units_ + ( Meter,)
132 def toDegrees(self, name=NN, **toDMS_kwds):
133 '''Convert this L{BetaOmega3Tuple} to L{Degrees} or C{toDMS}.
135 @kwarg name: Optional C{B{name}=NN} (C{str}).
137 @return: L{BetaOmega3Tuple}C{(beta, omega, height)} with
138 C{beta} and C{omega} both in L{Degrees} or as
139 L{toDMS} strings provided some B{C{toDMS_kwds}}
140 keyword arguments are specified.
141 '''
142 return self._toDegrees(name, **toDMS_kwds)
144 def toRadians(self, **name):
145 '''Convert this L{BetaOmega3Tuple} to L{Radians}.
147 @kwarg name: Optional C{B{name}=NN} (C{str}), overriding this name.
149 @return: L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta}
150 and C{omega} both in L{Radians}.
151 '''
152 return self._toRadians(name)
154 def to2Tuple(self, **name):
155 '''Reduce this L{BetaOmega3Tuple} to a L{BetaOmega2Tuple}.
157 @kwarg name: Optional C{B{name}=NN} (C{str}), overriding this name.
158 '''
159 return BetaOmega2Tuple(*self[:2], name=self._name__(name))
162class Jacobi2Tuple(_NamedTupleToX):
163 '''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y}
164 projection, both in L{Radians} (or L{Degrees}).
165 '''
166 _Names_ = (_x_, _y_)
167 _Units_ = (_Pass, _Pass)
169 def toDegrees(self, name=NN, **toDMS_kwds):
170 '''Convert this L{Jacobi2Tuple} to L{Degrees} or C{toDMS}.
172 @kwarg name: Optional C{B{name}=NN} (C{str}).
174 @return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y} both
175 in L{Degrees} or as L{toDMS} strings provided some
176 B{C{toDMS_kwds}} keyword arguments are specified.
177 '''
178 return self._toDegrees(name, **toDMS_kwds)
180 def toRadians(self, **name):
181 '''Convert this L{Jacobi2Tuple} to L{Radians}.
183 @kwarg name: Optional C{B{name}=NN} (C{str}).
185 @return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y} both in L{Radians}.
186 '''
187 return self._toRadians(name)
190class Triaxial_(_NamedEnumItem):
191 '''I{Unordered} triaxial ellipsoid and base class.
193 Triaxial ellipsoids with right-handed semi-axes C{a}, C{b} and C{c}, oriented
194 such that the large principal ellipse C{ab} is the equator I{Z}=0, I{beta}=0,
195 while the small principal ellipse C{ac} is the prime meridian, plane I{Y}=0,
196 I{omega}=0.
198 The four umbilic points, C{abs}(I{omega}) = C{abs}(I{beta}) = C{PI/2}, lie on
199 the middle principal ellipse C{bc} in plane I{X}=0, I{omega}=C{PI/2}.
201 @note: I{Geodetic} C{lat}- and C{lon}gitudes are in C{degrees}, I{geodetic}
202 C{phi} and C{lam}bda are in C{radians}, but I{ellipsoidal} lat- and
203 longitude C{beta} and C{omega} are in L{Radians} by default (or in
204 L{Degrees} if converted).
205 '''
206 _ijk = _kji = None
207 _unordered = True
209 def __init__(self, a_triaxial, b=None, c=None, **name):
210 '''New I{unordered} L{Triaxial_}.
212 @arg a_triaxial: Large, C{X} semi-axis (C{scalar}, conventionally in
213 C{meter}) or an other L{Triaxial} or L{Triaxial_} instance.
214 @kwarg b: Middle, C{Y} semi-axis (C{meter}, same units as B{C{a}}), required
215 if C{B{a_triaxial} is scalar}, ignored otherwise.
216 @kwarg c: Small, C{Z} semi-axis (C{meter}, like B{C{b}}).
217 @kwarg name: Optional C{B{name}=NN} (C{str}).
219 @raise TriaxialError: Invalid semi-axis or -axes.
220 '''
221 try:
222 try:
223 a = a_triaxial
224 t = a._abc3
225 except AttributeError:
226 t = Radius(a=a), Radius(b=b), Radius(c=c)
227 except (TypeError, ValueError) as x:
228 raise TriaxialError(a=a, b=b, c=c, cause=x)
229 if name:
230 self.name = name
232 a, b, c = self._abc3 = t
233 if self._unordered: # == not isinstance(self, Triaxial)
234 s, _, t = sorted(t)
235 if not (isfinite(t) and s > 0):
236 raise TriaxialError(a=a, b=b, c=c) # txt=_invalid_
237 elif not (isfinite(a) and a >= b >= c > 0):
238 raise TriaxialError(a=a, b=b, c=c, txt=_not_ordered_)
239 elif not (a > c and self._a2c2 > 0 and self.e2ac > 0):
240 raise TriaxialError(a=a, c=c, e2ac=self.e2ac, txt=_spherical_)
242 def __str__(self):
243 return self.toStr()
245 @Property_RO
246 def a(self):
247 '''Get the C{largest, x} semi-axis (C{meter}, conventionally).
248 '''
249 a, _, _ = self._abc3
250 return a
252 @Property_RO
253 def _a2b2(self):
254 '''(INTERNAL) Get C{a**2 - b**2} == E_sub_e**2.
255 '''
256 a, b, _ = self._abc3
257 return ((a - b) * (a + b)) if a != b else _0_0
259 @Property_RO
260 def _a2_b2(self):
261 '''(INTERNAL) Get C{(a / b)**2}.
262 '''
263 a, b, _ = self._abc3
264 return (a / b)**2 if a != b else _1_0
266 @Property_RO
267 def _a2c2(self):
268 '''(INTERNAL) Get C{a**2 - c**2} == E_sub_x**2.
269 '''
270 a, _, c = self._abc3
271 return ((a - c) * (a + c)) if a != c else _0_0
273 @Property_RO
274 def area(self):
275 '''Get the surface area (C{meter} I{squared}).
276 '''
277 c, b, a = sorted(self._abc3)
278 if a > c:
279 a = Triaxial(a, b, c).area if a > b else \
280 Ellipsoid(a, b=c).areax # a == b
281 else: # a == c == b
282 a = Meter2(area=a**2 * PI4)
283 return a
285 def area_p(self, p=1.6075):
286 '''I{Approximate} the surface area (C{meter} I{squared}).
288 @kwarg p: Exponent (C{scalar} > 0), 1.6 for near-spherical or 1.5849625007
289 for "near-flat" triaxials.
291 @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Approximate_formula>}.
292 '''
293 a, b, c = self._abc3
294 if a == b == c:
295 a *= a
296 else:
297 _p = pow
298 a = _p(fmean_(_p(a * b, p), _p(a * c, p), _p(b * c, p)), _1_0 / p)
299 return Meter2(area_p=a * PI4)
301 @Property_RO
302 def b(self):
303 '''Get the C{middle, y} semi-axis (C{meter}, same units as B{C{a}}).
304 '''
305 _, b, _ = self._abc3
306 return b
308 @Property_RO
309 def _b2c2(self):
310 '''(INTERNAL) Get C{b**2 - c**2} == E_sub_y**2.
311 '''
312 _, b, c = self._abc3
313 return ((b - c) * (b + c)) if b != c else _0_0
315 @Property_RO
316 def c(self):
317 '''Get the C{smallest, z} semi-axis (C{meter}, same units as B{C{a}}).
