Coverage for pygeodesy/ellipsoids.py: 97%
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« prev ^ index » next coverage.py v7.6.1, created at 2025-01-06 12:20 -0500
2# -*- coding: utf-8 -*-
4u'''Ellipsoidal and spherical earth models.
6Classes L{a_f2Tuple}, L{Ellipsoid} and L{Ellipsoid2}, an L{Ellipsoids} registry and
72 dozen functions to convert I{equatorial} radius, I{polar} radius, I{eccentricities},
8I{flattenings} and I{inverse flattening}.
10See module L{datums} for L{Datum} and L{Transform} information and other details.
12Following is the list of predefined L{Ellipsoid}s, all instantiated lazily.
14@var Ellipsoids.Airy1830: Ellipsoid(name='Airy1830', a=6377563.396, f=0.00334085, f_=299.3249646, b=6356256.90923729)
15@var Ellipsoids.AiryModified: Ellipsoid(name='AiryModified', a=6377340.189, f=0.00334085, f_=299.3249646, b=6356034.44793853)
16@var Ellipsoids.ATS1977: Ellipsoid(name='ATS1977', a=6378135, f=0.00335281, f_=298.257, b=6356750.30492159)
17@var Ellipsoids.Australia1966: Ellipsoid(name='Australia1966', a=6378160, f=0.00335289, f_=298.25, b=6356774.71919531)
18@var Ellipsoids.Bessel1841: Ellipsoid(name='Bessel1841', a=6377397.155, f=0.00334277, f_=299.1528128, b=6356078.962818)
19@var Ellipsoids.BesselModified: Ellipsoid(name='BesselModified', a=6377492.018, f=0.00334277, f_=299.1528128, b=6356173.5087127)
20@var Ellipsoids.CGCS2000: Ellipsoid(name='CGCS2000', a=6378137, f=0.00335281, f_=298.2572221, b=6356752.31414036)
21@var Ellipsoids.Clarke1866: Ellipsoid(name='Clarke1866', a=6378206.4, f=0.00339008, f_=294.97869821, b=6356583.8)
22@var Ellipsoids.Clarke1880: Ellipsoid(name='Clarke1880', a=6378249.145, f=0.00340756, f_=293.465, b=6356514.86954978)
23@var Ellipsoids.Clarke1880IGN: Ellipsoid(name='Clarke1880IGN', a=6378249.2, f=0.00340755, f_=293.46602129, b=6356515)
24@var Ellipsoids.Clarke1880Mod: Ellipsoid(name='Clarke1880Mod', a=6378249.145, f=0.00340755, f_=293.46630766, b=6356514.96639549)
25@var Ellipsoids.CPM1799: Ellipsoid(name='CPM1799', a=6375738.7, f=0.00299052, f_=334.39, b=6356671.92557493)
26@var Ellipsoids.Delambre1810: Ellipsoid(name='Delambre1810', a=6376428, f=0.00321027, f_=311.5, b=6355957.92616372)
27@var Ellipsoids.Engelis1985: Ellipsoid(name='Engelis1985', a=6378136.05, f=0.00335282, f_=298.2566, b=6356751.32272154)
28@var Ellipsoids.Everest1969: Ellipsoid(name='Everest1969', a=6377295.664, f=0.00332445, f_=300.8017, b=6356094.667915)
29@var Ellipsoids.Everest1975: Ellipsoid(name='Everest1975', a=6377299.151, f=0.00332445, f_=300.8017255, b=6356098.14512013)
30@var Ellipsoids.Fisher1968: Ellipsoid(name='Fisher1968', a=6378150, f=0.00335233, f_=298.3, b=6356768.33724438)
31@var Ellipsoids.GEM10C: Ellipsoid(name='GEM10C', a=6378137, f=0.00335281, f_=298.2572236, b=6356752.31424783)
32@var Ellipsoids.GPES: Ellipsoid(name='GPES', a=6378135, f=0, f_=0, b=6378135)
33@var Ellipsoids.GRS67: Ellipsoid(name='GRS67', a=6378160, f=0.00335292, f_=298.24716743, b=6356774.51609071)
34@var Ellipsoids.GRS80: Ellipsoid(name='GRS80', a=6378137, f=0.00335281, f_=298.2572221, b=6356752.31414035)
35@var Ellipsoids.Helmert1906: Ellipsoid(name='Helmert1906', a=6378200, f=0.00335233, f_=298.3, b=6356818.16962789)
36@var Ellipsoids.IAU76: Ellipsoid(name='IAU76', a=6378140, f=0.00335281, f_=298.257, b=6356755.28815753)
37@var Ellipsoids.IERS1989: Ellipsoid(name='IERS1989', a=6378136, f=0.00335281, f_=298.257, b=6356751.30156878)
38@var Ellipsoids.IERS1992TOPEX: Ellipsoid(name='IERS1992TOPEX', a=6378136.3, f=0.00335281, f_=298.25722356, b=6356751.61659215)
39@var Ellipsoids.IERS2003: Ellipsoid(name='IERS2003', a=6378136.6, f=0.00335282, f_=298.25642, b=6356751.85797165)
40@var Ellipsoids.Intl1924: Ellipsoid(name='Intl1924', a=6378388, f=0.003367, f_=297, b=6356911.94612795)
41@var Ellipsoids.Intl1967: Ellipsoid(name='Intl1967', a=6378157.5, f=0.0033529, f_=298.24961539, b=6356772.2)
42@var Ellipsoids.Krassovski1940: Ellipsoid(name='Krassovski1940', a=6378245, f=0.00335233, f_=298.3, b=6356863.01877305)
43@var Ellipsoids.Krassowsky1940: Ellipsoid(name='Krassowsky1940', a=6378245, f=0.00335233, f_=298.3, b=6356863.01877305)
44@var Ellipsoids.Maupertuis1738: Ellipsoid(name='Maupertuis1738', a=6397300, f=0.0052356, f_=191, b=6363806.28272251)
45@var Ellipsoids.Mercury1960: Ellipsoid(name='Mercury1960', a=6378166, f=0.00335233, f_=298.3, b=6356784.28360711)
46@var Ellipsoids.Mercury1968Mod: Ellipsoid(name='Mercury1968Mod', a=6378150, f=0.00335233, f_=298.3, b=6356768.33724438)
47@var Ellipsoids.NWL1965: Ellipsoid(name='NWL1965', a=6378145, f=0.00335289, f_=298.25, b=6356759.76948868)
48@var Ellipsoids.OSU86F: Ellipsoid(name='OSU86F', a=6378136.2, f=0.00335281, f_=298.2572236, b=6356751.51693008)
49@var Ellipsoids.OSU91A: Ellipsoid(name='OSU91A', a=6378136.3, f=0.00335281, f_=298.2572236, b=6356751.6165948)
50@var Ellipsoids.Plessis1817: Ellipsoid(name='Plessis1817', a=6376523, f=0.00324002, f_=308.64, b=6355862.93325557)
51@var Ellipsoids.PZ90: Ellipsoid(name='PZ90', a=6378136, f=0.0033528, f_=298.2578393, b=6356751.36174571)
52@var Ellipsoids.SGS85: Ellipsoid(name='SGS85', a=6378136, f=0.00335281, f_=298.257, b=6356751.30156878)
53@var Ellipsoids.SoAmerican1969: Ellipsoid(name='SoAmerican1969', a=6378160, f=0.00335289, f_=298.25, b=6356774.71919531)
54@var Ellipsoids.Sphere: Ellipsoid(name='Sphere', a=6371008.771415, f=0, f_=0, b=6371008.771415)
55@var Ellipsoids.SphereAuthalic: Ellipsoid(name='SphereAuthalic', a=6371000, f=0, f_=0, b=6371000)
56@var Ellipsoids.SpherePopular: Ellipsoid(name='SpherePopular', a=6378137, f=0, f_=0, b=6378137)
57@var Ellipsoids.Struve1860: Ellipsoid(name='Struve1860', a=6378298.3, f=0.00339294, f_=294.73, b=6356657.14266956)
58@var Ellipsoids.WGS60: Ellipsoid(name='WGS60', a=6378165, f=0.00335233, f_=298.3, b=6356783.28695944)
59@var Ellipsoids.WGS66: Ellipsoid(name='WGS66', a=6378145, f=0.00335289, f_=298.25, b=6356759.76948868)
60@var Ellipsoids.WGS72: Ellipsoid(name='WGS72', a=6378135, f=0.00335278, f_=298.26, b=6356750.52001609)
61@var Ellipsoids.WGS84: Ellipsoid(name='WGS84', a=6378137, f=0.00335281, f_=298.25722356, b=6356752.31424518)
62@var Ellipsoids.WGS84_NGS: Ellipsoid(name='WGS84_NGS', a=6378137, f=0.00335281, f_=298.2572221, b=6356752.31414035)
63'''
64# make sure int/int division yields float quotient, see .basics
65from __future__ import division as _; del _ # PYCHOK semicolon
67from pygeodesy.basics import copysign0, isbool, isint
68from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, INF, NINF, PI4, PI_2, PI_3, R_M, R_MA, R_FM, \
69 _EPSqrt, _EPStol as _TOL, _floatuple as _T, _isfinite, \
70 _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \
71 _0_25, _3_0 # PYCHOK used!
72from pygeodesy.errors import _AssertionError, IntersectionError, _ValueError, _xattr, _xkwds_not
73from pygeodesy.fmath import cbrt, cbrt2, fdot, Fhorner, fpowers, hypot, hypot_, \
74 hypot1, hypot2, sqrt3, Fsum
75# from pygeodesy.fsums import Fsum # from .fmath
76from pygeodesy.interns import NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, _beta_, \
77 _Clarke1866_, _Clarke1880IGN_, _DOT_, _f_, _GRS80_, _height_, \
78 _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \
79 _Krassowsky1940_, _lat_, _meridional_, _negative_, _not_finite_, \
80 _prime_vertical_, _radius_, _Sphere_, _SPACE_, _vs_, _WGS72_, _WGS84_
81# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .named
82from pygeodesy.named import _lazyNamedEnumItem as _lazy, _name__, _NamedEnum, \
83 _NamedEnumItem, _NamedTuple, _Pass, _ALL_LAZY, _MODS
84from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple
85from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_, \
86 deprecated_property_RO, property_RO, property_ROver
87from pygeodesy.streprs import Fmt, fstr, instr, strs, unstr
88# from pygeodesy.triaxials import _hartzell3 # _MODS
89from pygeodesy.units import Bearing_, Distance, Float, Float_, Height, Lamd, Lat, Meter, \
90 Meter2, Meter3, Phi, Phid, Radius, Radius_, Scalar
91from pygeodesy.utily import atan1, atan1d, atan2b, degrees90, m2radians, radians2m, sincos2d
93from math import asinh, atan, atanh, cos, degrees, exp, fabs, radians, sin, sinh, sqrt, tan # as _tan
95__all__ = _ALL_LAZY.ellipsoids
96__version__ = '24.11.26'
98_f_0_0 = Float(f =_0_0) # zero flattening
99_f__0_0 = Float(f_=_0_0) # zero inverse flattening
100# see U{WGS84_f<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Constants.html>}
101_f__WGS84 = Float(f_=_1_0 / (1000000000 / 298257223563)) # 298.25722356299997 vs 298.257223563
104def _aux(lat, inverse, auxLat, clip=90):
105 '''Return a named auxiliary latitude in C{degrees}.
