Coverage for pygeodesy/triaxials.py: 95%
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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>} (2008-2023). 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=798619018175.109863281)
21@var Triaxials.Europa: Triaxial(name='Europa', a=1564130, b=1561230, c=1560930, e2ab=0.003704694, e2bc=0.000384275, e2ac=0.004087546, volume=15966575194402123776, area=30663773697323.51953125, area_p=30663773794562.45703125)
22@var Triaxials.Io: Triaxial(name='Io', a=1829400, b=1819300, c=1815700, e2ab=0.011011391, e2bc=0.003953651, e2ac=0.014921506, volume=25313121117889765376, area=41691875849096.7421875, area_p=41691877397441.2109375)
23@var Triaxials.Mars: Triaxial(name='Mars', a=3394600, b=3393300, c=3376300, e2ab=0.000765776, e2bc=0.009994646, e2ac=0.010752768, volume=162907283585817247744, area=144249140795107.4375, area_p=144249144150662.15625)
24@var Triaxials.Mimas: Triaxial(name='Mimas', a=207400, b=196800, c=190600, e2ab=0.09960581, e2bc=0.062015624, e2ac=0.155444317, volume=32587072869017956, area=493855762247.691894531, area_p=493857714107.9375)
25@var Triaxials.Miranda: Triaxial(name='Miranda', a=240400, b=234200, c=232900, e2ab=0.050915557, e2bc=0.011070811, e2ac=0.061422691, volume=54926187094835456, area=698880863325.756958008, area_p=698881306767.950317383)
26@var Triaxials.Moon: Triaxial(name='Moon', a=1735550, b=1735324, c=1734898, e2ab=0.000260419, e2bc=0.000490914, e2ac=0.000751206, volume=21886698675223740416, area=37838824729886.09375, area_p=37838824733332.2265625)
27@var Triaxials.Tethys: Triaxial(name='Tethys', a=535600, b=528200, c=525800, e2ab=0.027441672, e2bc=0.009066821, e2ac=0.036259685, volume=623086233855821440, area=3528073490771.394042969, area_p=3528074261832.738769531)
28@var Triaxials.WGS84_35: Triaxial(name='WGS84_35', a=6378172, b=6378102, c=6356752.314245179, e2ab=0.00002195, e2bc=0.006683478, e2ac=0.006705281, volume=1083207319768789942272, area=510065621722018.125, area_p=510065626587483.3125)
29'''
30# make sure int/int division yields float quotient, see .basics
31from __future__ import division as _; del _ # PYCHOK semicolon
33from pygeodesy.basics import isLatLon, isscalar, _zip, _ValueError
34from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, INT0, PI2, PI_3, PI4, \
35 _EPS2e4, float0_, isfinite, isnear1, _0_0, _0_5, \
36 _1_0, _N_1_0, _N_2_0, _4_0 # PYCHOK used!
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
40# from pygeodesy.errors import _ValueError # from .basics
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_, _finite_, \
44 _height_, _inside_, _near_, _negative_, _not_, \
45 _NOTEQUAL_, _null_, _opposite_, _outside_, _SPACE_, \
46 _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, atan2d, km2m, m2km, SinCos2, sincos2d_
57from pygeodesy.vector3d import _otherV3d, Vector3d, _ALL_LAZY, _MODS
59from math import atan2, fabs, sqrt
61__all__ = _ALL_LAZY.triaxials
62__version__ = '24.07.29'
64_not_ordered_ = _not_('ordered')
65_omega_ = 'omega'
66_TRIPS = 269 # 48-55, 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 name (C{str}), overriding this name.
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}), overriding this name.
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 name (C{str}), overriding this name.
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 name (C{str}), overriding this name.
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}), overriding this name.
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}, 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 (largest) C{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 (middle) C{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 (smallest) C{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{normal, plumb} to the surface of,
386 otherwise the C{radial} line to the center of this triaxial (C{bool}).
387 @kwarg eps: Tolerance for root finding and validation (C{scalar}), use a negative
388 value to skip validation.
389 @kwarg name: Optional C{B{name}="heigh4"} (C{str}).
391 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x}, C{y}
392 and C{z} of the projection on or the intersection with and with the height
393 C{h} above or below the triaxial's surface in C{meter}, conventionally.
395 @raise TriaxialError: Non-cartesian B{C{xyz}}, invalid B{C{eps}}, no convergence in
396 root finding or validation failed.
398 @see: Method L{Ellipsoid.height4} and I{Eberly}'s U{Distance from a Point to ...
399 <https://www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}.
400 '''
401 v, r = _otherV3d_(x_xyz, y, z), self.isSpherical
403 i, h = None, v.length
404 if h < EPS0: # EPS
405 x = y = z = _0_0
406 h -= min(self._abc3) # nearest
407 elif r: # .isSpherical
408 x, y, z = v.times(r / h).xyz
409 h -= r
410 else:
411 x, y, z = v.xyz
412 try:
413 if normal: # plumb to surface
414 x, y, z, h, i = _plumbTo5(x, y, z, self, eps=eps)
415 else: # radial to center
416 x, y, z = self._radialTo3(z, hypot(x, y), y, x)
417 h = v.minus_(x, y, z).length
418 except Exception as e:
419 raise TriaxialError(x=x, y=y, z=z, cause=e)
420 if h > 0 and self.sideOf(v, eps=EPS0) < 0:
421 h = -h # below the surface
422 n = _name__(name, name__=self.height4)
423 return Vector4Tuple(x, y, z, h, iteration=i, name=n)
425 @Property_RO
426 def isOrdered(self):
427 '''Is this triaxial I{ordered} and I{not spherical} (C{bool})?
428 '''
429 a, b, c = self._abc3
430 return bool(a >= b > c) # b > c!
432 @Property_RO
433 def isSpherical(self):
434 '''Is this triaxial I{spherical} (C{Radius} or INT0)?
435 '''
436 a, b, c = self._abc3
437 return a if a == b == c else INT0
439 def _norm2(self, s, c, *a):
440 '''(INTERNAL) Normalize C{s} and C{c} iff not already.