318 '''
319 _, _, c = self._abc3
320 return c
322 @Property_RO
323 def _c2_b2(self):
324 '''(INTERNAL) Get C{(c / b)**2}.
325 '''
326 _, b, c = self._abc3
327 return (c / b)**2 if b != c else _1_0
329 @Property_RO
330 def e2ab(self):
331 '''Get the C{ab} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (b/a)**2}.
332 '''
333 return Float(e2ab=(_1_0 - self._1e2ab) or _0_0)
335 @Property_RO
336 def _1e2ab(self):
337 '''(INTERNAL) Get C{1 - e2ab} == C{(b/a)**2}.
338 '''
339 a, b, _ = self._abc3
340 return (b / a)**2 if a != b else _1_0
342 @Property_RO
343 def e2ac(self):
344 '''Get the C{ac} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/a)**2}.
345 '''
346 return Float(e2ac=(_1_0 - self._1e2ac) or _0_0)
348 @Property_RO
349 def _1e2ac(self):
350 '''(INTERNAL) Get C{1 - e2ac} == C{(c/a)**2}.
351 '''
352 a, _, c = self._abc3
353 return (c / a)**2 if a != c else _1_0
355 @Property_RO
356 def e2bc(self):
357 '''Get the C{bc} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/b)**2}.
358 '''
359 return Float(e2bc=(_1_0 - self._1e2bc) or _0_0)
361 _1e2bc = _c2_b2 # C{1 - e2bc} == C{(c/b)**2}
363 @property_ROver
364 def _Elliptic(self):
365 '''(INTERNAL) Get class L{Elliptic}, I{once}.
366 '''
367 return _MODS.elliptic.Elliptic # overwrite property_ROver
369 def hartzell4(self, pov, los=False, **name):
370 '''Compute the intersection of this triaxial's surface with a Line-Of-Sight
371 from a Point-Of-View in space.
373 @see: Function L{hartzell4<triaxials.hartzell4>} for further details.
374 '''
375 return hartzell4(pov, los=los, tri_biax=self, **name)
377 def height4(self, x_xyz, y=None, z=None, normal=True, eps=EPS, **name):
378 '''Compute the projection on and the height above or below this triaxial's surface.
380 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, L{Ecef9Tuple},
381 L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
382 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}, ignored
383 otherwise.
384 @kwarg z: Z component (C{scalar}), like B{C{y}}.
385 @kwarg normal: If C{True}, the projection is the I{perpendicular, plumb} to the
386 triaxial's surface, otherwise the C{radial} line to the center of
387 this triaxial (C{bool}).
388 @kwarg eps: Tolerance for root finding and validation (C{scalar}), use a negative
389 value to skip validation.
390 @kwarg name: Optional C{B{name}="heigh4"} (C{str}).
392 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x}, C{y}
393 and C{z} of the projection on or the intersection with and with the height
394 C{h} above or below the triaxial's surface in C{meter}, conventionally.
396 @raise TriaxialError: Non-cartesian B{C{xyz}}, invalid B{C{eps}}, no convergence in
397 root finding or validation failed.
399 @see: Methods L{Triaxial.normal3d} and L{Ellipsoid.height4}, I{Eberly}'s U{Distance from a Point to ...
400 <https://www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>} and I{Bektas}'
401 U{Shortest Distance from a Point to Triaxial Ellipsoid<https://www.ResearchGate.net/publication/
402 272149005_SHORTEST_DISTANCE_FROM_A_POINT_TO_TRIAXIAL_ELLIPSOID>}.
403 '''
404 v, r = _otherV3d_(x_xyz, y, z), self.isSpherical
406 i, h = None, v.length
407 if h < EPS0: # EPS
408 x = y = z = _0_0
409 h -= min(self._abc3) # nearest
410 elif r: # .isSpherical
411 x, y, z = v.times(r / h).xyz3
412 h -= r
413 else:
414 x, y, z = v.xyz3
415 try:
416 if normal: # plumb to surface
417 x, y, z, h, i = _plumbTo5(x, y, z, self, eps=eps)
418 else: # radial to center
419 x, y, z = self._radialTo3(z, hypot(x, y), y, x)
420 h = v.minus_(x, y, z).length
421 except Exception as e:
422 raise TriaxialError(x=x, y=y, z=z, cause=e)
423 if h > 0 and self.sideOf(v, eps=EPS0) < 0:
424 h = -h # inside
425 n = _name__(name, name__=self.height4) # typename
426 return Vector4Tuple(x, y, z, h, iteration=i, name=n)
428 @Property_RO
429 def isOrdered(self):
430 '''Is this triaxial I{ordered} and I{not spherical} (C{bool})?
431 '''
432 a, b, c = self._abc3
433 return bool(a >= b > c) # b > c!
435 @Property_RO
436 def isSpherical(self):
437 '''Is this triaxial I{spherical} (C{Radius} or INT0)?
438 '''
439 a, b, c = self._abc3
440 return a if a == b == c else INT0
442 def _norm2(self, s, c, *a):
443 '''(INTERNAL) Normalize C{s} and C{c} iff not already.
444 '''
445 if fabs(_hypot2_1(s, c)) > EPS02:
446 s, c = norm2(s, c)
447 if a:
448 s, c = norm2(s * self.b, c * a[0])
449 return float0_(s, c)
451 def normal3d(self, x_xyz, y=None, z=None, length=_1_0):
452 '''Get a 3-D vector at a cartesian I{on and perpendicular to} this triaxial's surface.
454 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
455 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
456 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}, ignored
457 otherwise.
458 @kwarg z: Z component (C{scalar}), like B{C{y}}.
459 @kwarg length: Optional, signed length and in-/outward direction (C{scalar}).
461 @return: A C{Vector3d(x_, y_, z_)} normalized to B{C{length}}, pointing in- or
462 outward for neg- respectively positive B{C{length}}.
464 @raise TriaxialError: Zero length cartesian or vector.
466 @note: Cartesian C{(B{x}, B{y}, B{z})} I{must be on} this triaxial's surface, use
467 method L{Triaxial.sideOf} to validate.
469 @see: Methods L{Triaxial.height4} and L{Triaxial.sideOf}.
470 '''
471 # n = 2 * (x / a2, y / b2, z / c2)
472 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered
473 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2
474 # == unit(x, y / _1e2ab, z / _1e2ac).times(length)
475 x, y, z = _otherV3d_(x_xyz, y, z).xyz3
476 n = Vector3d(x, y / self._1e2ab,
477 z / self._1e2ac, name__=self.normal3d)
478 u = n.length
479 if u < EPS0:
480 raise TriaxialError(x=x_xyz, y=y, z=z, txt=_null_)
481 return n.times(length / u)
483 def _order3(self, *abc, **reverse): # reverse=False
484 '''(INTERNAL) Un-/Order C{a}, C{b} and C{c}.
486 @return: 3-Tuple C{(a, b, c)} ordered by or un-ordered
487 (reverse-ordered) C{ijk} if C{B{reverse}=True}.
488 '''
489 ijk = self._order_ijk(**reverse)
490 return _getitems(abc, *ijk) if ijk else abc
492 def _order3d(self, v, **reverse): # reverse=False
493 '''(INTERNAL) Un-/Order a C{Vector3d}.
495 @return: Vector3d(x, y, z) un-/ordered.