106 '''
107 return Lat(lat, clip=clip, name=_lat_ if inverse else auxLat.__name__)
110def _s2_c2(phi):
111 '''(INTERNAL) Return 2-tuple C{(sin(B{phi})**2, cos(B{phi})**2)}.
112 '''
113 if phi:
114 s2 = sin(phi)**2
115 if s2 > EPS:
116 c2 = _1_0 - s2
117 if c2 > EPS:
118 if c2 < EPS1:
119 return s2, c2
120 else:
121 return _1_0, _0_0 # phi == PI_2
122 return _0_0, _1_0 # phi == 0
125class a_f2Tuple(_NamedTuple):
126 '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial}
127 radius C{a} in C{meter} and scalar I{flattening} C{f}.
129 @see: Class L{Ellipsoid2}.
130 '''
131 _Names_ = (_a_, _f_) # name 'f' not 'f_'
132 _Units_ = (_Pass, _Pass)
134 def __new__(cls, a, f, **name):
135 '''New L{a_f2Tuple} ellipsoid specification.
137 @arg a: Equatorial radius (C{scalar} > 0).
138 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
139 @kwarg name: Optional C{B{name}=NN} (C{str}).
141 @return: An L{a_f2Tuple}C{(a, f)} instance.
143 @raise UnitError: Invalid B{C{a}} or B{C{f}}.
145 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.
146 Negative C{B{f}} produces a I{prolate} ellipsoid.
147 '''
148 a = Radius_(a=a) # low=EPS, high=None
149 f = Float_( f=f, low=None, high=EPS1)
150 if fabs(f) < EPS: # force spherical
151 f = _f_0_0
152 return _NamedTuple.__new__(cls, a, f, **name)
154 @Property_RO
155 def b(self):
156 '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}.
157 '''
158 return a_f2b(self.a, self.f) # PYCHOK .a and .f
160 def ellipsoid(self, **name):
161 '''Return an L{Ellipsoid} for this 2-tuple C{(a, f)}.
163 @kwarg name: Optional C{B{name}=NN} (C{str}).
165 @raise NameError: A registered C{ellipsoid} with the
166 same B{C{name}} already exists.
167 '''
168 return Ellipsoid(self.a, f=self.f, name=self._name__(name)) # PYCHOK .a and .f
170 @Property_RO
171 def f_(self):
172 '''Get the I{inverse} flattening (C{scalar}), M{1 / f} == M{a / (a - b)}.
173 '''
174 return f2f_(self.f) # PYCHOK .f
177class Circle4Tuple(_NamedTuple):
178 '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height},
179 both conventionally in C{meter} of a parallel I{circle of latitude} at
180 (geodetic) latitude C{lat} and the I{parametric (or reduced) auxiliary
181 latitude} C{beta}, both in C{degrees90}.
183 The C{height} is the (signed) distance along the z-axis between the
184 parallel and the equator. At near-polar C{lat}s, the C{radius} is C{0},
185 the C{height} is the ellipsoid's (signed) polar radius and C{beta}
186 equals C{lat}.
187 '''
188 _Names_ = (_radius_, _height_, _lat_, _beta_)
189 _Units_ = ( Radius, Height, Lat, Lat)
192class Curvature2Tuple(_NamedTuple):
193 '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in
194 C{meter}, conventionally.
195 '''
196 _Names_ = (_meridional_, _prime_vertical_)
197 _Units_ = ( Meter, Meter)
199 @property_RO
200 def transverse(self):
201 '''Get this I{prime_vertical}, aka I{transverse} radius of curvature.
202 '''
203 return self.prime_vertical
206class Ellipsoid(_NamedEnumItem):
207 '''Ellipsoid with I{equatorial} and I{polar} radii, I{flattening}, I{inverse
208 flattening} and other, often used, I{cached} attributes, supporting
209 I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models.
210 '''
211 _a = 0 # equatorial radius, semi-axis (C{meter})
212 _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f
213 _f = 0 # (1st) flattening: (a - b) / a
214 _f_ = 0 # inverse flattening: 1 / f = a / (a - b)
216 _geodsolve = NN # means, use PYGEODESY_GEODSOLVE
217 _KsOrder = 8 # Krüger series order (4, 6 or 8)
218 _rhumbsolve = NN # means, use PYGEODESY_RHUMBSOLVE
220 def __init__(self, a, b=None, f_=None, f=None, **name):
221 '''New L{Ellipsoid} from the I{equatorial} radius I{and} either
222 the I{polar} radius or I{inverse flattening} or I{flattening}.
224 @arg a: Equatorial radius, semi-axis (C{meter}).
225 @arg b: Optional polar radius, semi-axis (C{meter}).
226 @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0).
227 @arg f: Flattening: M{(a - b) / a} (C{scalar}, near zero for
228 spherical).
229 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
231 @raise NameError: Ellipsoid with the same B{C{name}} already exists.
233 @raise ValueError: Invalid B{C{a}}, B{C{b}}, B{C{f_}} or B{C{f}} or
234 B{C{f_}} and B{C{f}} are incompatible.
236 @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced
237 to M{1 / f_ = 0}, spherical.
238 '''
239 ff_ = f, f_ # assertion below
240 n = _name__(**name) if name else NN
241 try:
242 a = Radius_(a=a) # low=EPS
243 if not _isfinite(a):
244 raise ValueError(_SPACE_(_a_, _not_finite_))
246 if b: # not in (_0_0, None)
247 b = Radius_(b=b) # low=EPS
248 f = a_b2f(a, b) if f is None else Float(f=f)
249 f_ = f2f_(f) if f_ is None else Float(f_=f_)
250 elif f is not None:
251 f = Float(f=f)
252 b = a_f2b(a, f)
253 f_ = f2f_(f) if f_ is None else Float(f_=f_)
254 elif f_:
255 f_ = Float(f_=f_)
256 b = a_f_2b(a, f_) # a * (f_ - 1) / f_
257 f = f_2f(f_)
258 else: # only a, spherical
259 f_ = f = 0
260 b = a # superfluous
262 if not f < _1_0: # sanity check, see .ecef.Ecef.__init__
263 raise ValueError(_SPACE_(_f_, _invalid_))
264 if not _isfinite(b):
265 raise ValueError(_SPACE_(_b_, _not_finite_))
267 if fabs(f) < EPS or a == b or not f_: # spherical
268 b = a
269 f = _f_0_0
270 f_ = _f__0_0
272 except (TypeError, ValueError) as x:
273 d = _xkwds_not(None, b=b, f_=f_, f=f)
274 t = instr(self, a=a, name=n, **d)
275 raise _ValueError(t, cause=x)
277 self._a = a
278 self._b = b
279 self._f = f
280 self._f_ = f_
282 self._register(Ellipsoids, n)
284 if f and f_: # see test/testEllipsoidal
285 d = dict(eps=_TOL)
286 if None in ff_: # both f_ and f given
287 d.update(Error=_ValueError, txt=_incompatible_)
288 self._assert(_1_0 / f, f_=f_, **d)
289 self._assert(_1_0 / f_, f =f, **d)
290 self._assert(self.b2_a2, e21=self.e21, eps=EPS)
292 def __eq__(self, other):
293 '''Compare this and an other ellipsoid.
295 @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}).
297 @return: C{True} if equal, C{False} otherwise.
298 '''
299 return self is other or (isinstance(other, Ellipsoid) and
300 self.a == other.a and
301 (self.f == other.f or self.b == other.b))
303 def __hash__(self):
304 return self._hash # memoized
306 @Property_RO
307 def a(self):
308 '''Get the I{equatorial} radius, semi-axis (C{meter}).
309 '''
310 return self._a
312 equatoradius = a # = Requatorial
314 @Property_RO
315 def a2(self):
316 '''Get the I{equatorial} radius I{squared} (C{meter} I{squared}), M{a**2}.
317 '''
318 return Meter2(a2=self.a**2)
320 @Property_RO
321 def a2_(self):
322 '''Get the inverse of the I{equatorial} radius I{squared} (C{meter} I{squared}), M{1 / a**2}.
323 '''
324 return Float(a2_=_1_0 / self.a2)
326 @Property_RO
327 def a_b(self):
328 '''Get the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b} == M{1 / (1 - f)}.
329 '''
330 return Float(a_b=self.a / self.b if self.f else _1_0)
332 @Property_RO
333 def a2_b(self):
334 '''Get the I{polar} meridional (or polar) radius of curvature (C{meter}), M{a**2 / b}.
336 @see: U{Radii of Curvature
337 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
338 and U{Moritz, H. (1980), Geodetic Reference System 1980
339 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}.
341 @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature
342 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{c2}
343 and L{R2} or L{Rauthalic}.
344 '''
345 return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar
347 @Property_RO
348 def a2_b2(self):
349 '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}),
350 M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e21}.
351 '''
352 return Float(a2_b2=self.a_b**2 if self.f else _1_0)
354 @Property_RO
355 def a_f(self):
356 '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}), see method C{toEllipsoid2}.
357 '''
358 return a_f2Tuple(self.a, self.f, name=self.name)
360 @Property_RO
361 def A(self):
362 '''Get the UTM I{meridional (or rectifying)} radius (C{meter}).
364 @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/>}.