441 '''
442 if fabs(_hypot21(s, c)) > EPS02:
443 s, c = norm2(s, c)
444 if a:
445 s, c = norm2(s * self.b, c * a[0])
446 return float0_(s, c)
448 def normal3d(self, x_xyz, y=None, z=None, length=_1_0):
449 '''Get a 3-D vector at a cartesian on and perpendicular to this triaxial's surface.
451 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
452 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
453 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}, ignored
454 otherwise.
455 @kwarg z: Z component (C{scalar}), like B{C{y}}.
456 @kwarg length: Optional length and in-/outward direction (C{scalar}).
458 @return: A C{Vector3d(x_, y_, z_)} normalized to B{C{length}}, pointing
459 in- or outward for neg- respectively positive B{C{length}}.
461 @raise TriaxialError: Zero length cartesian or vector.
463 @note: Cartesian location C{(B{x}, B{y}, B{z})} must be on this triaxial's
464 surface, use method L{Triaxial.sideOf} to validate.
465 '''
466 # n = 2 * (x / a2, y / b2, z / c2)
467 # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered
468 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2
469 # == unit(x, y / _1e2ab, z / _1e2ac).times(length)
470 n = self._normal3d.times_(*_otherV3d_(x_xyz, y, z).xyz)
471 if n.length < EPS0:
472 raise TriaxialError(x=x_xyz, y=y, z=z, txt=_null_)
473 return n.times(length / n.length)
475 @Property_RO
476 def _normal3d(self):
477 '''(INTERNAL) Get M{Vector3d((d/a)**2, (d/b)**2, (d/c)**2)}, M{d = max(a, b, c)}.
478 '''
479 d = max(self._abc3)
480 t = tuple(((d / x)**2 if x != d else _1_0) for x in self._abc3)
481 return Vector3d(*t, name__=self.normal3d)
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.xyz, *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 = 0, 1, 2 # 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 z, r = c * sa, c * ca
553 x, y = r * cb, r * sb
554 return x, y, z
556 def sideOf(self, x_xyz, y=None, z=None, eps=EPS4):
557 '''Is a cartesian above, below or on the surface of this triaxial?
559 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
560 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
561 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
562 ignored otherwise.
563 @kwarg z: Z component (C{scalar}), like B{C{y}}.
564 @kwarg eps: Near-surface tolerance (C{scalar}, distance I{squared}).
566 @return: C{INT0} if C{(B{x}, B{y}, B{z})} is near this triaxial's surface
567 within tolerance B{C{eps}}, otherwise a signed, radial, normalized
568 distance I{squared} (C{float}), negative or positive for in-
569 respectively outside this triaxial.
571 @see: Methods L{Triaxial.height4} and L{Triaxial.normal3d}.
572 '''
573 return _sideOf(_otherV3d_(x_xyz, y, z).xyz, self._abc3, eps=eps)
575 def toEllipsoid(self, **name):
576 '''Convert this triaxial to an L{Ellipsoid}, provided 2 axes match.
578 @kwarg name: Optional, overriding C{B{name}=NN} (C{str})=.
580 @return: An L{Ellipsoid} with north along this C{Z} axis if C{a == b},
581 this C{Y} axis if C{a == c} or this C{X} axis if C{b == c}.
583 @raise TriaxialError: This C{a != b}, C{b != c} and C{c != a}.
585 @see: Method L{Ellipsoid.toTriaxial}.
586 '''
587 a, b, c = self._abc3
588 if a == b:
589 b = c # N = c-Z
590 elif b == c: # N = a-X
591 a, b = b, a
592 elif a != c: # N = b-Y
593 t = _SPACE_(_a_, _NOTEQUAL_, _b_, _NOTEQUAL_, _c_)
594 raise TriaxialError(a=a, b=b, c=c, txt=t)
595 return Ellipsoid(a, b=b, name=self._name__(name))
597 def toStr(self, prec=9, **name): # PYCHOK signature
598 '''Return this C{Triaxial} as a string.
600 @kwarg prec: Precision, number of decimal digits (0..9).
601 @kwarg name: Optional, overriding C{B{name}=NN} (C{str})
602 or C{None} to exclude this triaxial's name.
604 @return: This C{Triaxial}'s attributes (C{str}).
605 '''
606 T = Triaxial_
607 t = T.a,
608 J = JacobiConformalSpherical
609 t += (J.ab, J.bc) if isinstance(self, J) else (T.b, T.c)
610 t += T.e2ab, T.e2bc, T.e2ac
611 J = JacobiConformal
612 if isinstance(self, J):
613 t += J.xyQ2,
614 t += T.volume, T.area
615 return self._instr(area_p=self.area_p(), prec=prec, props=t, **name)
617 @Property_RO
618 def volume(self):
619 '''Get the volume (C{meter**3}), M{4 / 3 * PI * a * b * c}.
620 '''
621 a, b, c = self._abc3
622 return Meter3(volume=a * b * c * PI_3 * _4_0)
625class Triaxial(Triaxial_):
626 '''I{Ordered} triaxial ellipsoid.
628 @see: L{Triaxial_} for more information.
629 '''
630 _unordered = False
632 def __init__(self, a_triaxial, b=None, c=None, **name):
633 '''New I{ordered} L{Triaxial}.
635 @arg a_triaxial: Largest semi-axis (C{scalar}, conventionally in C{meter})
636 or an other L{Triaxial} or L{Triaxial_} instance.
637 @kwarg b: Middle semi-axis (C{meter}, same units as B{C{a}}), required
638 if C{B{a_triaxial} is scalar}, ignored otherwise.
639 @kwarg c: Smallest semi-axis (C{meter}, like B{C{b}}).
640 @kwarg name: Optional C{B{name}=NN} (C{str}).
642 @note: The semi-axes must be ordered as C{B{a} >= B{b} >= B{c} > 0} and
643 must be ellipsoidal, C{B{a} > B{c}}.
645 @raise TriaxialError: Semi-axes unordered, spherical or invalid.
646 '''
647 Triaxial_.__init__(self, a_triaxial, b=b, c=c, **name)
649 @Property_RO
650 def _a2b2_a2c2(self):
651 '''@see: Methods C{.forwardBetaOmega} and C{._k2_kp2}.