496 '''
497 ijk = self._order_ijk(**reverse)
498 return v.classof(*_getitems(v.xyz3, *ijk)) if ijk else v
500 @Property_RO
501 def _ordered4(self):
502 '''(INTERNAL) Helper for C{_hartzell3} and C{_plumbTo5}.
503 '''
504 def _order2(reverse, a, b, c):
505 '''(INTERNAL) Un-Order C{a}, C{b} and C{c}.
507 @return: 2-Tuple C{((a, b, c), ijk)} with C{a} >= C{b} >= C{c}
508 and C{ijk} a 3-tuple with the initial indices.
509 '''
510 i, j, k = range(3)
511 if a < b:
512 a, b, i, j = b, a, j, i
513 if a < c:
514 a, c, i, k = c, a, k, i
515 if b < c:
516 b, c, j, k = c, b, k, j
517 # reverse (k, j, i) since (a, b, c) is reversed-sorted
518 ijk = (k, j, i) if reverse else (None if i < j < k else (i, j, k))
519 return (a, b, c), ijk
521 abc, T = self._abc3, self
522 if not self.isOrdered:
523 abc, ijk = _order2(False, *abc)
524 if ijk:
525 _, kji = _order2(True, *ijk)
526 T = Triaxial_(*abc)
527 T._ijk, T._kji = ijk, kji
528 return abc + (T,)
530 def _order_ijk(self, reverse=False):
531 '''(INTERNAL) Get the un-/order indices.
532 '''
533 return self._kji if reverse else self._ijk
535 def _radialTo3(self, sbeta, cbeta, somega, comega):
536 '''(INTERNAL) I{Unordered} helper for C{.height4}.
537 '''
538 def _rphi(a, b, sphi, cphi):
539 # <https://WikiPedia.org/wiki/Ellipse#Polar_form_relative_to_focus>
540 # polar form: radius(phi) = a * b / hypot(a * sphi, b * cphi)
541 return (b / hypot(sphi, b / a * cphi)) if a > b else (
542 (a / hypot(cphi, a / b * sphi)) if a < b else a)
544 sa, ca = self._norm2(sbeta, cbeta)
545 sb, cb = self._norm2(somega, comega)
547 a, b, c = self._abc3
548 if a != b:
549 a = _rphi(a, b, sb, cb)
550 if a != c:
551 c = _rphi(a, c, sa, ca)
552 t = c * ca
553 return (t * cb), (t * sb), (c * sa)
555 def sideOf(self, x_xyz, y=None, z=None, eps=EPS4):
556 '''Is a cartesian on, above or below the surface of this triaxial?
558 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
559 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
560 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
561 ignored otherwise.
562 @kwarg z: Z component (C{scalar}), like B{C{y}}.
563 @kwarg eps: On-surface tolerance (C{scalar}, distance I{squared}).
565 @return: C{INT0} if C{(B{x}, B{y}, B{z})} is near this triaxial's surface
566 within tolerance B{C{eps}}, otherwise the signed, radial distance
567 I{squared} (C{float}), negative for in- or positive for outside
568 this triaxial.
570 @see: Methods L{Triaxial.height4} and L{Triaxial.normal3d}.
571 '''
572 v = _otherV3d_(x_xyz, y, z)
573 s = fsumf_(_N_1_0, *map(_over02, v.xyz3, self._abc3))
574 return INT0 if fabs(s) < eps else s
576 def toEllipsoid(self, **name):
577 '''Convert this triaxial to an L{Ellipsoid}, provided 2 axes match.
579 @kwarg name: Optional C{B{name}=NN} (C{str}).
581 @return: An L{Ellipsoid} with north along this C{Z} axis if C{a == b},
582 this C{Y} axis if C{a == c} or this C{X} axis if C{b == c}.
584 @raise TriaxialError: This C{a != b}, C{b != c} and C{c != a}.
586 @see: Method L{Ellipsoid.toTriaxial}.
587 '''
588 a, b, c = self._abc3
589 if a == b:
590 b = c # N = c-Z
591 elif b == c: # N = a-X
592 a, b = b, a
593 elif a != c: # N = b-Y
594 t = _SPACE_(_a_, _NOTEQUAL_, _b_, _NOTEQUAL_, _c_)
595 raise TriaxialError(a=a, b=b, c=c, txt=t)
596 return Ellipsoid(a, b=b, name=self._name__(name))
598 def toStr(self, prec=9, **name): # PYCHOK signature
599 '''Return this C{Triaxial} as a string.
601 @kwarg prec: Precision, number of decimal digits (0..9).
602 @kwarg name: Optional name (C{str}), to override or C{None}
603 to exclude this triaxial's name.
605 @return: This C{Triaxial}'s attributes (C{str}).
606 '''
607 T = Triaxial_
608 t = T.a,
609 J = JacobiConformalSpherical
610 t += (J.ab, J.bc) if isinstance(self, J) else (T.b, T.c)
611 t += T.e2ab, T.e2bc, T.e2ac
612 J = JacobiConformal
613 if isinstance(self, J):
614 t += J.xyQ2,
615 t += T.volume, T.area
616 return self._instr(area_p=self.area_p(), prec=prec, props=t, **name)
618 @Property_RO
619 def unOrdered(self):
620 '''Is this triaxial I{un-ordered} and I{not spherical} (C{bool})?
621 '''
622 return not (self.isOrdered or bool(self.isSpherical))
624 @Property_RO
625 def volume(self):
626 '''Get the volume (C{meter**3}), M{4 / 3 * PI * a * b * c}.
627 '''
628 a, b, c = self._abc3
629 return Meter3(volume=a * b * c * PI_3 * _4_0)
632class Triaxial(Triaxial_):
633 '''I{Ordered} triaxial ellipsoid.
635 @see: L{Triaxial_} for more information.
636 '''
637 _unordered = False
639 def __init__(self, a_triaxial, b=None, c=None, **name):
640 '''New I{ordered} L{Triaxial}.
642 @arg a_triaxial: Largest semi-axis (C{scalar}, conventionally in C{meter})
643 or an other L{Triaxial} or L{Triaxial_} instance.
644 @kwarg b: Middle semi-axis (C{meter}, same units as B{C{a}}), required
645 if C{B{a_triaxial} is scalar}, ignored otherwise.
646 @kwarg c: Smallest semi-axis (C{meter}, like B{C{b}}).
647 @kwarg name: Optional C{B{name}=NN} (C{str}).
649 @note: The semi-axes must be ordered as C{B{a} >= B{b} >= B{c} > 0} and
650 must be ellipsoidal, C{B{a} > B{c}}.
652 @raise TriaxialError: Semi-axes unordered, spherical or invalid.
653 '''
654 Triaxial_.__init__(self, a_triaxial, b=b, c=c, **name)
656 @Property_RO
657 def _a2b2_a2c2(self):
658 '''@see: Methods C{.forwardBetaOmega} and C{._k2_kp2}.
659 '''
660 return self._a2b2 / self._a2c2
662 @Property_RO
663 def area(self):
664 '''Get the surface area (C{meter} I{squared}).
666 @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Surface_area>}.
667 '''
668 a, b, c = self._abc3
669 if a != b:
670 kp2, k2 = self._k2_kp2 # swapped!