365 '''
366 A, n = self.a, self.n
367 if n:
368 d = (n + _1_0) * 1048576 / A
369 if d: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8
370 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>
371 # <https://GeographicLib.SourceForge.io/C++/doc/transversemercator.html> and
372 # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> (3)
373 # A *= fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441) / 1048576) / (1 + n)
374 A = Radius(A=Fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441).fover(d))
375 return A
377 @Property_RO
378 def _albersCyl(self):
379 '''(INTERNAL) Helper for C{auxAuthalic}.
380 '''
381 return _MODS.albers.AlbersEqualAreaCylindrical(datum=self, name=self.name)
383 @Property_RO
384 def AlphaKs(self):
385 '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
386 '''
387 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon
388 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
389 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200),
390 _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned
391 _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned
392 _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned
393 _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned
394 _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned
395 _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned
396 _T(1424729850961/743921418240)) # PYCHOK unaligned
398 @Property_RO
399 def area(self):
400 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2}.
402 @see: Properties L{areax}, L{c2} and L{R2} and functions
403 L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
404 '''
405 return Meter2(area=self.c2 * PI4)
407 @Property_RO
408 def areax(self):
409 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more
410 accurate for very I{oblate} ellipsoids.
412 @see: Properties L{area}, L{c2x} and L{R2x}, class L{GeodesicExact} and
413 functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
414 '''
415 return Meter2(areax=self.c2x * PI4)
417 def _assert(self, val, eps=_TOL, f0=_0_0, Error=_AssertionError, txt=NN, **name_value):
418 '''(INTERNAL) Assert a C{name=value} vs C{val}.
419 '''
420 for n, v in name_value.items():
421 if fabs(v - val) > eps: # PYCHOK no cover
422 t = (v, _vs_, val)
423 t = _SPACE_.join(strs(t, prec=12, fmt=Fmt.g))
424 t = Fmt.EQUAL(self._DOT_(n), t)
425 raise Error(t, txt=txt or Fmt.exceeds_eps(eps))
426 return Float(v if self.f else f0, name=n)
427 raise Error(unstr(self._DOT_(self._assert.__name__), val,
428 eps=eps, f0=f0, **name_value))
430 def auxAuthalic(self, lat, inverse=False):
431 '''Compute the I{authalic} auxiliary latitude or the I{inverse} thereof.
433 @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}).
434 @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and
435 return the geodetic latitude (C{bool}).
437 @return: The I{authalic} (or geodetic) latitude in C{degrees90}.
439 @see: U{Inverse-/AuthalicLatitude<https://GeographicLib.SourceForge.io/
440 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude
441 <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and
442 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16.
443 '''
444 if self.f:
445 f = self._albersCyl._tanf if inverse else self._albersCyl._txif # PYCHOK attr
446 lat = atan1d(f(tan(Phid(lat)))) # PYCHOK attr
447 return _aux(lat, inverse, Ellipsoid.auxAuthalic)
449 def auxConformal(self, lat, inverse=False):
450 '''Compute the I{conformal} auxiliary latitude or the I{inverse} thereof.
452 @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}).
453 @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and
454 return the geodetic latitude (C{bool}).
456 @return: The I{conformal} (or geodetic) latitude in C{degrees90}.
458 @see: U{Inverse-/ConformalLatitude<https://GeographicLib.SourceForge.io/
459 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude
460 <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and
461 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
462 '''
463 if self.f:
464 f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr
465 lat = atan1d(f(tan(Phid(lat)))) # PYCHOK attr
466 return _aux(lat, inverse, Ellipsoid.auxConformal)
468 def auxGeocentric(self, lat, inverse=False):
469 '''Compute the I{geocentric} auxiliary latitude or the I{inverse} thereof.
471 @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}).
472 @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and
473 return the I{geocentric} latitude (C{bool}).
475 @return: The I{geocentric} (or geodetic) latitude in C{degrees90}.
477 @see: U{Inverse-/GeocentricLatitude<https://GeographicLib.SourceForge.io/
478 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude
479 <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and
480 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18.
481 '''
482 if self.f:
483 f = self.a2_b2 if inverse else self.b2_a2
484 lat = atan1d(f * tan(Phid(lat)))
485 return _aux(lat, inverse, Ellipsoid.auxGeocentric)
487 def auxIsometric(self, lat, inverse=False):
488 '''Compute the I{isometric} auxiliary latitude or the I{inverse} thereof.
490 @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}).
491 @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and
492 return the geodetic latitude (C{bool}).
494 @return: The I{isometric} (or geodetic) latitude in C{degrees}.
496 @note: The I{isometric} latitude for geodetic C{+/-90} is far
497 outside the C{[-90..+90]} range but the inverse
498 thereof is the original geodetic latitude.
500 @see: U{Inverse-/IsometricLatitude<https://GeographicLib.SourceForge.io/
501 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude
502 <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and
503 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
504 '''
505 if self.f:
506 r = Phid(lat, clip=0)
507 lat = degrees(atan1(self.es_tauf(sinh(r))) if inverse else
508 asinh(self.es_taupf(tan(r))))
509 # clip=0, since auxIsometric(+/-90) is far outside [-90..+90]
510 return _aux(lat, inverse, Ellipsoid.auxIsometric, clip=0)
512 def auxParametric(self, lat, inverse=False):
513 '''Compute the I{parametric} auxiliary latitude or the I{inverse} thereof.
515 @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}).
516 @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and
517 return the geodetic latitude (C{bool}).
519 @return: The I{parametric} (or geodetic) latitude in C{degrees90}.
521 @see: U{Inverse-/ParametricLatitude<https://GeographicLib.SourceForge.io/
522 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude
523 <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>},
524 and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18.
525 '''
526 if self.f:
527 lat = self._beta(Lat(lat), inverse=inverse)
528 return _aux(lat, inverse, Ellipsoid.auxParametric)
530 auxReduced = auxParametric # synonymous
532 def auxRectifying(self, lat, inverse=False):
533 '''Compute the I{rectifying} auxiliary latitude or the I{inverse} thereof.
535 @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}).
536 @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and
537 return the geodetic latitude (C{bool}).
539 @return: The I{rectifying} (or geodetic) latitude in C{degrees90}.
541 @see: U{Inverse-/RectifyingLatitude<https://GeographicLib.SourceForge.io/
542 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude
543 <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and
544 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17.
545 '''
546 if self.f:
547 lat = Lat(lat)
548 if 0 < fabs(lat) < _90_0:
549 if inverse:
550 e = self._elliptic_e22
551 d = degrees90(e.fEinv(e.cE * lat / _90_0))
552 lat = self.auxParametric(d, inverse=True)
553 else:
554 lat = _90_0 * self.Llat(lat) / self.L
555 return _aux(lat, inverse, Ellipsoid.auxRectifying)
557 @Property_RO
558 def b(self):
559 '''Get the I{polar} radius, semi-axis (C{meter}).
560 '''
561 return self._b
563 polaradius = b # = Rpolar
565 @Property_RO
566 def b_a(self):
567 '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a == f1 == 1 - f}.
569 @see: Property L{f1}.
570 '''
571 return self._assert(self.b / self.a, b_a=self.f1, f0=_1_0)
573 @Property_RO
574 def b2(self):
575 '''Get the I{polar} radius I{squared} (C{float}), M{b**2}.
576 '''
577 return Meter2(b2=self.b**2)
579 @Property_RO
580 def b2_a(self):
581 '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see C{rocMeridional}C{(0)}.
583 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
584 '''
585 return Radius(b2_a=self.b2 / self.a if self.f else self.b)
587 @Property_RO
588 def b2_a2(self):
589 '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2}
590 == M{(1 - f)**2} == M{1 - e**2} == C{e21}.
591 '''
592 return Float(b2_a2=self.b_a**2 if self.f else _1_0)
594 def _beta(self, lat, inverse=False):
595 '''(INTERNAL) Get the I{parametric (or reduced) auxiliary latitude} or inverse thereof.
596 '''
597 s, c = sincos2d(lat) # like Karney's tand(lat)
598 s *= self.a_b if inverse else self.b_a
599 return atan1d(s, c)
601 @Property_RO
602 def BetaKs(self):
603 '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
604 '''
605 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon
606 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
607 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600),
608 _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned
609 _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned
610 _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned
611 _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned
612 _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne
613 _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned
614 _T(191773887257/3719607091200)) # PYCHOK unaligned
616 @deprecated_Property_RO
617 def c(self): # PYCHOK no cover
618 '''DEPRECATED, use property C{R2} or C{Rauthalic}.'''
619 return self.R2
621 @Property_RO
622 def c2(self):
623 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}).
625 @see: Properties L{c2x}, L{area}, L{R2}, L{Rauthalic}, I{Karney's} U{equation (60)
626 <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area
627 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>},
628 U{Authalic radius<https://WikiPedia.org/wiki/Earth_radius#Authalic_radius>}, U{Surface area
629 <https://WikiPedia.org/wiki/Ellipsoid>} and U{surface area
630 <https://www.Numericana.com/answer/geometry.htm#oblate>}.
631 '''
632 return self._c2f(False)
634 @Property_RO
635 def c2x(self):
636 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}), more accurate for very I{oblate}
637 ellipsoids.
639 @see: Properties L{c2}, L{areax}, L{R2x}, L{Rauthalicx}, class L{GeodesicExact} and I{Karney}'s comments at C++
640 attribute U{GeodesicExact._c2<https://GeographicLib.SourceForge.io/C++/doc/GeodesicExact_8cpp_source.html>}.
641 '''
642 return self._c2f(True)
644 def _c2f(self, c2x):
645 '''(INTERNAL) Helper for C{.c2} and C{.c2x}.
646 '''
647 f, c2 = self.f, self.b2
648 if f:
649 e = self.e
650 if e > EPS0:
651 if f > 0: # .isOblate
652 c2 *= (asinh(sqrt(self.e22abs)) if c2x else atanh(e)) / e
653 elif f < 0: # .isProlate
654 c2 *= atan1(e) / e # XXX asin?
655 c2 = Meter2(c2=(self.a2 + c2) * _0_5)
656 return c2
658 def circle4(self, lat):
659 '''Get the equatorial or a parallel I{circle of latitude}.
661 @arg lat: Geodetic latitude (C{degrees90}, C{str}).
663 @return: A L{Circle4Tuple}C{(radius, height, lat, beta)}.
665 @raise RangeError: Latitude B{C{lat}} outside valid range and
666 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
668 @raise TypeError: Invalid B{C{lat}}.