652 '''
653 return self._a2b2 / self._a2c2
655 @Property_RO
656 def area(self):
657 '''Get the surface area (C{meter} I{squared}).
659 @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Surface_area>}.
660 '''
661 a, b, c = self._abc3
662 if a != b:
663 kp2, k2 = self._k2_kp2 # swapped!
664 aE = self._Elliptic(k2, _0_0, kp2, _1_0)
665 c2 = self._1e2ac # cos(phi)**2 = (c/a)**2
666 s = sqrt(self.e2ac) # sin(phi)**2 = 1 - c2
667 r = asin1(s) # phi = atan2(sqrt(c2), s)
668 b *= fsum1f_(aE.fE(r) * s, c / a * c / b,
669 aE.fF(r) * c2 / s)
670 a = Meter2(area=a * b * PI2)
671 else: # a == b > c
672 a = Ellipsoid(a, b=c).areax
673 return a
675 def _exyz3(self, u):
676 '''(INTERNAL) Helper for C{.forwardBetOmg}.
677 '''
678 if u > 0:
679 u2 = u**2
680 x = u * sqrt0(_1_0 + self._a2c2 / u2, Error=TriaxialError)
681 y = u * sqrt0(_1_0 + self._b2c2 / u2, Error=TriaxialError)
682 else:
683 x = y = u = _0_0
684 return x, y, u
686 def forwardBetaOmega(self, beta, omega, height=0, **name):
687 '''Convert I{ellipsoidal} lat- and longitude C{beta}, C{omega}
688 and height to cartesian.
690 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
691 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
692 @arg height: Height above or below the ellipsoid's surface (C{meter}, same
693 units as this triaxial's C{a}, C{b} and C{c} semi-axes).
694 @kwarg name: Optional C{B{name}=NN} (C{str}).
696 @return: A L{Vector3Tuple}C{(x, y, z)}.
698 @see: Method L{Triaxial.reverseBetaOmega} and U{Expressions (23-25)<https://
699 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
700 '''
701 if height:
702 h = self._Height(height)
703 x, y, z = self._exyz3(h + self.c)
704 else:
705 x, y, z = self._abc3 # == self._exyz3(self.c)
706 if z: # and x and y:
707 sa, ca = SinCos2(beta)
708 sb, cb = SinCos2(omega)
710 r = self._a2b2_a2c2
711 x *= cb * sqrt0(ca**2 + r * sa**2, Error=TriaxialError)
712 y *= ca * sb
713 z *= sa * sqrt0(_1_0 - r * cb**2, Error=TriaxialError)
714 return Vector3Tuple(x, y, z, **name)
716 def forwardBetaOmega_(self, sbeta, cbeta, somega, comega, **name):
717 '''Convert I{ellipsoidal} lat- and longitude C{beta} and C{omega}
718 to cartesian coordinates I{on the triaxial's surface}.
720 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
721 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
722 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
723 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
724 @kwarg name: Optional C{B{name}=NN} (C{str}).
726 @return: A L{Vector3Tuple}C{(x, y, z)} on the surface.
728 @raise TriaxialError: This triaxial is near-spherical.
730 @see: Method L{Triaxial.reverseBetaOmega}, U{Triaxial ellipsoid coordinate
731 system<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid#
732 Triaxial_ellipsoid_coordinate_system>} and U{expressions (23-25)<https://
733 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
734 '''
735 t = self._radialTo3(sbeta, cbeta, somega, comega)
736 return Vector3Tuple(*t, **name)
738 def forwardCartesian(self, x_xyz, y=None, z=None, **normal_eps_name):
739 '''Project a cartesian on this triaxial.
741 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
742 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
743 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
744 ignored otherwise.
745 @kwarg z: Z component (C{scalar}), like B{C{y}}.
746 @kwarg normal_eps_name: Optional keyword arguments C{B{normal}=True},
747 C{B{eps}=EPS} and overriding C{B{name}="height4"} (C{str}),
748 see method L{Triaxial.height4}.
750 @see: Method L{Triaxial.height4} for further information and method
751 L{Triaxial.reverseCartesian} to reverse the projection.
752 '''
753 return self.height4(x_xyz, y, z, **normal_eps_name)
755 def forwardLatLon(self, lat, lon, height=0, **name):
756 '''Convert I{geodetic} lat-, longitude and height to cartesian.
758 @arg lat: Geodetic latitude (C{degrees}).
759 @arg lon: Geodetic longitude (C{degrees}).
760 @arg height: Height above the ellipsoid (C{meter}, same units
761 as this triaxial's C{a}, C{b} and C{c} axes).
762 @kwarg name: Optional C{B{name}=NN} (C{str}).
764 @return: A L{Vector3Tuple}C{(x, y, z)}.
766 @see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https://
767 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
768 '''
769 return self._forwardLatLon3(height, name, *sincos2d_(lat, lon))
771 def forwardLatLon_(self, slat, clat, slon, clon, height=0, **name):
772 '''Convert I{geodetic} lat-, longitude and height to cartesian.
774 @arg slat: Geodetic latitude C{sin(lat)} (C{scalar}).
775 @arg clat: Geodetic latitude C{cos(lat)} (C{scalar}).
776 @arg slon: Geodetic longitude C{sin(lon)} (C{scalar}).
777 @arg clon: Geodetic longitude C{cos(lon)} (C{scalar}).
778 @arg height: Height above the ellipsoid (C{meter}, same units
779 as this triaxial's axes C{a}, C{b} and C{c}).
780 @kwarg name: Optional C{B{name}=NN} (C{str}).
782 @return: A L{Vector3Tuple}C{(x, y, z)}.
784 @see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https://
785 www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}.
786 '''
787 sa, ca = self._norm2(slat, clat)
788 sb, cb = self._norm2(slon, clon)
789 return self._forwardLatLon3(height, name, sa, ca, sb, cb)
791 def _forwardLatLon3(self, height, name, sa, ca, sb, cb): # name always **name
792 '''(INTERNAL) Helper for C{.forwardLatLon} and C{.forwardLatLon_}.