671 aE = self._Elliptic(k2, _0_0, kp2, _1_0)
672 c2 = self._1e2ac # cos(phi)**2 = (c/a)**2
673 s = sqrt(self.e2ac) # sin(phi)**2 = 1 - c2
674 r = asin1(s) # phi = atan2(sqrt(c2), s)
675 b *= fsum1f_(aE.fE(r) * s, (c / a) * (c / b),
676 aE.fF(r) * c2 / s)
677 a = Meter2(area=a * b * PI2)
678 else: # a == b > c
679 a = Ellipsoid(a, b=c).areax
680 return a
682 def forwardBetaOmega(self, beta, omega, height=0, **name):
683 '''Convert I{ellipsoidal} lat- C{beta}, longitude C{omega} and C{height}
684 to cartesian.
686 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
687 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
688 @arg height: Height above or below the ellipsoid's surface (C{meter}, same
689 units as this triaxial's C{a}, C{b} and C{c} semi-axes).
690 @kwarg name: Optional C{B{name}=NN} (C{str}).
692 @return: A L{Vector3Tuple}C{(x, y, z)}.
694 @see: Method L{Triaxial.reverseBetaOmega} and U{expressions (23-25)<https://
695 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
696 '''
697 if height:
698 z = self._Height(height) + self.c
699 if z > 0:
700 z2 = z**2
701 x = z * _sqrt0(_1_0 + self._a2c2 / z2)
702 y = z * _sqrt0(_1_0 + self._b2c2 / z2)
703 else:
704 x = y = z = _0_0
705 else:
706 x, y, z = self._abc3
707 if z: # and x and y:
708 sa, ca = SinCos2(beta)
709 sb, cb = SinCos2(omega)
711 r = self._a2b2_a2c2
712 x *= cb * _sqrt0(ca**2 + sa**2 * r)
713 y *= ca * sb
714 z *= sa * _sqrt0(_1_0 - cb**2 * r)
715 return Vector3Tuple(x, y, z, **name)
717 def forwardBetaOmega_(self, sbeta, cbeta, somega, comega, **name):
718 '''Convert I{ellipsoidal} lat- and longitude C{beta} and C{omega}
719 to cartesian coordinates I{on the triaxial's surface}.
721 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
722 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
723 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
724 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
725 @kwarg name: Optional C{B{name}=NN} (C{str}).
727 @return: A L{Vector3Tuple}C{(x, y, z)} on the surface.
729 @raise TriaxialError: This triaxial is near-spherical.
731 @see: Method L{Triaxial.reverseBetaOmega}, U{Triaxial ellipsoid coordinate
732 system<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid#
733 Triaxial_ellipsoid_coordinate_system>} and U{expressions (23-25)<https://
734 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
735 '''
736 t = self._radialTo3(sbeta, cbeta, somega, comega)
737 return Vector3Tuple(*t, **name)
739 def forwardCartesian(self, x_xyz, y=None, z=None, **normal_eps_name):
740 '''Project a cartesian on this triaxial.
742 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
743 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
744 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
745 ignored otherwise.
746 @kwarg z: Z component (C{scalar}), like B{C{y}}.
747 @kwarg normal_eps_name: Optional keyword arguments C{B{normal}=True},
748 C{B{eps}=EPS} and overriding C{B{name}="height4"} (C{str}),
749 see method L{Triaxial.height4}.
751 @see: Method L{Triaxial.height4} for further information and method
752 L{Triaxial.reverseCartesian} to reverse the projection.
753 '''
754 return self.height4(x_xyz, y, z, **normal_eps_name)
756 def forwardLatLon(self, lat, lon, height=0, **name):
757 '''Convert I{geodetic} lat-, longitude and height to cartesian.
759 @arg lat: Geodetic latitude (C{degrees}).
760 @arg lon: Geodetic longitude (C{degrees}).
761 @arg height: Height above the ellipsoid (C{meter}, same units
762 as this triaxial's C{a}, C{b} and C{c} axes).
763 @kwarg name: Optional C{B{name}=NN} (C{str}).
765 @return: A L{Vector3Tuple}C{(x, y, z)}.
767 @see: Method L{Triaxial.reverseLatLon} and U{expressions (9-11)<https://
768 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
769 '''
770 return self._forwardLatLon3(height, name, *sincos2d_(lat, lon))
772 def forwardLatLon_(self, slat, clat, slon, clon, height=0, **name):
773 '''Convert I{geodetic} lat-, longitude and height to cartesian.
775 @arg slat: Geodetic latitude C{sin(lat)} (C{scalar}).
776 @arg clat: Geodetic latitude C{cos(lat)} (C{scalar}).
777 @arg slon: Geodetic longitude C{sin(lon)} (C{scalar}).
778 @arg clon: Geodetic longitude C{cos(lon)} (C{scalar}).
779 @arg height: Height above the ellipsoid (C{meter}, same units
780 as this triaxial's axes C{a}, C{b} and C{c}).
781 @kwarg name: Optional C{B{name}=NN} (C{str}).
783 @return: A L{Vector3Tuple}C{(x, y, z)}.
785 @see: Method L{Triaxial.reverseLatLon} and U{expressions (9-11)<https://
786 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
787 '''
788 sa, ca = self._norm2(slat, clat)
789 sb, cb = self._norm2(slon, clon)
790 return self._forwardLatLon3(height, name, sa, ca, sb, cb)
792 def _forwardLatLon3(self, height, name, sa, ca, sb, cb): # name always **name
793 '''(INTERNAL) Helper for C{.forwardLatLon} and C{.forwardLatLon_}.
794 '''
795 ca_x_sb = ca * sb
796 h = self._Height(height)
797 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2
798 t = fsumf_(_1_0, -self.e2ac * sa**2, -self.e2ab * ca_x_sb**2)
799 n = self.a / _sqrt0(t) # prime vertical
800 x = (h + n) * ca * cb
801 y = (h + n * self._1e2ab) * ca_x_sb
802 z = (h + n * self._1e2ac) * sa
803 return Vector3Tuple(x, y, z, **name)
805 def _Height(self, height):
806 '''(INTERNAL) Validate a C{height}.
807 '''
808 return Height_(height=height, low=-self.c, Error=TriaxialError)
810 @Property_RO
811 def _k2_kp2(self):
812 '''(INTERNAL) Get C{k2} and C{kp2} for C{._xE}, C{._yE} and C{.area}.
813 '''
814 # k2 = a2b2 / a2c2 * c2_b2
815 # kp2 = b2c2 / a2c2 * a2_b2
816 # b2 = b**2
817 # xE = Elliptic(k2, -a2b2 / b2, kp2, a2_b2)
818 # yE = Elliptic(kp2, +b2c2 / b2, k2, c2_b2)
819 # aE = Elliptic(kp2, 0, k2, 1)
820 return (self._a2b2_a2c2 * self._c2_b2,
821 self._b2c2 / self._a2c2 * self._a2_b2)
823 def _radialTo3(self, sbeta, cbeta, somega, comega):
824 '''(INTERNAL) Convert I{ellipsoidal} lat- and longitude C{beta} and
825 C{omega} to cartesian coordinates I{on the triaxial's surface},
826 also I{ordered} helper for C{.height4}.