670 @raise ValueError: Invalid B{C{lat}}.
672 @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude}
673 <https://WikiPedia.org/wiki/Latitude>}, I{Karney's} C++ U{CircleRadius and CircleHeight
674 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>}
675 and method C{Rlat}.
676 '''
677 lat = Lat(lat)
678 if lat:
679 b = lat
680 if fabs(lat) < _90_0:
681 if self.f:
682 b = self._beta(lat)
683 z, r = sincos2d(b)
684 r *= self.a
685 z *= self.b
686 else: # near-polar
687 r, z = _0_0, copysign0(self.b, lat)
688 else: # equator
689 r = self.a
690 z = lat = b = _0_0
691 return Circle4Tuple(r, z, lat, b)
693 def degrees2m(self, deg, lat=0):
694 '''Convert an angle to the distance along the equator or
695 along a parallel of (geodetic) latitude.
697 @arg deg: The angle (C{degrees}).
698 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
700 @return: Distance (C{meter}, same units as the equatorial
701 and polar radii) or C{0} for near-polar B{C{lat}}.
703 @raise RangeError: Latitude B{C{lat}} outside valid range and
704 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
706 @raise ValueError: Invalid B{C{deg}} or B{C{lat}}.
707 '''
708 return self.radians2m(radians(deg), lat=lat)
710 def distance2(self, lat0, lon0, lat1, lon1):
711 '''I{Approximate} the distance and (initial) bearing between
712 two points based on the U{local, flat earth approximation
713 <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny
714 <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
716 I{Suitable only for distances of several hundred Km or Miles
717 and only between points not near-polar}.
719 @arg lat0: From latitude (C{degrees}).
720 @arg lon0: From longitude (C{degrees}).
721 @arg lat1: To latitude (C{degrees}).
722 @arg lon1: To longitude (C{degrees}).
724 @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance}
725 in same units as this ellipsoid's axes.
727 @note: The meridional and prime_vertical radii of curvature are
728 taken and scaled I{at the initial latitude}, see C{roc2}.
730 @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}.
731 '''
732 phi0 = Phid(lat0=lat0)
733 m, n = self.roc2_(phi0, scaled=True)
734 m *= Phid(lat1=lat1) - phi0
735 n *= Lamd(lon1=lon1) - Lamd(lon0=lon0)
736 return Distance2Tuple(hypot(m, n), atan2b(n, m))
738 @Property_RO
739 def e(self):
740 '''Get the I{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}.
742 @see: Property L{es}.
743 '''
744 return Float(e=sqrt(self.e2abs) if self.e2 else _0_0)
746 @deprecated_Property_RO
747 def e12(self): # see property ._e12
748 '''DEPRECATED, use property C{e21}.'''
749 return self.e21
751# @Property_RO
752# def _e12(self): # see property ._elliptic_e12
753# # (INTERNAL) until e12 above can be replaced with e21.
754# return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2)
756 @Property_RO
757 def e2(self):
758 '''Get the I{signed, (1st) eccentricity squared} (C{float}), M{f * (2 - f)
759 == 1 - (b / a)**2}, see C{a_b2e2}.
760 '''
761 return self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f))
763 @Property_RO
764 def e2abs(self):
765 '''Get the I{unsigned, (1st) eccentricity squared} (C{float}).
766 '''
767 return fabs(self.e2)
769 @Property_RO
770 def e21(self):
771 '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2}
772 == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}.
773 '''
774 return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0)
776# _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2
777 _1_e21 = a2_b2 # == M{1 / e21} == M{1 / (1 - e**2)}
779 @Property_RO
780 def e22(self):
781 '''Get the I{signed, 2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)
782 == e2 / (1 - f)**2 == (a / b)**2 - 1}, see C{a_b2e22}.
783 '''
784 return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f))
786 @Property_RO
787 def e22abs(self):
788 '''Get the I{unsigned, 2nd eccentricity squared} (C{float}).
789 '''
790 return fabs(self.e22)
792 @Property_RO
793 def e32(self):
794 '''Get the I{signed, 3rd eccentricity squared} (C{float}), M{e2 / (2 - e2)
795 == (a**2 - b**2) / (a**2 + b**2)}, see C{a_b2e32}.
796 '''
797 return self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f))
799 @Property_RO
800 def e32abs(self):
801 '''Get the I{unsigned, 3rd eccentricity squared} (C{float}).
802 '''
803 return fabs(self.e32)
805 @Property_RO
806 def e4(self):
807 '''Get the I{unsignd, (1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}.
808 '''
809 return Float(e4=self.e2**2 if self.e2 else _0_0)
811 eccentricity = e # eccentricity
812# eccentricity2 = e2 # eccentricity squared
813 eccentricity1st2 = e2 # first eccentricity squared, signed
814 eccentricity2nd2 = e22 # second eccentricity squared, signed
815 eccentricity3rd2 = e32 # third eccentricity squared, signed
817 def ecef(self, Ecef=None):
818 '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter.
820 @kwarg Ecef: ECEF class to use, default L{EcefKarney}.
822 @return: An ECEF converter for this C{ellipsoid}.
824 @raise TypeError: Invalid B{C{Ecef}}.
826 @see: Module L{pygeodesy.ecef}.
827 '''
828 return _MODS.ecef._4Ecef(self, Ecef)
830 @Property_RO
831 def _elliptic_e12(self): # see I{Karney}'s Ellipsoid._e12
832 '''(INTERNAL) Elliptic helper for C{Rhumb}.
833 '''
834 e12 = self.e2 / (self.e2 - _1_0) # NOT DEPRECATED .e12!
835 return _MODS.elliptic.Elliptic(e12)
837 @Property_RO
838 def _elliptic_e22(self): # aka ._elliptic_ep2
839 '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}.
840 '''
841 return _MODS.elliptic.Elliptic(-self.e22abs) # complex
843 equatoradius = a # Requatorial
845 def e2s(self, s):
846 '''Compute norm M{sqrt(1 - e2 * s**2)}.
848 @arg s: Sine value (C{scalar}).
850 @return: Norm (C{float}).
852 @raise ValueError: Invalid B{C{s}}.
853 '''
854 return sqrt(self.e2s2(s)) if self.e2 else _1_0
856 def e2s2(self, s):
857 '''Compute M{1 - e2 * s**2}.
859 @arg s: Sine value (C{scalar}).
861 @return: Result (C{float}).
863 @raise ValueError: Invalid B{C{s}}.
864 '''
865 r = _1_0
866 if self.e2:
867 try:
868 r -= self.e2 * Scalar(s=s)**2
869 if r < 0:
870 raise ValueError(_negative_)
871 except (TypeError, ValueError) as x:
872 t = self._DOT_(Ellipsoid.e2s2.__name__)
873 raise _ValueError(t, s, cause=x)
874 return r
876 @Property_RO
877 def es(self):
878 '''Get the I{signed (1st) eccentricity} (C{float}).
880 @see: Property L{e}.
881 '''
882 # note, self.e is always non-negative
883 return Float(es=copysign0(self.e, self.f)) # see .ups
885 def es_atanh(self, x):
886 '''Compute M{es * atanh(es * x)} or M{-es * atan(es * x)}
887 for I{oblate} respectively I{prolate} ellipsoids where
888 I{es} is the I{signed} (1st) eccentricity.
890 @raise ValueError: Invalid B{C{x}}.
892 @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/
893 C++/doc/classGeographicLib_1_1Math.html>}.
894 '''
895 return self._es_atanh(Scalar(x=x)) if self.f else _0_0
897 def _es_atanh(self, x): # see .albers._atanhee, .AuxLat._atanhee
898 '''(INTERNAL) Helper for .es_atanh, ._es_taupf2 and ._exp_es_atanh.
899 '''
900 es = self.es # signOf(es) == signOf(f)
901 return es * (atanh(es * x) if es > 0 else # .isOblate
902 (-atan(es * x) if es < 0 else # .isProlate
903 _0_0)) # .isSpherical
905 @Property_RO
906 def es_c(self):
907 '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}.
908 '''
909 return Float(es_c=(self._exp_es_atanh_1 * self.b_a) if self.f else _1_0)
911 def es_tauf(self, taup):
912 '''Compute I{Karney}'s U{equations (19), (20) and (21)
913 <https://ArXiv.org/abs/1002.1417>}.
915 @see: I{Karney}'s C++ method U{Math::tauf<https://GeographicLib.
916 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>} and
917 and I{Veness}' JavaScript method U{toLatLon<https://www.
918 Movable-Type.co.UK/scripts/latlong-utm-mgrs.html>}.
919 '''
920 t = Scalar(taup=taup)
921 if self.f: # .isEllipsoidal
922 a = fabs(t)
923 T = t * (self._exp_es_atanh_1 if a > 70 else self._1_e21)
924 if fabs(T * _EPSqrt) < _2_0: # handles +/- INF and NAN
925 s = (a * _TOL) if a > _1_0 else _TOL
926 for T, _, d in self._es_tauf3(t, T): # max 2
927 if fabs(d) < s:
928 break
929 t = Scalar(tauf=T)
930 return t
932 def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB
933 '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most
934 B{C{N}} Newton iterations, converging rapidly except when C{delta}
935 toggles on +/-1.12e-16 or +/-4.47e-16, see C{.utm.Utm._toLLEB}.
936 '''
937 e = self._1_e21
938 _F2_ = Fsum(T).fsum2f_ # τ0
939 _tf2 = self._es_taupf2
940 for i in range(1, N + 1):
941 a, h = _tf2(T)
942 d = (taup - a) * (e + T**2) / (hypot1(a) * h)
943 # = (taup - a) / hypot1(a) / ((e + T**2) / h)
944 T, d = _F2_(d) # τi, (τi - τi-1)
945 yield T, i, d
947 def es_taupf(self, tau):
948 '''Compute I{Karney}'s U{equations (7), (8) and (9)
949 <https://ArXiv.org/abs/1002.1417>}.
951 @see: I{Karney}'s C++ method U{Math::taupf<https://GeographicLib.
952 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}.
953 '''
954 t = Scalar(tau=tau)
955 if self.f: # .isEllipsoidal
956 t, _ = self._es_taupf2(t)
957 t = Scalar(taupf=t)
958 return t
960 def _es_taupf2(self, tau):
961 '''(INTERNAL) Return 2-tuple C{(es_taupf(tau), hypot1(tau))}.