793 '''
794 ca_x_sb = ca * sb
795 h = self._Height(height)
796 # 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2
797 t = fsumf_(_1_0, -self.e2ac * sa**2, -self.e2ab * ca_x_sb**2)
798 n = self.a / sqrt0(t, Error=TriaxialError) # prime vertical
799 x = (h + n) * ca * cb
800 y = (h + n * self._1e2ab) * ca_x_sb
801 z = (h + n * self._1e2ac) * sa
802 return Vector3Tuple(x, y, z, **name)
804 def _Height(self, height):
805 '''(INTERNAL) Validate a C{height}.
806 '''
807 return Height_(height=height, low=-self.c, Error=TriaxialError)
809 @Property_RO
810 def _k2_kp2(self):
811 '''(INTERNAL) Get C{k2} and C{kp2} for C{._xE}, C{._yE} and C{.area}.
812 '''
813 # k2 = a2b2 / a2c2 * c2_b2
814 # kp2 = b2c2 / a2c2 * a2_b2
815 # b2 = b**2
816 # xE = Elliptic(k2, -a2b2 / b2, kp2, a2_b2)
817 # yE = Elliptic(kp2, +b2c2 / b2, k2, c2_b2)
818 # aE = Elliptic(kp2, 0, k2, 1)
819 return (self._a2b2_a2c2 * self._c2_b2,
820 self._b2c2 / self._a2c2 * self._a2_b2)
822 def _radialTo3(self, sbeta, cbeta, somega, comega):
823 '''(INTERNAL) Convert I{ellipsoidal} lat- and longitude C{beta} and
824 C{omega} to cartesian coordinates I{on the triaxial's surface},
825 also I{ordered} helper for C{.height4}.
826 '''
827 sa, ca = self._norm2(sbeta, cbeta)
828 sb, cb = self._norm2(somega, comega)
830 b2_a2 = self._1e2ab # == (b/a)**2
831 c2_a2 = -self._1e2ac # == -(c/a)**2
832 a2c2_a2 = self. e2ac # (a**2 - c**2) / a**2 == 1 - (c/a)**2
834 x2 = _Fsumf_(_1_0, -b2_a2 * sa**2, c2_a2 * ca**2).fover(a2c2_a2)
835 z2 = _Fsumf_(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2)
837 x, y, z = self._abc3
838 x *= cb * sqrt0(x2, Error=TriaxialError)
839 y *= ca * sb
840 z *= sa * sqrt0(z2, Error=TriaxialError)
841 return x, y, z
843 def reverseBetaOmega(self, x_xyz, y=None, z=None, **name):
844 '''Convert cartesian to I{ellipsoidal} lat- and longitude, C{beta}, C{omega}
845 and height.
847 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
848 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
849 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
850 ignored otherwise.
851 @kwarg z: Z component (C{scalar}), like B{C{y}}.
852 @kwarg name: Optional C{B{name}=NN} (C{str}).
854 @return: A L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and
855 C{omega} in L{Radians} and (radial) C{height} in C{meter}, same
856 units as this triaxial's axes.
858 @see: Methods L{Triaxial.forwardBetaOmega} and L{Triaxial.forwardBetaOmega_}
859 and U{Expressions (21-22)<https://www.Topo.Auth.GR/wp-content/uploads/
860 sites/111/2021/12/09_Panou.pdf>}.
861 '''
862 v = _otherV3d_(x_xyz, y, z)
863 a, b, h = self._reverseLatLon3(v, atan2, v, self.forwardBetaOmega_)
864 return BetaOmega3Tuple(Radians(beta=a), Radians(omega=b), h, **name)
866 def reverseCartesian(self, x_xyz, y=None, z=None, h=0, normal=True, eps=_EPS2e4, **name):
867 '''"Unproject" a cartesian on to a cartesion I{off} this triaxial's surface.
869 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
870 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
871 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
872 ignored otherwise.
873 @kwarg z: Z component (C{scalar}), like B{C{y}}.
874 @arg h: Height above or below this triaxial's surface (C{meter}, same units
875 as the axes).
876 @kwarg normal: If C{True} the height is C{normal} to the surface, otherwise
877 C{radially} to the center of this triaxial (C{bool}).
878 @kwarg eps: Tolerance for surface test (C{scalar}).
879 @kwarg name: Optional C{B{name}=NN} (C{str}).
881 @return: A L{Vector3Tuple}C{(x, y, z)}.
883 @raise TrialError: Cartesian C{(x, y, z)} not on this triaxial's surface.
885 @see: Methods L{Triaxial.forwardCartesian} and L{Triaxial.height4}.
886 '''
887 v = _otherV3d_(x_xyz, y, z, **name)
888 s = _sideOf(v.xyz, self._abc3, eps=eps)
889 if s: # PYCHOK no cover
890 t = _SPACE_((_inside_ if s < 0 else _outside_), self.toRepr())
891 raise TriaxialError(eps=eps, sideOf=s, x=v.x, y=v.y, z=v.z, txt=t)
893 if h:
894 if normal:
895 v = v.plus(self.normal3d(*v.xyz, length=h))
896 elif v.length > EPS0:
897 v = v.times(_1_0 + (h / v.length))
898 return v.xyz # Vector3Tuple
900 def reverseLatLon(self, x_xyz, y=None, z=None, **name):
901 '''Convert cartesian to I{geodetic} lat-, longitude and height.
903 @arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian},
904 L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}).
905 @kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar},
906 ignored otherwise.
907 @kwarg z: Z component (C{scalar}), like B{C{y}}.
908 @kwarg name: Optional C{B{name}=NN} (C{str}).
910 @return: A L{LatLon3Tuple}C{(lat, lon, height)} with C{lat} and C{lon}
911 in C{degrees} and (radial) C{height} in C{meter}, same units
912 as this triaxial's axes.
914 @see: Methods L{Triaxial.forwardLatLon} and L{Triaxial.forwardLatLon_}
915 and U{Expressions (4-5)<https://www.Topo.Auth.GR/wp-content/uploads/
916 sites/111/2021/12/09_Panou.pdf>}.