827 '''
828 sa, ca = self._norm2(sbeta, cbeta)
829 sb, cb = self._norm2(somega, comega)
831 b2_a2 = self._1e2ab # == (b/a)**2
832 c2_a2 = -self._1e2ac # == -(c/a)**2
833 a2c2_a2 = self. e2ac # (a**2 - c**2) / a**2 == 1 - (c/a)**2
835 x2 = _Fsumf_(_1_0, -b2_a2 * sa**2, c2_a2 * ca**2).fover(a2c2_a2)
836 z2 = _Fsumf_(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2)
838 x, y, z = self._abc3
839 x *= cb * _sqrt0(x2)
840 y *= ca * sb
841 z *= sa * _sqrt0(z2)
842 return x, y, z
844 def reverseBetaOmega(self, x_xyz, y=None, z=None, **name):
845 '''Convert cartesian to I{ellipsoidal} lat- and longitude, C{beta}, C{omega}
846 and height.
848 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
849 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
850 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
851 ignored otherwise.
852 @kwarg z: Z component (C{scalar}), like B{C{y}}.
853 @kwarg name: Optional C{B{name}=NN} (C{str}).
855 @return: A L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and
856 C{omega} in L{Radians} and (radial) C{height} in C{meter}, same
857 units as this triaxial's axes.
859 @see: Methods L{Triaxial.forwardBetaOmega} and L{Triaxial.forwardBetaOmega_}
860 and U{expressions (21-22)<https://www.Topo.Auth.GR/wp-content/uploads/
861 sites/111/2021/12/09_Panou.pdf>}.
862 '''
863 v = _otherV3d_(x_xyz, y, z)
864 a, b, h = self._reverseLatLon3(v, atan2, v, self.forwardBetaOmega_)
865 return BetaOmega3Tuple(Radians(beta=a), Radians(omega=b), h, **name)
867 def reverseCartesian(self, x_xyz, y=None, z=None, h=0, normal=True, eps=_EPS2e4, **name):
868 '''Unproject" a cartesian I{on} this triaxial's surface to a cartesion I{off}
869 this triaxial's surface.
871 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
872 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
873 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
874 ignored otherwise.
875 @kwarg z: Z component (C{scalar}), like B{C{y}}.
876 @arg h: Height above or below this triaxial's surface (C{meter}, same units
877 as this triaxial's axes).
878 @kwarg normal: If C{True}, the height is C{normal} to the surface, otherwise
879 C{radially} to the center of this triaxial (C{bool}).
880 @kwarg eps: Tolerance for on-surface test (C{scalar}), see method L{Triaxial.sideOf}.
881 @kwarg name: Optional C{B{name}=NN} (C{str}).
883 @return: A L{Vector3Tuple}C{(x, y, z)}.
885 @raise TrialError: Cartesian C{(x, y, z)} not on this triaxial's surface.
887 @see: Methods L{Triaxial.forwardCartesian} and L{Triaxial.height4}.
888 '''
889 v = _otherV3d_(x_xyz, y, z, **name)
890 s = self.sideOf(v.xyz, eps=eps)
891 if s: # PYCHOK no cover
892 t = _SPACE_((_inside_ if s < 0 else _outside_), self.toRepr())
893 raise TriaxialError(eps=eps, sideOf=s, x=v.x, y=v.y, z=v.z, txt=t)
895 if h:
896 if normal:
897 v = v.plus(self.normal3d(*v.xyz, length=h))
898 elif v.length > EPS0:
899 v = v.times(_1_0 + (h / v.length))
900 return v.xyz # Vector3Tuple
902 def reverseLatLon(self, x_xyz, y=None, z=None, **name):
903 '''Convert cartesian to I{geodetic} lat-, longitude and height.
905 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
906 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
907 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
908 ignored otherwise.
909 @kwarg z: Z component (C{scalar}), like B{C{y}}.
910 @kwarg name: Optional C{B{name}=NN} (C{str}).
912 @return: A L{LatLon3Tuple}C{(lat, lon, height)} with C{lat} and C{lon}
913 in C{degrees} and (radial) C{height} in C{meter}, same units
914 as this triaxial's axes.
916 @see: Methods L{Triaxial.forwardLatLon} and L{Triaxial.forwardLatLon_}
917 and U{expressions (4-5)<https://www.Topo.Auth.GR/wp-content/uploads/
918 sites/111/2021/12/09_Panou.pdf>}.
919 '''
920 v = _otherV3d_(x_xyz, y, z)
921 s = v.times_(self._1e2ac, # == 1 - e_sub_x**2
922 self._1e2bc, # == 1 - e_sub_y**2
923 _1_0)
924 a, b, h = self._reverseLatLon3(s, atan2d, v, self.forwardLatLon_)
925 return LatLon3Tuple(Degrees(lat=a), Degrees(lon=b), h, **name)
927 def _reverseLatLon3(self, s, atan2_, v, forward_):
928 '''(INTERNAL) Helper for C{.reverseBetOmg} and C{.reverseLatLon}.
929 '''
930 x, y, z = s.xyz3
931 d = hypot( x, y)
932 a = atan2_(z, d)
933 b = atan2_(y, x)
934 h = v.minus_(*forward_(z, d, y, x)).length
935 return a, b, h
938class JacobiConformal(Triaxial):
939 '''This is a conformal projection of a triaxial ellipsoid to a plane in which the
940 C{X} and C{Y} grid lines are straight.
942 Ellipsoidal coordinates I{beta} and I{omega} are converted to Jacobi Conformal
943 I{y} respectively I{x} separately. Jacobi's coordinates have been multiplied
944 by C{sqrt(B{a}**2 - B{c}**2) / (2 * B{b})} so that the customary results are
945 returned in the case of an ellipsoid of revolution.
947 Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2014-2024) and
948 licensed under the MIT/X11 License.
950 @note: This constructor can I{not be used to specify a sphere}, see alternate
951 L{JacobiConformalSpherical}.
953 @see: L{Triaxial}, C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/
954 C++/doc/classGeographicLib_1_1JacobiConformal.html#details>}, U{Jacobi's conformal
955 projection<https://GeographicLib.SourceForge.io/C++/doc/jacobi.html>} and Jacobi,
956 C. G. J. I{U{Vorlesungen über Dynamik<https://Books.Google.com/books?
957 id=ryEOAAAAQAAJ&pg=PA212>}}, page 212ff.
958 '''
960 @Property_RO
961 def _xE(self):
962 '''(INTERNAL) Get the x-elliptic function.
963 '''
964 k2, kp2 = self._k2_kp2
965 # -a2b2 / b2 == (b2 - a2) / b2 == 1 - a2 / b2 == 1 - a2_b2
966 return self._Elliptic(k2, _1_0 - self._a2_b2, kp2, self._a2_b2)
968 def xR(self, omega):
969 '''Compute a Jacobi Conformal C{x} projection.
971 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
973 @return: The C{x} projection (L{Radians}).
974 '''
975 return self.xR_(*SinCos2(omega))
977 def xR_(self, somega, comega):
978 '''Compute a Jacobi Conformal C{x} projection.
980 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
981 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
983 @return: The C{x} projection (L{Radians}).
984 '''
985 s, c = self._norm2(somega, comega, self.a)
986 return Radians(x=self._xE.fPi(s, c) * self._a2_b2)
988 @Property_RO
989 def xyQ2(self):
990 '''Get the Jacobi Conformal quadrant size (L{Jacobi2Tuple}C{(x, y)}).
991 '''
992 return Jacobi2Tuple(Radians(x=self._a2_b2 * self._xE.cPi),
993 Radians(y=self._c2_b2 * self._yE.cPi),
994 name=JacobiConformal.xyQ2.name)
996 def xyR2(self, beta, omega, **name):
997 '''Compute a Jacobi Conformal C{x} and C{y} projection.