962 '''
963 if _isfinite(tau):
964 h = hypot1(tau)
965 s = sinh(self._es_atanh(tau / h))
966 a = hypot1(s) * tau - h * s
967 else:
968 a, h = tau, INF
969 return a, h
971 @Property_RO
972 def _exp_es_atanh_1(self):
973 '''(INTERNAL) Helper for .es_c and .es_tauf.
974 '''
975 return exp(self._es_atanh(_1_0)) if self.es else _1_0
977 @Property_RO
978 def f(self):
979 '''Get the I{flattening} (C{scalar}), M{(a - b) / a}, C{0} for spherical, negative for prolate.
980 '''
981 return self._f
983 @Property_RO
984 def f_(self):
985 '''Get the I{inverse flattening} (C{scalar}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}.
986 '''
987 return self._f_
989 @Property_RO
990 def f1(self):
991 '''Get the I{1 - flattening} (C{float}), M{f1 == 1 - f == b / a}.
993 @see: Property L{b_a}.
994 '''
995 return Float(f1=_1_0 - self.f)
997 @Property_RO
998 def f2(self):
999 '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}.
1000 '''
1001 return self._assert(self.a_b - _1_0, f2=f2f2(self.f))
1003 @deprecated_Property_RO
1004 def geodesic(self):
1005 '''DEPRECATED, use property C{geodesicw}.'''
1006 return self.geodesicw
1008 def geodesic_(self, exact=True):
1009 '''Get the an I{exact} C{Geodesic...} instance for this ellipsoid.
1011 @kwarg exact: If C{bool} return L{GeodesicExact}C{(exact=B{exact}, ...)},
1012 otherwise a L{Geodesic}, L{GeodesicExact} or L{GeodesicSolve}
1013 instance for I{this} ellipsoid.
1015 @return: The C{exact} geodesic (C{Geodesic...}).
1017 @raise TypeError: Invalid B{C{exact}}.
1019 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1020 '''
1021 if isbool(exact): # for consistenccy with C{.rhumb_}
1022 g = _MODS.geodesicx.GeodesicExact(self, C4order=30 if exact else 24,
1023 name=self.name)
1024 else:
1025 g = exact
1026 E = _xattr(g, ellipsoid=None)
1027 if not (E is self and isinstance(g, self._Geodesics)):
1028 raise _ValueError(exact=g, ellipsoid=E, txt_not_=self.name)
1029 return g
1031 @property_ROver
1032 def _Geodesics(self):
1033 '''(INTERNAL) Get all C{Geodesic...} classes, I{once}.
1034 '''
1035 t = (_MODS.geodesicx.GeodesicExact,
1036 _MODS.geodsolve.GeodesicSolve)
1037 try:
1038 t += (_MODS.geodesicw.Geodesic,
1039 _MODS.geodesicw._wrapped.Geodesic)
1040 except ImportError:
1041 pass
1042 return t # overwrite property_ROver
1044 @property_RO
1045 def geodesicw(self):
1046 '''Get this ellipsoid's I{wrapped} U{geodesicw.Geodesic
1047 <https://GeographicLib.SourceForge.io/Python/doc/code.html>}, provided
1048 I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1049 package is installed.
1050 '''
1051 # if not self.isEllipsoidal:
1052 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1053 return _MODS.geodesicw.Geodesic(self)
1055 @property_RO
1056 def geodesicx(self):
1057 '''Get this ellipsoid's I{exact} L{GeodesicExact}.
1058 '''
1059 # if not self.isEllipsoidal:
1060 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1061 return _MODS.geodesicx.GeodesicExact(self, name=self.name)
1063 @property
1064 def geodsolve(self):
1065 '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility
1066 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1067 provided the path to the C{GeodSolve} executable is specified with env
1068 variable C{PYGEODESY_GEODSOLVE} or re-/set with this property..
1069 '''
1070 # if not self.isEllipsoidal:
1071 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1072 return _MODS.geodsolve.GeodesicSolve(self, path=self._geodsolve, name=self.name)
1074 @geodsolve.setter # PYCHOK setter!
1075 def geodsolve(self, path):
1076 '''Re-/set the (fully qualified) path to the U{GeodSolve
1077 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1078 overriding env variable C{PYGEODESY_GEODSOLVE} (C{str}).
1079 '''
1080 self._geodsolve = path
1082 def hartzell4(self, pov, los=None):
1083 '''Compute the intersection of this ellipsoid's surface and a Line-Of-Sight
1084 from a Point-Of-View in space.
1086 @arg pov: Point-Of-View outside this ellipsoid (C{Cartesian}, L{Ecef9Tuple}
1087 or L{Vector3d}).
1088 @kwarg los: Line-Of-Sight, I{direction} to this ellipsoid (L{Vector3d}) or
1089 C{None} to point to this ellipsoid's center.
1091 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1092 C{y} and C{z} of the projection on or the intersection with this
1093 ellipsoid and the I{distance} C{h} from B{C{pov}} to C{(x, y, z)}
1094 along B{C{los}}, all in C{meter}, conventionally.
1096 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, or B{C{pov}}
1097 is inside this ellipsoid or B{C{los}} points
1098 outside this ellipsoid or points in an opposite
1099 direction.
1101 @raise TypeError: Invalid B{C{pov}} or B{C{los}}.
1103 @see: U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.
1104 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and
1105 methods L{Ellipsoid.height4} and L{Triaxial.hartzell4}.
1106 '''
1107 try:
1108 v, d, _ = _MODS.triaxials._hartzell3(pov, los, self._triaxial)
1109 except Exception as x:
1110 raise IntersectionError(pov=pov, los=los, cause=x)
1111 return Vector4Tuple(v.x, v.y, v.z, d, name__=self.hartzell4)
1113 @Property_RO
1114 def _hash(self):
1115 return hash((self.a, self.f))
1117 def height4(self, xyz, normal=True):
1118 '''Compute the projection on and the height of a cartesian above or below
1119 this ellipsoid's surface.
1121 @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d},
1122 L{Vector3Tuple} or L{Vector4Tuple}).
1123 @kwarg normal: If C{True}, the projection is perpendicular to (the nearest
1124 point on) this ellipsoid's surface, otherwise the C{radial}
1125 line to this ellipsoid's center (C{bool}).
1127 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1128 C{y} and C{z} of the projection on and the height C{h} above or
1129 below this ellipsoid's surface, all in C{meter}, conventionally.
1131 @raise ValueError: Null B{C{xyz}}.
1133 @raise TypeError: Non-cartesian B{C{xyz}}.
1135 @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>}
1136 and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse and
1137 methods L{Ellipsoid.hartzell4} and L{Triaxial.height4}.
1138 '''
1139 v = _MODS.vector3d._otherV3d(xyz=xyz)
1140 r = v.length
1142 a, b, i = self.a, self.b, None
1143 if r < EPS0: # EPS
1144 v = v.times(_0_0)
1145 h = -a
1147 elif self.isSpherical:
1148 v = v.times(a / r)
1149 h = r - a
1151 elif normal: # perpendicular to ellipsoid
1152 x, y = hypot(v.x, v.y), fabs(v.z)
1153 if x < EPS0: # PYCHOK no cover
1154 z = copysign0(b, v.z)
1155 v = Vector3Tuple(v.x, v.y, z)
1156 h = y - b # polar
1157 elif y < EPS0: # PYCHOK no cover
1158 t = a / r
1159 v = v.times_(t, t, 0) # force z=0.0
1160 h = x - a # equatorial
1161 else: # normal in 1st quadrant
1162 x, y, i = _MODS.triaxials._plumbTo3(x, y, self)
1163 t, v = v, v.times_(x, x, y)
1164 h = t.minus(v).length
1166 else: # radial to ellipsoid's center
1167 h = hypot_(a * v.z, b * v.x, b * v.y)
1168 t = (a * b / h) if h > EPS0 else _0_0 # EPS
1169 v = v.times(t)
1170 h = r * (_1_0 - t)
1172 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name__=self.height4)
1174 def _hubeny_2(self, phi2, phi1, lam21, scaled=True, squared=True):
1175 '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_},
1176 returning the I{angular} distance in C{radians squared} or C{radians}
1177 '''
1178 m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=scaled)
1179 h, r = (hypot2, self.a2_) if squared else (hypot, _1_0 / self.a)
1180 return h(m * (phi2 - phi1), n * lam21) * r
1182 @Property_RO
1183 def isEllipsoidal(self):
1184 '''Is this model I{ellipsoidal} (C{bool})?
1185 '''
1186 return self.f != 0
1188 @Property_RO
1189 def isOblate(self):
1190 '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or
1191 spherical otherwise.
1192 '''
1193 return self.f > 0
1195 @Property_RO
1196 def isProlate(self):
1197 '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or
1198 spherical otherwise.
1199 '''
1200 return self.f < 0
1202 @Property_RO
1203 def isSpherical(self):
1204 '''Is this ellipsoid I{spherical} (C{bool})?
1205 '''
1206 return self.f == 0
1208 def _Kseries(self, *AB8Ks):
1209 '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha
1210 or Beta series coefficients per I{Karney}'s U{equations (35)
1211 and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}.
1213 @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series
1214 coefficient tuples.
1216 @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}).
1218 @see: I{Karney}'s 30-th order U{TMseries30
1219 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}.
1220 '''
1221 k = self.KsOrder
1222 if self.n:
1223 ns = fpowers(self.n, k)
1224 ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k))
1225 else:
1226 ks = _0_0s(k)
1227 return ks
1229 @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''')
1230 def KsOrder(self):
1231 '''Get the I{Krüger} series' order (C{int} 4, 6 or 8).
1232 '''
1233 return self._KsOrder
1235 @KsOrder.setter # PYCHOK setter!
1236 def KsOrder(self, order):
1237 '''Set the I{Krüger} series' order (C{int} 4, 6 or 8).
1239 @raise ValueError: Invalid B{C{order}}.