917 '''
918 v = _otherV3d_(x_xyz, y, z)
919 s = v.times_(self._1e2ac, # == 1 - e_sub_x**2
920 self._1e2bc, # == 1 - e_sub_y**2
921 _1_0)
922 a, b, h = self._reverseLatLon3(s, atan2d, v, self.forwardLatLon_)
923 return LatLon3Tuple(Degrees(lat=a), Degrees(lon=b), h, **name)
925 def _reverseLatLon3(self, s, atan2_, v, forward_):
926 '''(INTERNAL) Helper for C{.reverseBetOmg} and C{.reverseLatLon}.
927 '''
928 x, y, z = s.xyz
929 d = hypot( x, y)
930 a = atan2_(z, d)
931 b = atan2_(y, x)
932 h = v.minus_(*forward_(z, d, y, x)).length
933 return a, b, h
936class JacobiConformal(Triaxial):
937 '''This is a conformal projection of a triaxial ellipsoid to a plane in which the
938 C{X} and C{Y} grid lines are straight.
940 Ellipsoidal coordinates I{beta} and I{omega} are converted to Jacobi Conformal
941 I{y} respectively I{x} separately. Jacobi's coordinates have been multiplied
942 by C{sqrt(B{a}**2 - B{c}**2) / (2 * B{b})} so that the customary results are
943 returned in the case of an ellipsoid of revolution.
945 Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2014-2023) and
946 licensed under the MIT/X11 License.
948 @note: This constructor can I{not be used to specify a sphere}, see alternate
949 L{JacobiConformalSpherical}.
951 @see: L{Triaxial}, C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/
952 C++/doc/classGeographicLib_1_1JacobiConformal.html#details>}, U{Jacobi's conformal
953 projection<https://GeographicLib.SourceForge.io/C++/doc/jacobi.html>} and Jacobi,
954 C. G. J. I{U{Vorlesungen über Dynamik<https://Books.Google.com/books?
955 id=ryEOAAAAQAAJ&pg=PA212>}}, page 212ff.
956 '''
958 @Property_RO
959 def _xE(self):
960 '''(INTERNAL) Get the x-elliptic function.
961 '''
962 k2, kp2 = self._k2_kp2
963 # -a2b2 / b2 == (b2 - a2) / b2 == 1 - a2 / b2 == 1 - a2_b2
964 return self._Elliptic(k2, _1_0 - self._a2_b2, kp2, self._a2_b2)
966 def xR(self, omega):
967 '''Compute a Jacobi Conformal C{x} projection.
969 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
971 @return: The C{x} projection (L{Radians}).
972 '''
973 return self.xR_(*SinCos2(omega))
975 def xR_(self, somega, comega):
976 '''Compute a Jacobi Conformal C{x} projection.
978 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
979 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
981 @return: The C{x} projection (L{Radians}).
982 '''
983 s, c = self._norm2(somega, comega, self.a)
984 return Radians(x=self._xE.fPi(s, c) * self._a2_b2)
986 @Property_RO
987 def xyQ2(self):
988 '''Get the Jacobi Conformal quadrant size (L{Jacobi2Tuple}C{(x, y)}).
989 '''
990 return Jacobi2Tuple(Radians(x=self._a2_b2 * self._xE.cPi),
991 Radians(y=self._c2_b2 * self._yE.cPi),
992 name=JacobiConformal.xyQ2.name)
994 def xyR2(self, beta, omega, **name):
995 '''Compute a Jacobi Conformal C{x} and C{y} projection.
997 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
998 @arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}).
999 @kwarg name: Optional, overriding C{B{name}="xyR2"} (C{str}).
1001 @return: A L{Jacobi2Tuple}C{(x, y)}.
1002 '''
1003 return self.xyR2_(*(SinCos2(beta) + SinCos2(omega)),
1004 name=_name__(name, name__=self.xyR2))
1006 def xyR2_(self, sbeta, cbeta, somega, comega, **name):
1007 '''Compute a Jacobi Conformal C{x} and C{y} projection.
1009 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
1010 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
1011 @arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}).
1012 @arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}).
1013 @kwarg name: Optional, overriding C{B{name}="xyR2_"} (C{str}).
1015 @return: A L{Jacobi2Tuple}C{(x, y)}.
1016 '''
1017 return Jacobi2Tuple(self.xR_(somega, comega),
1018 self.yR_(sbeta, cbeta),
1019 name=_name__(name, name__=self.xyR2_))
1021 @Property_RO
1022 def _yE(self):
1023 '''(INTERNAL) Get the x-elliptic function.
1024 '''
1025 kp2, k2 = self._k2_kp2 # swapped!
1026 # b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc
1027 return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2)
1029 def yR(self, beta):
1030 '''Compute a Jacobi Conformal C{y} projection.
1032 @arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}).
1034 @return: The C{y} projection (L{Radians}).
1035 '''
1036 return self.yR_(*SinCos2(beta))
1038 def yR_(self, sbeta, cbeta):
1039 '''Compute a Jacobi Conformal C{y} projection.
1041 @arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}).
1042 @arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}).
1044 @return: The C{y} projection (L{Radians}).
1045 '''
1046 s, c = self._norm2(sbeta, cbeta, self.c)
1047 return Radians(y=self._yE.fPi(s, c) * self._c2_b2)
1050class JacobiConformalSpherical(JacobiConformal):
1051 '''An alternate, I{spherical} L{JacobiConformal} projection.
1053 @see: L{JacobiConformal} for other and more details.
1054 '''
1055 _ab = _bc = 0
1057 def __init__(self, radius_triaxial, ab=0, bc=0, **name):
1058 '''New L{JacobiConformalSpherical}.
1060 @arg radius_triaxial: Radius (C{scalar}, conventionally in
1061 C{meter}) or an other L{JacobiConformalSpherical},
1062 L{JacobiConformal} or ordered L{Triaxial}.
1063 @kwarg ab: Relative magnitude of C{B{a} - B{b}} (C{meter},
1064 same units as C{scalar B{radius}}.
1065 @kwarg bc: Relative magnitude of C{B{b} - B{c}} (C{meter},
1066 same units as C{scalar B{radius}}.
1067 @kwarg name: Optional C{B{name}=NN} (C{str}).
1069 @raise TriaxialError: Invalid B{C{radius_triaxial}}, negative
1070 B{C{ab}}, negative B{C{bc}} or C{(B{ab}
1071 + B{bc})} not positive.