999 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
1000 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
1001 @kwarg name: Optional name (C{str}), overriding C{B{name}="xyR2"}.
1003 @return: A L{Jacobi2Tuple}C{(x, y)}.
1004 '''
1005 return self.xyR2_(*(SinCos2(beta) + SinCos2(omega)),
1006 name=_name__(name, name__=self.xyR2))
1008 def xyR2_(self, sbeta, cbeta, somega, comega, **name):
1009 '''Compute a Jacobi Conformal C{x} and C{y} projection.
1011 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
1012 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
1013 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
1014 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
1015 @kwarg name: Optional name (C{str}), overriding C{B{name}="xyR2_"}.
1017 @return: A L{Jacobi2Tuple}C{(x, y)}.
1018 '''
1019 return Jacobi2Tuple(self.xR_(somega, comega),
1020 self.yR_(sbeta, cbeta),
1021 name=_name__(name, name__=self.xyR2_))
1023 @Property_RO
1024 def _yE(self):
1025 '''(INTERNAL) Get the x-elliptic function.
1026 '''
1027 kp2, k2 = self._k2_kp2 # swapped!
1028 # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc
1029 return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2)
1031 def yR(self, beta):
1032 '''Compute a Jacobi Conformal C{y} projection.
1034 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
1036 @return: The C{y} projection (L{Radians}).
1037 '''
1038 return self.yR_(*SinCos2(beta))
1040 def yR_(self, sbeta, cbeta):
1041 '''Compute a Jacobi Conformal C{y} projection.
1043 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
1044 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
1046 @return: The C{y} projection (L{Radians}).
1047 '''
1048 s, c = self._norm2(sbeta, cbeta, self.c)
1049 return Radians(y=self._yE.fPi(s, c) * self._c2_b2)
1052class JacobiConformalSpherical(JacobiConformal):
1053 '''An alternate, I{spherical} L{JacobiConformal} projection.
1055 @see: L{JacobiConformal} for other and more details.
1056 '''
1057 _ab = _bc = 0
1059 def __init__(self, radius_triaxial, ab=0, bc=0, **name):
1060 '''New L{JacobiConformalSpherical}.
1062 @arg radius_triaxial: Radius (C{scalar}, conventionally in
1063 C{meter}) or an other L{JacobiConformalSpherical},
1064 L{JacobiConformal} or ordered L{Triaxial}.
1065 @kwarg ab: Relative magnitude of C{B{a} - B{b}} (C{meter},
1066 same units as C{scalar B{radius}}.
1067 @kwarg bc: Relative magnitude of C{B{b} - B{c}} (C{meter},
1068 same units as C{scalar B{radius}}.
1069 @kwarg name: Optional C{B{name}=NN} (C{str}).
1071 @raise TriaxialError: Invalid B{C{radius_triaxial}}, negative
1072 B{C{ab}}, negative B{C{bc}} or C{(B{ab}
1073 + B{bc})} not positive.
1075 @note: If B{C{radius_triaxial}} is a L{JacobiConformalSpherical}
1076 and if B{C{ab}} and B{C{bc}} are both zero or C{None},
1077 the B{C{radius_triaxial}}'s C{ab}, C{bc}, C{a}, C{b}
1078 and C{c} are copied.
1079 '''
1080 try:
1081 r = radius_triaxial
1082 if isinstance(r, Triaxial): # ordered only
1083 t = r._abc3
1084 j = isinstance(r, JacobiConformalSpherical) and not bool(ab or bc)
1085 else:
1086 t = (Radius(radius=r),) * 3
1087 j = False
1088 self._ab = r.ab if j else Scalar_(ab=ab) # low=0
1089 self._bc = r.bc if j else Scalar_(bc=bc) # low=0
1090 if (self.ab + self.bc) <= 0:
1091 raise ValueError(_negative_)
1092 a, _, c = self._abc3 = t
1093 if not (a >= c and isfinite(self._a2b2)
1094 and isfinite(self._a2c2)):
1095 raise ValueError(_not_(_finite_))
1096 except (TypeError, ValueError) as x:
1097 raise TriaxialError(radius_triaxial=r, ab=ab, bc=bc, cause=x)
1098 if name:
1099 self.name = name
1101 @Property_RO
1102 def ab(self):
1103 '''Get relative magnitude C{a - b} (C{meter}, same units as B{C{a}}).
1104 '''
1105 return self._ab
1107 @Property_RO
1108 def _a2b2(self):
1109 '''(INTERNAL) Get C{a**2 - b**2} == ab * (a + b).
1110 '''
1111 a, b, _ = self._abc3
1112 return self.ab * (a + b)
1114 @Property_RO
1115 def _a2c2(self):
1116 '''(INTERNAL) Get C{a**2 - c**2} == a2b2 + b2c2.
1117 '''
1118 return self._a2b2 + self._b2c2
1120 @Property_RO
1121 def bc(self):
1122 '''Get relative magnitude C{b - c} (C{meter}, same units as B{C{a}}).
1123 '''
1124 return self._bc
1126 @Property_RO
1127 def _b2c2(self):
1128 '''(INTERNAL) Get C{b**2 - c**2} == bc * (b + c).
1129 '''
1130 _, b, c = self._abc3
1131 return self.bc * (b + c)
1133 @Property_RO
1134 def radius(self):
1135 '''Get radius (C{meter}, conventionally).
1136 '''
1137 return self.a
1140class TriaxialError(_ValueError):
1141 '''Raised for L{Triaxial} issues.
1142 '''
1143 pass # ...
1146class Triaxials(_NamedEnum):
1147 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class
1148 to accommodate the L{_LazyNamedEnumItem} properties.
1149 '''
1150 def _Lazy(self, *abc, **name):
1151 '''(INTERNAL) Instantiate the C{Triaxial}.
1152 '''
1153 a, b, c = map(km2m, abc)
1154 return Triaxial(a, b, c, **name)
1156Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton
1157'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.'''
1158# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in Km
1159# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf>
1160# <https://link.Springer.com/article/10.1007/s00190-022-01650-9>
1161_abc84_35 = (_EWGS84.a + 35), (_EWGS84.a - 35), _EWGS84.b
1162Triaxials._assert( # a (Km) b (Km) c (Km) planet
1163 Amalthea = _lazy('Amalthea', 125.0, 73.0, _64_0), # Jupiter
1164 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus
1165 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544),
1166 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn
1167 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter
1168 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter
1169 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3),
1170 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn
1171 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus
1172 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth
1173 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn
1174 WGS84_35 = _lazy('WGS84_35', *map(m2km, _abc84_35)))
1175del _abc84_35, _EWGS84
1178def _getitems(items, *indices):
1179 '''(INTERNAL) Get the C{items} at the given I{indices}.
1181 @return: C{Type(items[i] for i in indices)} with
1182 C{Type = type(items)}, any C{type} having
1183 the special method C{__getitem__}.
1184 '''
1185 return type(items)(map(items.__getitem__, indices))
1188def _hartzell3(pov, los, Tun): # in .Ellipsoid.hartzell4, .formy.hartzell
1189 '''(INTERNAL) Hartzell's "Satellite Line-of-Sight Intersection ...",
1190 formula from a Point-Of-View to an I{un-/ordered} Triaxial.