1240 '''
1241 if not (isint(order) and order in (4, 6, 8)):
1242 raise _ValueError(order=order)
1243 if self._KsOrder != order:
1244 Ellipsoid.AlphaKs._update(self)
1245 Ellipsoid.BetaKs._update(self)
1246 self._KsOrder = order
1248 @Property_RO
1249 def L(self):
1250 '''Get the I{quarter meridian} C{L}, aka the C{polar distance}
1251 along a meridian between the equator and a pole (C{meter}),
1252 M{b * Elliptic(-e2 / (1 - e2)).cE} or M{b * PI / 2}.
1253 '''
1254 r = self._elliptic_e22.cE if self.f else PI_2
1255 return Distance(L=self.b * r)
1257 def Llat(self, lat):
1258 '''Return the I{meridional length}, the distance along a meridian
1259 between the equator and a (geodetic) latitude, see C{L}.
1261 @arg lat: Geodetic latitude (C{degrees90}).
1263 @return: The meridional length at B{C{lat}}, negative on southern
1264 hemisphere (C{meter}).
1265 '''
1266 r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phid(lat)
1267 return Distance(Llat=self.b * r)
1269 Lmeridian = Llat # meridional distance
1271 @property_RO
1272 def _Lpd(self):
1273 '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}.
1274 '''
1275 return Meter(_Lpd=self.L / _90_0)
1277 @property_RO
1278 def _Lpr(self):
1279 '''Get the I{quarter meridian} per radian (C{meter}), M{self.L / PI_2}.
1280 '''
1281 return Meter(_Lpr=self.L / PI_2)
1283 @deprecated_Property_RO
1284 def majoradius(self): # PYCHOK no cover
1285 '''DEPRECATED, use property C{a} or C{Requatorial}.'''
1286 return self.a
1288 def m2degrees(self, distance, lat=0):
1289 '''Convert a distance to an angle along the equator or along
1290 a parallel of (geodetic) latitude.
1292 @arg distance: Distance (C{meter}).
1293 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1295 @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}.
1297 @raise RangeError: Latitude B{C{lat}} outside valid range and
1298 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1300 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1301 '''
1302 return degrees(self.m2radians(distance, lat=lat))
1304 def m2radians(self, distance, lat=0):
1305 '''Convert a distance to an angle along the equator or
1306 along a parallel of (geodetic) latitude.
1308 @arg distance: Distance (C{meter}).
1309 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1311 @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}.
1313 @raise RangeError: Latitude B{C{lat}} outside valid range and
1314 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1316 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1317 '''
1318 r = self.circle4(lat).radius if lat else self.a
1319 return m2radians(distance, radius=r, lat=0)
1321 @deprecated_Property_RO
1322 def minoradius(self): # PYCHOK no cover
1323 '''DEPRECATED, use property C{b}, C{polaradius} or C{Rpolar}.'''
1324 return self.b
1326 @Property_RO
1327 def n(self):
1328 '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}.
1329 '''
1330 return self._assert(a_b2n(self.a, self.b), n=f2n(self.f))
1332 flattening = f
1333 flattening1st = f
1334 flattening2nd = f2
1335 flattening3rd = n
1337 polaradius = b # Rpolar
1339# @Property_RO
1340# def Q(self):
1341# '''Get the I{meridian arc unit} C{Q}, the mean, meridional length I{per radian} C({float}).
1342#
1343# @note: C{Q * PI / 2} ≈ C{L}, the I{quarter meridian}.
1344#
1345# @see: Property C{A} and U{Engsager, K., Poder, K.<https://StudyLib.net/doc/7443565/
1346# a-highly-accurate-world-wide-algorithm-for-the-transverse...>}.
1347# '''
1348# n = self.n
1349# d = (n + _1_0) / self.a
1350# return Float(Q=Fhorner(n**2, _1_0, _0_25, _1_16th, _0_25).fover(d) if d else self.b)
1352# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1353# # Q = (1 - 3/4 * e'2 + 45/64 * e'4 - 175/256 * e'6 + 11025/16384 * e'8) * rocPolar
1354# # = (4 + e'2 * (-3 + e'2 * (45/16 + e'2 * (-175/64 + e'2 * 11025/4096)))) * rocPolar / 4
1355# return Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(4 / self.rocPolar)
1357 @deprecated_Property_RO
1358 def quarteradius(self): # PYCHOK no cover
1359 '''DEPRECATED, use property C{L} or method C{Llat}.'''
1360 return self.L
1362 @Property_RO
1363 def R1(self):
1364 '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}.
1366 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}
1367 and method C{Rgeometric}.
1368 '''
1369 r = Fsum(self.a, self.a, self.b).fover(_3_0) if self.f else self.a
1370 return Radius(R1=r)
1372 Rmean = R1
1374 @Property_RO
1375 def R2(self):
1376 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}.
1378 @see: C{R2x}, C{c2}, C{area} and U{Earth radius
1379 <https://WikiPedia.org/wiki/Earth_radius>}.
1380 '''
1381 return Radius(R2=sqrt(self.c2) if self.f else self.a)
1383 Rauthalic = R2
1385# @Property_RO
1386# def R2(self):
1387# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1388# # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar
1389# # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3
1390# return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar)
1392 @Property_RO
1393 def R2x(self):
1394 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}.
1396 @see: C{R2}, C{c2x} and C{areax}.
1397 '''
1398 return Radius(R2x=sqrt(self.c2x) if self.f else self.a)
1400 Rauthalicx = R2x
1402 @Property_RO
1403 def R3(self):
1404 '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}.
1406 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}.
1407 '''
1408 r = (cbrt(self.b_a) * self.a) if self.f else self.a
1409 return Radius(R3=r)
1411 Rvolumetric = R3
1413 def radians2m(self, rad, lat=0):
1414 '''Convert an angle to the distance along the equator or along
1415 a parallel of (geodetic) latitude.
1417 @arg rad: The angle (C{radians}).
1418 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1420 @return: Distance (C{meter}, same units as the equatorial
1421 and polar radii) or C{0} for near-polar B{C{lat}}.
1423 @raise RangeError: Latitude B{C{lat}} outside valid range and
1424 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1426 @raise ValueError: Invalid B{C{rad}} or B{C{lat}}.
1427 '''
1428 r = self.circle4(lat).radius if lat else self.a
1429 return radians2m(rad, radius=r, lat=0)
1431 @Property_RO
1432 def Rbiaxial(self):
1433 '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}.
1435 @see: C{Rtriaxial}
1436 '''
1437 a, b = self.a, self.b
1438 if b < a:
1439 b = sqrt(_0_5 + self.b2_a2 * _0_5) * a
1440 elif b > a:
1441 b *= sqrt(_0_5 + self.a2_b2 * _0_5)
1442 return Radius(Rbiaxial=b)
1444 Requatorial = a # for consistent naming
1446 def Rgeocentric(self, lat):
1447 '''Compute the I{geocentric} earth radius of (geodetic) latitude.
1449 @arg lat: Latitude (C{degrees90}).
1451 @return: Geocentric earth radius (C{meter}).
1453 @raise ValueError: Invalid B{C{lat}}.
1455 @see: U{Geocentric Radius
1456 <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>}
1457 '''
1458 r, a = self.a, Phid(lat)
1459 if a and self.f:
1460 if fabs(a) < PI_2:
1461 s2, c2 = _s2_c2(a)
1462 b2_a2_s2 = self.b2_a2 * s2
1463 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2))
1464 # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2)))
1465 # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2))
1466 # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2))
1467 # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2))
1468 r *= sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2))
1469 else:
1470 r = self.b
1471 return Radius(Rgeocentric=r)
1473 @Property_RO
1474 def Rgeometric(self):
1475 '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}.
1477 @see: C{R1}.
1478 '''
1479 g = sqrt(self.a * self.b) if self.f else self.a
1480 return Radius(Rgeometric=g)
1482 def rhumb_(self, exact=True):
1483 '''Get the an I{exact} C{Rhumb...} instance for this ellipsoid.
1485 @kwarg exact: If C{bool} or C{None} return L{Rhumb}C{(exact=B{exact}, ...)},
1486 otherwise a L{Rhumb}, L{RhumbAux} or L{RhumbSolve} instance
1487 for I{this} ellipsoid.
1489 @return: The C{exact} rhumb (C{Rhumb...}).
1491 @raise TypeError: Invalid B{C{exact}}.
1493 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1494 '''
1495 if isbool(exact): # use Rhumb for backward compatibility
1496 r = _MODS.rhumb.ekx.Rhumb(self, exact=exact, name=self.name)
1497 else:
1498 r = exact
1499 E = _xattr(r, ellipsoid=None)
1500 if not (E is self and isinstance(r, self._Rhumbs)):
1501 raise _ValueError(exact=r, ellipsosid=E, txt_not_=self.name)
1502 return r
1504 @property_RO
1505 def rhumbaux(self):
1506 '''Get this ellipsoid's I{Auxiliary} C{rhumb.RhumbAux}.
1507 '''
1508 # if not self.isEllipsoidal:
1509 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1510 return _MODS.rhumb.aux_.RhumbAux(self, name=self.name)
1512 @property_RO
1513 def rhumbekx(self):
1514 '''Get this ellipsoid's I{Elliptic, Krüger} C{rhumb.Rhumb}.
1515 '''
1516 # if not self.isEllipsoidal:
1517 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1518 return _MODS.rhumb.ekx.Rhumb(self, name=self.name)
1520 @property_ROver
1521 def _Rhumbs(self):
1522 '''(INTERNAL) Get all C{Rhumb...} classes, I{once}.
1523 '''
1524 r = _MODS.rhumb
1525 return (r.aux_.RhumbAux, # overwrite property_ROver
1526 r.ekx.Rhumb, r.solve.RhumbSolve)
1528 @property
1529 def rhumbsolve(self):
1530 '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility
1531 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1532 provided the path to the C{RhumbSolve} executable is specified with env
1533 variable C{PYGEODESY_RHUMBSOLVE} or re-/set with this property.
1534 '''
1535 # if not self.isEllipsoidal:
1536 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1537 return _MODS.rhumb.solve.RhumbSolve(self, path=self._rhumbsolve, name=self.name)
1539 @rhumbsolve.setter # PYCHOK setter!
1540 def rhumbsolve(self, path):
1541 '''Re-/set the (fully qualified) path to the U{RhumbSolve
1542 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1543 overriding env variable C{PYGEODESY_RHUMBSOLVE} (C{str}).
1544 '''
1545 self._rhumbsolve = path
1547 @deprecated_property_RO
1548 def rhumbx(self):
1549 '''DEPRECATED on 2023.11.28, use property C{rhumbekx}. '''
1550 return self.rhumbekx
1552 def Rlat(self, lat):
1553 '''I{Approximate} the earth radius of (geodetic) latitude.