1073 @note: If B{C{radius_triaxial}} is a L{JacobiConformalSpherical}
1074 and if B{C{ab}} and B{C{bc}} are both zero or C{None},
1075 the B{C{radius_triaxial}}'s C{ab}, C{bc}, C{a}, C{b}
1076 and C{c} are copied.
1077 '''
1078 try:
1079 r = radius_triaxial
1080 if isinstance(r, Triaxial): # ordered only
1081 t = r._abc3
1082 j = isinstance(r, JacobiConformalSpherical) and not bool(ab or bc)
1083 else:
1084 t = (Radius(radius=r),) * 3
1085 j = False
1086 self._ab = r.ab if j else Scalar_(ab=ab) # low=0
1087 self._bc = r.bc if j else Scalar_(bc=bc) # low=0
1088 if (self.ab + self.bc) <= 0:
1089 raise ValueError(_negative_)
1090 a, _, c = self._abc3 = t
1091 if not (a >= c and isfinite(self._a2b2)
1092 and isfinite(self._a2c2)):
1093 raise ValueError(_not_(_finite_))
1094 except (TypeError, ValueError) as x:
1095 raise TriaxialError(radius_triaxial=r, ab=ab, bc=bc, cause=x)
1096 if name:
1097 self.name = name
1099 @Property_RO
1100 def ab(self):
1101 '''Get relative magnitude C{ab} (C{meter}, same units as B{C{a}}).
1102 '''
1103 return self._ab
1105 @Property_RO
1106 def _a2b2(self):
1107 '''(INTERNAL) Get C{a**2 - b**2} == ab * (a + b).
1108 '''
1109 a, b, _ = self._abc3
1110 return self.ab * (a + b)
1112 @Property_RO
1113 def _a2c2(self):
1114 '''(INTERNAL) Get C{a**2 - c**2} == a2b2 + b2c2.
1115 '''
1116 return self._a2b2 + self._b2c2
1118 @Property_RO
1119 def bc(self):
1120 '''Get relative magnitude C{bc} (C{meter}, same units as B{C{a}}).
1121 '''
1122 return self._bc
1124 @Property_RO
1125 def _b2c2(self):
1126 '''(INTERNAL) Get C{b**2 - c**2} == bc * (b + c).
1127 '''
1128 _, b, c = self._abc3
1129 return self.bc * (b + c)
1131 @Property_RO
1132 def radius(self):
1133 '''Get radius (C{meter}, conventionally).
1134 '''
1135 return self.a
1138class TriaxialError(_ValueError):
1139 '''Raised for L{Triaxial} issues.
1140 '''
1141 pass # ...
1144class Triaxials(_NamedEnum):
1145 '''(INTERNAL) L{Triaxial} registry, I{must} be a sub-class
1146 to accommodate the L{_LazyNamedEnumItem} properties.
1147 '''
1148 def _Lazy(self, *abc, **name):
1149 '''(INTERNAL) Instantiate the C{Triaxial}.
1150 '''
1151 a, b, c = map(km2m, abc)
1152 return Triaxial(a, b, c, **name)
1154Triaxials = Triaxials(Triaxial, Triaxial_) # PYCHOK singleton
1155'''Some pre-defined L{Triaxial}s, all I{lazily} instantiated.'''
1156# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in Km
1157# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf>
1158# <https://link.Springer.com/article/10.1007/s00190-022-01650-9>
1159_abc84_35 = (_EWGS84.a + 35), (_EWGS84.a - 35), _EWGS84.b
1160Triaxials._assert( # a (Km) b (Km) c (Km) planet
1161 Amalthea = _lazy('Amalthea', 125.0, 73.0, 64), # Jupiter
1162 Ariel = _lazy('Ariel', 581.1, 577.9, 577.7), # Uranus
1163 Earth = _lazy('Earth', 6378.173435, 6378.1039, 6356.7544),
1164 Enceladus = _lazy('Enceladus', 256.6, 251.4, 248.3), # Saturn
1165 Europa = _lazy('Europa', 1564.13, 1561.23, 1560.93), # Jupiter
1166 Io = _lazy('Io', 1829.4, 1819.3, 1815.7), # Jupiter
1167 Mars = _lazy('Mars', 3394.6, 3393.3, 3376.3),
1168 Mimas = _lazy('Mimas', 207.4, 196.8, 190.6), # Saturn
1169 Miranda = _lazy('Miranda', 240.4, 234.2, 232.9), # Uranus
1170 Moon = _lazy('Moon', 1735.55, 1735.324, 1734.898), # Earth
1171 Tethys = _lazy('Tethys', 535.6, 528.2, 525.8), # Saturn
1172 WGS84_35 = _lazy('WGS84_35', *map(m2km, _abc84_35)))
1173del _abc84_35, _EWGS84
1176def _getitems(items, *indices):
1177 '''(INTERNAL) Get the C{items} at the given I{indices}.
1179 @return: C{Type(items[i] for i in indices)} with
1180 C{Type = type(items)}, any C{type} having
1181 the special method C{__getitem__}.
1182 '''
1183 return type(items)(map(items.__getitem__, indices))
1186def _hartzell3(pov, los, Tun): # in .Ellipsoid.hartzell4, .formy.hartzell
1187 '''(INTERNAL) Hartzell's "Satellite Line-of-Sight Intersection ...",
1188 formula from a Point-Of-View to an I{un-/ordered} Triaxial.