1191 '''
1192 def _toUvwV3d(los, pov):
1193 try: # pov must be LatLon or Cartesian if los is a Los
1194 v = los.toUvw(pov)
1195 except (AttributeError, TypeError):
1196 v = _otherV3d(los=los)
1197 return v
1199 p3 = _otherV3d(pov=pov.toCartesian() if isLatLon(pov) else pov)
1200 if los is True: # normal
1201 a, b, c, d, i = _plumbTo5(p3.x, p3.y, p3.z, Tun)
1202 return type(p3)(a, b, c), d, i
1204 u3 = p3.negate() if los is False or los is None else _toUvwV3d(los, pov)
1206 a, b, c, T = Tun._ordered4
1208 a2 = a**2 # largest, factored out
1209 b2, p2 = (b**2, T._1e2ab) if b != a else (a2, _1_0)
1210 c2, q2 = (c**2, T._1e2ac) if c != a else (a2, _1_0)
1212 p3 = T._order3d(p3)
1213 u3 = T._order3d(u3).unit() # unit vector, opposing signs
1215 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz3
1216 ux, vy, wz = u3.times_(p3).xyz3
1217 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz3
1219 t = (p2 * c2), c2, b2
1220 m = fdot(t, u2, v2, w2) # a2 factored out
1221 if m < EPS0: # zero or near-null LOS vector
1222 raise _ValueError(_near_(_null_))
1224 r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,
1225 -w2 * y2, -u2 * y2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2,
1226 -w2 * x2 * p2, b2 * u2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2)
1227 if r > 0: # a2 factored out
1228 r = sqrt(r) * b * c # == a * a * b * c / a2
1229 elif r < 0: # LOS pointing away from or missing the triaxial
1230 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_)
1232 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out
1233 if d > 0: # POV inside or LOS outside or missing the triaxial
1234 s = fsumf_(_N_1_0, _over(x2, a2), _over(y2, b2), _over(z2, c2)) # like _sideOf
1235 raise _ValueError(_outside_ if s > 0 else _inside_)
1236 elif fsum1f_(x2, y2, z2, -d*d) < 0: # d past triaxial's center
1237 raise _ValueError(_too_(_distant_))
1239 v = p3.minus(u3.times(d)) # cartesian type(pov) or Vector3d
1240 h = p3.minus(v).length # distance to pov == -d
1241 return T._order3d(v, reverse=True), h, None
1244def hartzell4(pov, los=False, tri_biax=_WGS84, **name):
1245 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight from
1246 a Point-Of-View outside.
1248 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple},
1249 C{LatLon} or L{Vector3d}).
1250 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Los}, L{Vector3d})
1251 or C{True} for the I{normal, perpendicular, plumb} to the surface of
1252 the tri-/biaxial or C{False} or C{None} to point to its center.
1253 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal} or
1254 L{JacobiConformalSpherical}) or biaxial ellipsoid (L{Datum},
1255 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple}) or spherical earth
1256 radius (C{scalar}, conventionally in C{meter}).
1257 @kwarg name: Optional name (C{str}), overriding C{B{name}="hartzell4"}.
1259 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h}
1260 the distance from B{C{pov}} to C{(x, y, z)} I{along the} B{C{los}}, all
1261 in C{meter}, conventionally.
1263 @raise TriaxialError: Invalid B{C{pov}} or B{C{pov}} inside the tri-/biaxial or
1264 invalid B{C{los}} or B{C{los}} points outside or away from
1265 the tri-/biaxial.
1267 @raise TypeError: Invalid B{C{tri_biax}}, C{ellipsoid} or C{datum}.
1269 @see: Class L{pygeodesy3.Los}, functions L{pygeodesy.tyr3d} and L{pygeodesy.hartzell}
1270 and U{lookAtSpheroid<https://PyPI.org/project/pymap3d>} and U{"Satellite
1271 Line-of-Sight Intersection with Earth"<https://StephenHartzell.Medium.com/
1272 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
1273 '''
1274 if isinstance(tri_biax, Triaxial_):
1275 T = tri_biax
1276 else:
1277 D = tri_biax if isinstance(tri_biax, Datum) else \
1278 _spherical_datum(tri_biax, name__=hartzell4) # typename
1279 T = D.ellipsoid._triaxial
1280 try:
1281 v, h, i = _hartzell3(pov, los, T)
1282 except Exception as x:
1283 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x)
1284 n = _name__(name, name__=hartzell4) # typename
1285 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name=n)
1288def _hypot2_1(x, y, z=0):
1289 '''(INTERNAL) Compute M{x**2 + y**2 + z**2 - 1} with C{max(fabs(x), fabs(y),
1290 fabs(z))} rarely greater than 1.0.
1291 '''
1292 return fsumf_(_N_1_0, x*x, y*y, z*z)
1295def _otherV3d_(x_xyz, y, z, **name):
1296 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}.
1297 '''
1298 return Vector3d(x_xyz, y, z, **name) if isscalar(x_xyz) else \
1299 _otherV3d(x_xyz=x_xyz, **name)
1302def _over0(p, q):
1303 '''(INTERNAL) Return C{p / q} or C{0}.
1304 '''
1305 return (p / q) if q > fabs(p) else _0_0
1308def _over02(p, q):
1309 '''(INTERNAL) Return C{(p / q)**2} or C{0}.
1310 '''
1311 return (p / q)**2 if p and q else _0_0
1314def _plumbTo3(px, py, E, eps=EPS): # in .ellipsoids.Ellipsoid.height4
1315 '''(INTERNAL) Nearest point on a 2-D ellipse in 1st quadrant.
1316 '''
1317 a, b = E.a, E.b
1318 if min(px, py, a, b) < EPS0:
1319 raise _AssertionError(px=px, py=py, a=a, b=b, E=E)
1321 a2 = a - b * E.b_a
1322 b2 = b - a * E.a_b
1323 tx = ty = _SQRT2_2
1324 for i in range(16): # max 5
1325 ex = tx**3 * a2
1326 ey = ty**3 * b2
1328 qx = px - ex
1329 qy = py - ey
1330 q = hypot(qx, qy)
1331 if q < EPS0: # PYCHOK no cover
1332 break
1333 r = hypot(ex - tx * a,
1334 ey - ty * b) / q
1336 sx, tx = tx, min(_1_0, max(0, (ex + qx * r) / a))
1337 sy, ty = ty, min(_1_0, max(0, (ey + qy * r) / b))
1338 t = hypot(ty, tx)
1339 if t < EPS0: # PYCHOK no cover
1340 break
1341 tx = tx / t # /= chokes PyChecker
1342 ty = ty / t
1343 if fabs(sx - tx) < eps and \
1344 fabs(sy - ty) < eps:
1345 break
1347 tx *= a / px
1348 ty *= b / py
1349 return tx, ty, i # x and y as fractions
1352def _plumbTo4(x, y, a, b, eps=EPS):
1353 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}.
1355 @see: Function C{_plumbTo3} and I{Eberly}'s U{Distance from a Point to ... an Ellipse ...
1356 <https://www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}.