1555 @arg lat: Latitude (C{degrees90}).
1557 @return: Approximate earth radius (C{meter}).
1559 @raise RangeError: Latitude B{C{lat}} outside valid range and
1560 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1562 @raise TypeError: Invalid B{C{lat}}.
1564 @raise ValueError: Invalid B{C{lat}}.
1566 @note: C{Rlat(B{90})} equals C{Rpolar}.
1568 @see: Method C{circle4}.
1569 '''
1570 # r = a - (a - b) * |lat| / 90
1571 r = self.a
1572 if self.f and lat: # .isEllipsoidal
1573 r -= (r - self.b) * fabs(Lat(lat)) / _90_0
1574 r = Radius(Rlat=r)
1575 return r
1577 Rpolar = b # for consistent naming
1579 def roc1_(self, sa, ca=None):
1580 '''Compute the I{prime-vertical}, I{normal} radius of curvature
1581 of (geodetic) latitude, I{unscaled}.
1583 @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]).
1584 @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0])
1585 to use an alternate formula.
1587 @return: The prime-vertical radius of curvature (C{float}).
1589 @note: The delta between both formulae with C{Ellipsoids.WGS84}
1590 is less than 2 nanometer over the entire latitude range.
1592 @see: Method L{roc2_} and class L{EcefYou}.
1593 '''
1594 if not self.f: # .isSpherical
1595 n = self.a
1596 elif ca is None:
1597 r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward
1598 n = sqrt(self.a2 / r) if r > EPS02 else _0_0
1599 elif ca: # derived from EcefYou.forward
1600 h = hypot(ca, self.b_a * sa) if sa else fabs(ca)
1601 n = self.a / h
1602 elif sa:
1603 n = self.a2_b / fabs(sa)
1604 else:
1605 n = self.a
1606 return n
1608 def roc2(self, lat, scaled=False):
1609 '''Compute the I{meridional} and I{prime-vertical}, I{normal}
1610 radii of curvature of (geodetic) latitude.
1612 @arg lat: Latitude (C{degrees90}).
1613 @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}).
1615 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with
1616 the radii of curvature.
1618 @raise ValueError: Invalid B{C{lat}}.
1620 @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation
1621 <https://www.EdWilliams.org/avform.htm#flat>} and meridional and
1622 prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/
1623 Earth_radius#Radii_of_curvature>}.
1624 '''
1625 return self.roc2_(Phid(lat), scaled=scaled)
1627 def roc2_(self, phi, scaled=False):
1628 '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of
1629 curvature of (geodetic) latitude.
1631 @arg phi: Latitude (C{radians}).
1632 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}).
1634 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the
1635 radii of curvature.
1637 @raise ValueError: Invalid B{C{phi}}.
1639 @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local,
1640 flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>}
1641 and the meridional and prime vertical U{Radii of Curvature
1642 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1643 '''
1644 a = fabs(Phi(phi))
1645 if self.f:
1646 r = self.e2s2(sin(a))
1647 if r > EPS02:
1648 n = self.a / sqrt(r)
1649 m = n * self.e21 / r
1650 else:
1651 m = n = _0_0
1652 else:
1653 m = n = self.a
1654 if scaled and a:
1655 n *= cos(a) if a < PI_2 else _0_0
1656 return Curvature2Tuple(Radius(rocMeridional=m),
1657 Radius(rocPrimeVertical=n))
1659 def rocBearing(self, lat, bearing):
1660 '''Compute the I{directional} radius of curvature of (geodetic)
1661 latitude and compass direction.
1663 @arg lat: Latitude (C{degrees90}).
1664 @arg bearing: Direction (compass C{degrees360}).
1666 @return: Directional radius of curvature (C{meter}).
1668 @raise RangeError: Latitude B{C{lat}} outside valid range and
1669 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1671 @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}.
1673 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
1674 '''
1675 if self.f:
1676 s2, c2 = _s2_c2(Bearing_(bearing))
1677 m, n = self.roc2_(Phid(lat))
1678 if n < m: # == n / (c2 * n / m + s2)
1679 c2 *= n / m
1680 elif m < n: # == m / (c2 + s2 * m / n)
1681 s2 *= m / n
1682 n = m
1683 b = n / (c2 + s2) # == 1 / (c2 / m + s2 / n)
1684 else:
1685 b = self.b # == self.a
1686 return Radius(rocBearing=b)
1688 @Property_RO
1689 def rocEquatorial2(self):
1690 '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature
1691 at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}.
1693 @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a},
1694 L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature
1695 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1696 '''
1697 return Curvature2Tuple(Radius(rocMeridional0=self.b2_a if self.f else self.a),
1698 Radius(rocPrimeVertical0=self.a))
1700 def rocGauss(self, lat):
1701 '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude.
1703 @arg lat: Latitude (C{degrees90}).
1705 @return: Gaussian radius of curvature (C{meter}).
1707 @raise ValueError: Invalid B{C{lat}}.
1709 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1710 Earth_radius#Radii_of_curvature>}
1711 '''
1712 # using ...
1713 # m, n = self.roc2_(Phid(lat))
1714 # return sqrt(m * n)
1715 # ... requires 1 or 2 sqrt
1716 g = self.b
1717 if self.f:
1718 s2, c2 = _s2_c2(Phid(lat))
1719 g = g / (c2 + self.b2_a2 * s2)
1720 return Radius(rocGauss=g)
1722 def rocMean(self, lat):
1723 '''Compute the I{mean} radius of curvature of (geodetic) latitude.
1725 @arg lat: Latitude (C{degrees90}).
1727 @return: Mean radius of curvature (C{meter}).
1729 @raise ValueError: Invalid B{C{lat}}.
1731 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1732 Earth_radius#Radii_of_curvature>}
1733 '''
1734 if self.f:
1735 m, n = self.roc2_(Phid(lat))
1736 m *= n * _2_0 / (m + n) # == 2 / (1 / m + 1 / n)
1737 else:
1738 m = self.a
1739 return Radius(rocMean=m)
1741 def rocMeridional(self, lat):
1742 '''Compute the I{meridional} radius of curvature of (geodetic) latitude.
1744 @arg lat: Latitude (C{degrees90}).
1746 @return: Meridional radius of curvature (C{meter}).
1748 @raise ValueError: Invalid B{C{lat}}.
1750 @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation
1751 <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of
1752 Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1753 '''
1754 return self.roc2_(Phid(lat)).meridional if lat else \
1755 self.rocEquatorial2.meridional
1757 rocPolar = a2_b # synonymous
1759 def rocPrimeVertical(self, lat):
1760 '''Compute the I{prime-vertical}, I{normal} radius of curvature of
1761 (geodetic) latitude, aka the I{transverse} radius of curvature.
1763 @arg lat: Latitude (C{degrees90}).
1765 @return: Prime-vertical radius of curvature (C{meter}).
1767 @raise ValueError: Invalid B{C{lat}}.
1769 @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth
1770 approximation<https://www.EdWilliams.org/avform.htm#flat>} and
1771 U{Radii of Curvature<https://WikiPedia.org/wiki/
1772 Earth_radius#Radii_of_curvature>}.
1773 '''
1774 return self.roc2_(Phid(lat)).prime_vertical if lat else \
1775 self.rocEquatorial2.prime_vertical
1777 rocTransverse = rocPrimeVertical # synonymous
1779 @deprecated_Property_RO
1780 def Rquadratic(self): # PYCHOK no cover
1781 '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.'''
1782 return self.Rbiaxial
1784 @deprecated_Property_RO
1785 def Rr(self): # PYCHOK no cover
1786 '''DEPRECATED, use property C{Rrectifying}.'''
1787 return self.Rrectifying
1789 @Property_RO
1790 def Rrectifying(self):
1791 '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}.
1793 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}.
1794 '''
1795 r = (cbrt2((_1_0 + sqrt3(self.b_a)) * _0_5) * self.a) if self.f else self.a
1796 return Radius(Rrectifying=r)
1798 @deprecated_Property_RO
1799 def Rs(self): # PYCHOK no cover
1800 '''DEPRECATED, use property C{Rgeometric}.'''
1801 return self.Rgeometric
1803 @Property_RO
1804 def Rtriaxial(self):
1805 '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}.
1807 @see: C{Rbiaxial}
1808 '''
1809 a, b = self.a, self.b
1810 q = (sqrt((_3_0 + self.b2_a2) * _0_25) * a) if a > b else (
1811 (sqrt((_3_0 * self.a2_b2 + _1_0) * _0_25) * b) if a < b else a)
1812 return Radius(Rtriaxial=q)
1814 def toEllipsoid2(self, **name):
1815 '''Get a copy of this ellipsoid as an L{Ellipsoid2}.
1817 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
1819 @see: Property C{a_f}.
1820 '''
1821 return Ellipsoid2(self, None, **name)
1823 def toStr(self, prec=8, terse=4, **sep_name): # PYCHOK expected
1824 '''Return this ellipsoid as a text string.
1826 @kwarg prec: Number of decimal digits, unstripped (C{int}).
1827 @kwarg terse: Limit the number of items (C{int}, 0...18),
1828 use C{B{terse}=0} or C{=None} for all.
1829 @kwarg sep_name: Optional C{B{name}=NN} (C{str}) or C{None}
1830 to exclude this ellipsoid's name and separator
1831 C{B{sep}=", "} to join the items (C{str}).
1833 @return: This C{Ellipsoid}'s attributes (C{str}).
1834 '''
1835 E = Ellipsoid
1836 t = (E.a, E.f, E.f_, E.b, E.f2, E.n, E.e,
1837 E.e2, E.e21, E.e22, E.e32,
1838 E.A, E.L, E.R1, E.R2, E.R3,
1839 E.Rbiaxial, E.Rtriaxial)
1840 if terse:
1841 t = t[:terse]
1842 return self._instr(prec=prec, props=t, **sep_name)
1844 def toTriaxial(self, **name):
1845 '''Convert this ellipsoid to a L{Triaxial_}.
1847 @kwarg name: Optional C{B{name}=NN} (C{str}).
1849 @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis
1850 pointing east and C{Z} pointing north.
1852 @see: Method L{Triaxial_.toEllipsoid}.
1853 '''
1854 T = self._triaxial
1855 return T.copy(**name) if name else T
1857 @property_RO
1858 def _triaxial(self):
1859 '''(INTERNAL) Get this ellipsoid's un-/ordered C{Triaxial/_}.