1189 '''
1190 def _toUvwV3d(los, pov):
1191 try: # pov must be LatLon or Cartesian if los is a Los
1192 v = los.toUvw(pov)
1193 except (AttributeError, TypeError):
1194 v = _otherV3d(los=los)
1195 return v
1197 p3 = _otherV3d(pov=pov.toCartesian() if isLatLon(pov) else pov)
1198 if los is True: # normal
1199 a, b, c, d, i = _plumbTo5(p3.x, p3.y, p3.z, Tun)
1200 return type(p3)(a, b, c), d, i
1202 u3 = p3.negate() if los is False or los is None else _toUvwV3d(los, pov)
1204 a, b, c, T = Tun._ordered4
1206 a2 = a**2 # largest, factored out
1207 b2, p2 = (b**2, T._1e2ab) if b != a else (a2, _1_0)
1208 c2, q2 = (c**2, T._1e2ac) if c != a else (a2, _1_0)
1210 p3 = T._order3d(p3)
1211 u3 = T._order3d(u3).unit() # unit vector, opposing signs
1213 x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz
1214 ux, vy, wz = u3.times_(p3).xyz
1215 u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz
1217 t = (p2 * c2), c2, b2
1218 m = fdot(t, u2, v2, w2) # a2 factored out
1219 if m < EPS0: # zero or near-null LOS vector
1220 raise _ValueError(_near_(_null_))
1222 r = fsumf_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2,
1223 -w2 * y2, -u2 * y2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2,
1224 -w2 * x2 * p2, b2 * u2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2)
1225 if r > 0: # a2 factored out
1226 r = sqrt(r) * b * c # == a * a * b * c / a2
1227 elif r < 0: # LOS pointing away from or missing the triaxial
1228 raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_)
1230 d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out
1231 if d > 0: # POV inside or LOS outside or missing the triaxial
1232 s = fsumf_(_N_1_0, x2 / a2, y2 / b2, z2 / c2) # like _sideOf
1233 raise _ValueError(_outside_ if s > 0 else _inside_)
1234 elif fsum1f_(x2, y2, z2) < d**2: # d past triaxial's center
1235 raise _ValueError(_too_(_distant_))
1237 v = p3.minus(u3.times(d)) # cartesian type(pov) or Vector3d
1238 h = p3.minus(v).length # distance to pov == -d
1239 return T._order3d(v, reverse=True), h, None
1242def hartzell4(pov, los=False, tri_biax=_WGS84, **name):
1243 '''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight
1244 from a Point-Of-View outside.
1246 @arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple}
1247 C{LatLon} or L{Vector3d}).
1248 @kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Los}, L{Vector3d}),
1249 C{True} for the I{normal, plumb} onto the surface or C{False} or
1250 C{None} to point to the center of the tri-/biaxial.
1251 @kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal} or
1252 L{JacobiConformalSpherical}) or biaxial ellipsoid (L{Datum},
1253 L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple} or C{scalar} radius,
1254 conventionally in C{meter}).
1255 @kwarg name: Optional, overriding C{B{name}="hartzell4"} (C{str}).
1257 @return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h}
1258 the distance from B{C{pov}} to C{(x, y, z)} I{along the} B{C{los}}, all
1259 in C{meter}, conventionally.
1261 @raise TriaxialError: Invalid B{C{pov}} or B{C{pov}} inside the tri-/biaxial or
1262 invalid B{C{los}} or B{C{los}} points outside or away from
1263 the tri-/biaxial.
1265 @raise TypeError: Invalid B{C{tri_biax}}, C{ellipsoid} or C{datum}.
1267 @see: Class L{pygeodesy3.Los}, functions L{pygeodesy.tyr3d} and L{pygeodesy.hartzell}
1268 and U{lookAtSpheroid<https://PyPI.org/project/pymap3d>} and U{"Satellite
1269 Line-of-Sight Intersection with Earth"<https://StephenHartzell.Medium.com/
1270 satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}.
1271 '''
1272 if isinstance(tri_biax, Triaxial_):
1273 T = tri_biax
1274 else:
1275 D = tri_biax if isinstance(tri_biax, Datum) else \
1276 _spherical_datum(tri_biax, name__=hartzell4) # _dunder_nameof
1277 T = D.ellipsoid._triaxial
1278 try:
1279 v, h, i = _hartzell3(pov, los, T)
1280 except Exception as x:
1281 raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x)
1282 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, # _dunder_nameof
1283 name=_name__(name, name__=hartzell4))
1286def _hypot21(x, y, z=0):
1287 '''(INTERNAL) Compute M{x**2 + y**2 + z**2 - 1} with C{max(fabs(x),
1288 fabs(y), fabs(z))} rarely greater than 1.0.
1289 '''
1290 return fsumf_(_1_0, x**2, y**2, (z**2 if z else _0_0), _N_2_0)
1293def _otherV3d_(x_xyz, y, z, **name):
1294 '''(INTERNAL) Get a Vector3d from C{x_xyz}, C{y} and C{z}.
1295 '''
1296 return Vector3d(x_xyz, y, z, **name) if isscalar(x_xyz) else \
1297 _otherV3d(x_xyz=x_xyz, **name)
1300def _plumbTo4(x, y, a, b, eps=EPS):
1301 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}.
1303 @see: Function C{pygeodesy.ellipsoids._plumbTo3} and I{Eberly}'s U{Distance
1304 from a Point to ... an Ellipse ...<https://www.GeometricTools.com/
1305 Documentation/DistancePointEllipseEllipsoid.pdf>}.
1306 '''
1307 if b > a:
1308 b, a, d, i = _plumbTo4(y, x, b, a, eps=eps)
1309 return a, b, d, i
1311 if not (b > 0 and isfinite(a)):
1312 raise _ValueError(a=a, b=b)
1314 i, _a = None, fabs
1315 if y:
1316 if x:
1317 u = _a(x / a)
1318 v = _a(y / b)
1319 g = _hypot21(u, v)
1320 if _a(g) < EPS02: # on the ellipse
1321 a, b, d = x, y, _0_0
1322 else:
1323 r = (b / a)**2
1324 t, i = _rootXd(_1_0 / r, 0, u, 0, v, g, eps)
1325 a = x / (t * r + _1_0)
1326 b = y / (t + _1_0)
1327 d = hypot(x - a, y - b)
1328 else: # x == 0
1329 if y < 0:
1330 b = -b
1331 a, d = x, _a(y - b)
1333 else: # y == 0
1334 n = a * x
1335 d = (a + b) * (a - b)
1336 if d > _a(n): # PYCHOK no cover
1337 r = n / d
1338 a *= r
1339 r = _1_0 - r**2
1340 if r > EPS02:
1341 b *= sqrt(r)
1342 d = hypot(x - a, b)
1343 else:
1344 b = _0_0
1345 d = _a(x - a)
1346 else:
1347 if x < 0:
1348 a = -a
1349 b, d = y, _a(x - a)
1350 return a, b, d, i
1353def _plumbTo5(x, y, z, Tun, eps=EPS): # MCCABE 19 in .testTriaxials
1354 '''(INTERNAL) Nearest point on and distance to an I{un-/ordered} triaxial.