1357 '''
1358 if b > a:
1359 b, a, d, i = _plumbTo4(y, x, b, a, eps=eps)
1360 return a, b, d, i
1362 if not (b > 0 and isfinite(a)):
1363 raise _ValueError(a=a, b=b)
1365 i = None
1366 if y:
1367 if x:
1368 u = fabs(x / a)
1369 w = fabs(y / b)
1370 g = _hypot2_1(u, w)
1371 if fabs(g) > EPS02:
1372 r = (b / a)**2
1373 t, i = _rootNd(_1_0 / r, 0, u, 0, w, g) # eps
1374 a = _over(x, t * r + _1_0)
1375 b = _over(y, t + _1_0)
1376 d = hypot(x - a, y - b)
1377 else: # on the ellipse
1378 a, b, d = x, y, _0_0
1379 else: # x == 0
1380 if y < 0:
1381 b = -b
1382 a = x # signed-0
1383 d = fabs(y - b)
1385 elif x: # y == 0
1386 d, r = None, _over0(a * x, (a + b) * (a - b))
1387 if r:
1388 a *= r
1389 r = _1_0 - r**2
1390 if r > EPS02:
1391 b *= sqrt(r)
1392 d = hypot(x - a, y - b)
1393 elif x < 0:
1394 a = -a
1395 if d is None:
1396 b = y # signed-0
1397 d = fabs(x - a)
1399 else: # x == y == 0
1400 a = x # signed-0
1401 d = b
1403 return a, b, d, i
1406def _plumbTo5(x, y, z, Tun, eps=EPS): # in .testTriaxials
1407 '''(INTERNAL) Nearest point on and distance to an I{un-/ordered} triaxial.
1409 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https://
1410 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}.
1411 '''
1412 a, b, c, T = Tun._ordered4
1413 if Tun is not T: # T is ordered, Tun isn't
1414 t = T._order3(x, y, z) + (T,)
1415 a, b, c, d, i = _plumbTo5(*t, eps=eps)
1416 return T._order3(a, b, c, reverse=True) + (d, i)
1418 if not (c > 0 and isfinite(a)):
1419 raise _ValueError(a=a, b=b, c=c)
1421 if eps > 0:
1422 val = max(eps * 1e8, EPS)
1423 else: # no validation
1424 val, eps = 0, max(EPS0, -eps)
1426 i = None
1427 if z:
1428 if y:
1429 if x:
1430 u = fabs(x / a)
1431 v = fabs(y / b)
1432 w = fabs(z / c)
1433 g = _hypot2_1(u, v, w)
1434 if fabs(g) > EPS02:
1435 r = T._1e2ac # (c / a)**2
1436 s = T._1e2bc # (c / b)**2
1437 t, i = _rootNd(_1_0 / r, _1_0 / s, u, v, w, g) # eps
1438 a = _over(x, t * r + _1_0)
1439 b = _over(y, t * s + _1_0)
1440 c = _over(z, t + _1_0)
1441 d = hypot_(x - a, y - b, z - c)
1442 else: # on the ellipsoid
1443 a, b, c, d = x, y, z, _0_0
1444 else: # x == 0
1445 a = x # 0
1446 b, c, d, i = _plumbTo4(y, z, b, c, eps=eps)
1447 elif x: # y == 0
1448 b = y # 0
1449 a, c, d, i = _plumbTo4(x, z, a, c, eps=eps)
1450 else: # x == y == 0
1451 if z < 0:
1452 c = -c
1453 a, b, d = x, y, fabs(z - c)
1455 else: # z == 0
1456 u = _over0(a * x, T._a2c2) # (a + c) * (a - c)
1457 v = _over0(b * y, T._b2c2) # (b + c) * (b - c)
1458 s = _hypot2_1(u, v)
1459 if u and v and s < 0:
1460 a *= u
1461 b *= v
1462 c *= sqrt(-s)
1463 d = hypot_(x - a, y - b, c)
1464 else:
1465 c = z # signed-0
1466 a, b, d, i = _plumbTo4(x, y, a, b, eps=eps)
1468 if val > 0:
1469 _validate(a, b, c, d, T, x, y, z, val)
1470 return a, b, c, d, i
1473def _rootNd(r, s, u, v, w, g, eps=EPS0):
1474 '''(INTERNAL) Robust 2-D or 3-D root finder: 2-D if C{s == v == 0} else 3-D root.
1476 @see: I{Eberly}'s U{Robust Root Finders ... and Listing 4<https://
1477 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}.
1478 '''
1479 u *= r
1480 v *= s # 0 for 2-D root
1481 t0 = w - _1_0
1482 t1 = _0_0 if g < 0 else (hypot_(u, w, v) - _1_0)
1483 # assert t0 <= t1
1484 for i in range(1, _TRIPS): # 48..58
1485 t = (t1 + t0) * _0_5
1486 e = t1 - t0
1487 if eps > e > -eps or _isin(t, t0, t1):
1488 break
1489 g = fsumf_(_N_1_0, # ~= _hypot2_1
1490 _over02(u, t + r),
1491 _over02(w, t + _1_0), (
1492 _over02(v, t + s) if v else _0_0))
1493 if g > 0:
1494 t0 = t
1495 elif g < 0:
1496 t1 = t
1497 else:
1498 break
1499 else: # PYCHOK no cover
1500 t = Fmt.no_convergence(e, eps)
1501 raise _ValueError(t, txt__=_rootNd)
1502 return t, i
1505def _sqrt0(x):
1506 '''(INTERNAL) C{sqrt0} with C{TriaxialError}.
1507 '''
1508 return sqrt0(x, Error=TriaxialError)
1511def _validate(a, b, c, d, T, x, y, z, val):
1512 '''(INTERNAL) Validate an C{_plumTo5} result.
1513 '''
1514 e = T.sideOf(a, b, c, eps=val)
1515 if e: # not near the ellipsoid's surface
1516 raise _ValueError(a=a, b=b, c=c, d=d,
1517 sideOf=e, eps=val)
1518 if d: # angle of delta and normal vector
1519 m = Vector3d(x, y, z).minus_(a, b, c)
1520 if m.euclid > val:
1521 m = m.unit()
1522 n = T.normal3d(a, b, c)
1523 e = n.dot(m) # n.negate().dot(m)
1524 if not isnear1(fabs(e), eps1=val):
1525 raise _ValueError(n=n, m=m,
1526 dot=e, eps=val)
1529if __name__ == _DMAIN_:
1531 from pygeodesy import printf
1532 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_
1534 t = Triaxial_(6378388.0, 6378318.0, 6356911.9461)
1535 t = t.height4(3909863.9271, 3909778.123, 3170932.5016)
1536 printf('# Bektas: %r', t)
1538 # __doc__ of this file, force all into registery
1539 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_)
1540 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))
1542# % python3 -m pygeodesy.triaxials
1543#
1544# Bektas: height4(x=3909251.554667, y=3909165.750567, z=3170432.501602, h=999.999996)
1546# **) MIT License
1547#
1548# Copyright (C) 2022-2025 -- mrJean1 at Gmail -- All Rights Reserved.
1549#
1550# Permission is hereby granted, free of charge, to any person obtaining a
1551# copy of this software and associated documentation files (the "Software"),
1552# to deal in the Software without restriction, including without limitation
1553# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1554# and/or sell copies of the Software, and to permit persons to whom the
1555# Software is furnished to do so, subject to the following conditions:
1556#
1557# The above copyright notice and this permission notice shall be included
1558# in all copies or substantial portions of the Software.
1559#
1560# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1561# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1562# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1563# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1564# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1565# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1566# OTHER DEALINGS IN THE SOFTWARE.