1356 @see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https://
1357 www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}.
1358 '''
1359 a, b, c, T = Tun._ordered4
1360 if Tun is not T: # T is ordered, Tun isn't
1361 t = T._order3(x, y, z) + (T,)
1362 a, b, c, d, i = _plumbTo5(*t, eps=eps)
1363 return T._order3(a, b, c, reverse=True) + (d, i)
1365 if not (c > 0 and isfinite(a)):
1366 raise _ValueError(a=a, b=b, c=c)
1368 if eps > 0:
1369 val = max(eps * 1e8, EPS)
1370 else: # no validation
1371 val, eps = 0, max(EPS0, -eps)
1373 i, _a = None, fabs
1374 if z:
1375 if y:
1376 if x:
1377 u = _a(x / a)
1378 v = _a(y / b)
1379 w = _a(z / c)
1380 g = _hypot21(u, v, w)
1381 if _a(g) < EPS02: # on the ellipsoid
1382 a, b, c, d = x, y, z, _0_0
1383 else:
1384 r = T._1e2ac # (c / a)**2
1385 s = T._1e2bc # (c / b)**2
1386 t, i = _rootXd(_1_0 / r, _1_0 / s, u, v, w, g, eps)
1387 a = x / (t * r + _1_0)
1388 b = y / (t * s + _1_0)
1389 c = z / (t + _1_0)
1390 d = hypot_(x - a, y - b, z - c)
1391 else: # x == 0
1392 a = x # 0
1393 b, c, d, i = _plumbTo4(y, z, b, c, eps=eps)
1394 elif x: # y == 0
1395 b = y # 0
1396 a, c, d, i = _plumbTo4(x, z, a, c, eps=eps)
1397 else: # x == y == 0
1398 if z < 0:
1399 c = -c
1400 a, b, d = x, y, _a(z - c)
1402 else: # z == 0
1403 t = True
1404 d = T._a2c2 # (a + c) * (a - c)
1405 n = a * x
1406 if d > _a(n):
1407 u = n / d
1408 d = T._b2c2 # (b + c) * (b - c)
1409 n = b * y
1410 if d > _a(n):
1411 v = n / d
1412 n = _hypot21(u, v)
1413 if n < 0:
1414 a *= u
1415 b *= v
1416 c *= sqrt(-n)
1417 d = hypot_(x - a, y - b, c)
1418 t = False
1419 if t:
1420 c = z # signed-0
1421 a, b, d, i = _plumbTo4(x, y, a, b, eps=eps)
1423 if val > 0: # validate
1424 e = T.sideOf(a, b, c, eps=val)
1425 if e: # not near the ellipsoid's surface
1426 raise _ValueError(a=a, b=b, c=c, d=d,
1427 sideOf=e, eps=val)
1428 if d: # angle of delta and normal vector
1429 m = Vector3d(x, y, z).minus_(a, b, c)
1430 if m.euclid > val:
1431 m = m.unit()
1432 n = T.normal3d(a, b, c)
1433 e = n.dot(m) # n.negate().dot(m)
1434 if not isnear1(_a(e), eps1=val):
1435 raise _ValueError(n=n, m=m,
1436 dot=e, eps=val)
1437 return a, b, c, d, i
1440def _rootXd(r, s, u, v, w, g, eps):
1441 '''(INTERNAL) Robust 2d- or 3d-root finder: 2d- if C{s == v == 0} else 3d-root.
1443 @see: I{Eberly}'s U{Robust Root Finders ...<https://www.GeometricTools.com/
1444 Documentation/DistancePointEllipseEllipsoid.pdf>}.
1445 '''
1446 _1, __2 = _1_0, _0_5
1447 _a, _h21 = fabs, _hypot21
1449 u *= r
1450 v *= s # 0 for 2d-root
1451 t0 = w - _1
1452 t1 = _0_0 if g < 0 else (hypot_(u, w, v) - _1)
1453 # assert t0 <= t1
1454 for i in range(1, _TRIPS): # 48-55
1455 e = _a(t0 - t1)
1456 if e < eps:
1457 break
1458 t = (t0 + t1) * __2
1459 if t in (t0, t1):
1460 break
1461 g = _h21(u / (t + r), w / (t + _1),
1462 (v / (t + s)) if v else 0)
1463 if g > 0:
1464 t0 = t
1465 elif g < 0:
1466 t1 = t
1467 else:
1468 break
1469 else: # PYCHOK no cover
1470 t = Fmt.no_convergence(e, eps)
1471 raise _ValueError(t, txt__=_rootXd)
1472 return t, i
1475def _sideOf(xyz, abc, eps=EPS):
1476 '''(INTERNAL) Helper for C{_hartzell3}, M{.sideOf} and M{.reverseCartesian}.
1478 @return: M{sum((x / a)**2 for x, a in zip(xyz, abc)) - 1} or C{INT0}.
1479 '''
1480 s = fsumf_(_N_1_0, *((x / a)**2 for x, a in _zip(xyz, abc) if a)) # strict=True
1481 return INT0 if fabs(s) < eps else s
1484if __name__ == '__main__':
1486 from pygeodesy import printf
1487 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_
1489 # __doc__ of this file, force all into registery
1490 t = [NN] + Triaxials.toRepr(all=True, asorted=True).split(_NL_)
1491 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))
1493# **) MIT License
1494#
1495# Copyright (C) 2022-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1496#
1497# Permission is hereby granted, free of charge, to any person obtaining a
1498# copy of this software and associated documentation files (the "Software"),
1499# to deal in the Software without restriction, including without limitation
1500# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1501# and/or sell copies of the Software, and to permit persons to whom the
1502# Software is furnished to do so, subject to the following conditions:
1503#
1504# The above copyright notice and this permission notice shall be included
1505# in all copies or substantial portions of the Software.
1506#
1507# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1508# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1509# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1510# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1511# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1512# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1513# OTHER DEALINGS IN THE SOFTWARE.