1860 '''
1861 a, b, m = self.a, self.b, _MODS.triaxials
1862 T = m.Triaxial if a > b else m.Triaxial_
1863 return T(a, a, b, name=self.name)
1865 @Property_RO
1866 def volume(self):
1867 '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}.
1869 @see: C{R3}.
1870 '''
1871 return Meter3(volume=self.a2 * self.b * PI_3 * _4_0)
1874class Ellipsoid2(Ellipsoid):
1875 '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}.
1876 '''
1877 def __init__(self, a, f=None, **name):
1878 '''New L{Ellipsoid2}.
1880 @arg a: Equatorial radius, semi-axis (C{meter}) or a previous
1881 L{Ellipsoid} instance.
1882 @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}),
1883 if B{C{a}} is in C{meter}.
1884 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
1886 @raise NameError: Ellipsoid with that B{C{name}} already exists.
1888 @raise ValueError: Invalid B{C{a}} or B{C{f}}.
1890 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.
1891 Negative C{B{f}} produces a I{prolate} ellipsoid.
1892 '''
1893 if f is None and isinstance(a, Ellipsoid):
1894 Ellipsoid.__init__(self, a.a, f =a.f,
1895 b=a.b, f_=a.f_, **name)
1896 else:
1897 Ellipsoid.__init__(self, a, f=f, **name)
1900def _spherical_a_b(a, b):
1901 '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}.
1902 '''
1903 return a < EPS0 or b < EPS0 or fabs(a - b) < EPS0
1906def _spherical_f(f):
1907 '''(INTERNAL) C{True} for spherical or invalid C{f}.
1908 '''
1909 return fabs(f) < EPS or f > EPS1
1912def _spherical_f_(f_):
1913 '''(INTERNAL) C{True} for spherical or invalid C{f_}.
1914 '''
1915 return fabs(f_) < EPS or fabs(f_) > _1_EPS
1918def a_b2e(a, b):
1919 '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius.
1921 @arg a: Equatorial radius (C{scalar} > 0).
1922 @arg b: Polar radius (C{scalar} > 0).
1924 @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}),
1925 M{sqrt(1 - (b / a)**2)}.
1927 @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids.
1928 '''
1929 return Float(e=sqrt(fabs(a_b2e2(a, b)))) # == sqrt(fabs(a - b) * (a + b)) / a)
1932def a_b2e2(a, b):
1933 '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius.
1935 @arg a: Equatorial radius (C{scalar} > 0).
1936 @arg b: Polar radius (C{scalar} > 0).
1938 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}),
1939 M{1 - (b / a)**2}.
1941 @note: The result is positive for I{oblate}, negative for I{prolate}
1942 or C{0} for I{near-spherical} ellipsoids.
1943 '''
1944 return Float(e2=_0_0 if _spherical_a_b(a, b) else ((a - b) * (a + b) / a**2))
1947def a_b2e22(a, b):
1948 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1950 @arg a: Equatorial radius (C{scalar} > 0).
1951 @arg b: Polar radius (C{scalar} > 0).
1953 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}),
1954 M{(a / b)**2 - 1}.
1956 @note: The result is positive for I{oblate}, negative for I{prolate}
1957 or C{0} for I{near-spherical} ellipsoids.
1958 '''
1959 return Float(e22=_0_0 if _spherical_a_b(a, b) else ((a - b) * (a + b) / b**2))
1962def a_b2e32(a, b):
1963 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1965 @arg a: Equatorial radius (C{scalar} > 0).
1966 @arg b: Polar radius (C{scalar} > 0).
1968 @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}),
1969 M{(a**2 - b**2) / (a**2 + b**2)}.
1971 @note: The result is positive for I{oblate}, negative for I{prolate}
1972 or C{0} for I{near-spherical} ellipsoids.
1973 '''
1974 a2, b2 = a**2, b**2
1975 return Float(e32=_0_0 if _spherical_a_b(a2, b2) else ((a2 - b2) / (a2 + b2)))
1978def a_b2f(a, b):
1979 '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius.
1981 @arg a: Equatorial radius (C{scalar} > 0).
1982 @arg b: Polar radius (C{scalar} > 0).
1984 @return: The flattening (C{scalar} or C{0}), M{(a - b) / a}.
1986 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1987 for I{near-spherical} ellipsoids.
1988 '''
1989 f = 0 if _spherical_a_b(a, b) else ((a - b) / a)
1990 return _f_0_0 if _spherical_f(f) else Float(f=f)
1993def a_b2f_(a, b):
1994 '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius.
1996 @arg a: Equatorial radius (C{scalar} > 0).
1997 @arg b: Polar radius (C{scalar} > 0).
1999 @return: The inverse flattening (C{scalar} or C{0}), M{a / (a - b)}.
2001 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2002 for I{near-spherical} ellipsoids.
2003 '''
2004 f_ = 0 if _spherical_a_b(a, b) else (a / float(a - b))
2005 return _f__0_0 if _spherical_f_(f_) else Float(f_=f_)
2008def a_b2f2(a, b):
2009 '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius.
2011 @arg a: Equatorial radius (C{scalar} > 0).
2012 @arg b: Polar radius (C{scalar} > 0).
2014 @return: The I{signed}, 2nd flattening (C{scalar} or C{0}), M{(a - b) / b}.
2016 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2017 for I{near-spherical} ellipsoids.
2018 '''
2019 t = 0 if _spherical_a_b(a, b) else float(a - b)
2020 return Float(f2=_0_0 if fabs(t) < EPS0 else (t / b))
2023def a_b2n(a, b):
2024 '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius.
2026 @arg a: Equatorial radius (C{scalar} > 0).
2027 @arg b: Polar radius (C{scalar} > 0).
2029 @return: The I{signed}, 3rd flattening (C{scalar} or C{0}), M{(a - b) / (a + b)}.
2031 @note: The result is positive for I{oblate}, negative for I{prolate}
2032 or C{0} for I{near-spherical} ellipsoids.
2033 '''
2034 t = 0 if _spherical_a_b(a, b) else float(a - b)
2035 return Float(n=_0_0 if fabs(t) < EPS0 else (t / (a + b)))
2038def a_f2b(a, f):
2039 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}.
2041 @arg a: Equatorial radius (C{scalar} > 0).
2042 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2044 @return: The polar radius (C{float}), M{a * (1 - f)}.
2045 '''
2046 b = a if _spherical_f(f) else (a * (_1_0 - f))
2047 return Radius_(b=a if _spherical_a_b(a, b) else b)
2050def a_f_2b(a, f_):
2051 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}.
2053 @arg a: Equatorial radius (C{scalar} > 0).
2054 @arg f_: Inverse flattening (C{scalar} >>> 1).
2056 @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}.
2057 '''
2058 b = a if _spherical_f_(f_) else (a * (f_ - _1_0) / f_)
2059 return Radius_(b=a if _spherical_a_b(a, b) else b)
2062def b_f2a(b, f):
2063 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}.
2065 @arg b: Polar radius (C{scalar} > 0).
2066 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2068 @return: The equatorial radius (C{float}), M{b / (1 - f)}.
2069 '''
2070 t = _1_0 - f
2071 a = b if fabs(t) < EPS0 else (b / t)
2072 return Radius_(a=b if _spherical_a_b(a, b) else a)
2075def b_f_2a(b, f_):
2076 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}.
2078 @arg b: Polar radius (C{scalar} > 0).
2079 @arg f_: Inverse flattening (C{scalar} >>> 1).
2081 @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}.
2082 '''
2083 t = f_ - _1_0
2084 a = b if _spherical_f_(f_) or fabs(t - f_) < EPS0 \
2085 or fabs(t) < EPS0 else (b * f_ / t)
2086 return Radius_(a=b if _spherical_a_b(a, b) else a)
2089def e2f(e):
2090 '''Return C{f}, the I{flattening} for a given I{1st eccentricity}.
2092 @arg e: The (1st) eccentricity (0 <= C{float} < 1)
2094 @return: The flattening (C{scalar} or C{0}).
2096 @see: Function L{e22f}.
2097 '''
2098 return e22f(e**2)
2101def e22f(e2):
2102 '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}.
2104 @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1)
2106 @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(e2 - 1) + 1)}.
2107 '''
2108 return Float(f=e2 / (sqrt(_1_0 - e2) + _1_0)) if e2 else _f_0_0
2111def f2e2(f):
2112 '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}.
2114 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2116 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1),
2117 M{f * (2 - f)}.
2119 @note: The result is positive for I{oblate}, negative for I{prolate}
2120 or C{0} for I{near-spherical} ellipsoids.
2122 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2123 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2124 <https://WikiPedia.org/wiki/Flattening>}.
2125 '''
2126 return Float(e2=_0_0 if _spherical_f(f) else (f * (_2_0 - f)))
2129def f2e22(f):
2130 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}.
2132 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2134 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or
2135 C{INF}), M{f * (2 - f) / (1 - f)**2}.
2137 @note: The result is positive for I{oblate}, negative for I{prolate}
2138 or C{0} for near-spherical ellipsoids.
2140 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2141 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.
2142 '''
2143 # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2
2144 t = (_1_0 - f)**2
2145 return Float(e22=INF if t < EPS0 else (f2e2(f) / t)) # PYCHOK type
2148def f2e32(f):
2149 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}.
2151 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2153 @return: The I{signed}, 3rd eccentricity I{squared} (C{float}),
2154 M{f * (2 - f) / (1 + (1 - f)**2)}.
2156 @note: The result is positive for I{oblate}, negative for I{prolate}
2157 or C{0} for I{near-spherical} ellipsoids.
2159 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2160 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.
2161 '''
2162 # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2)
2163 e2 = f2e2(f)
2164 return Float(e32=e2 / (_2_0 - e2))
2167def f_2f(f_):
2168 '''Return C{f}, the I{flattening} for a given I{inverse flattening}.
2170 @arg f_: Inverse flattening (C{scalar} >>> 1).
2172 @return: The flattening (C{scalar} or C{0}), M{1 / f_}.
2174 @note: The result is positive for I{oblate}, negative for I{prolate}
2175 or C{0} for I{near-spherical} ellipsoids.
2176 '''
2177 f = 0 if _spherical_f_(f_) else _1_0 / f_
2178 return _f_0_0 if _spherical_f(f) else Float(f=f) # PYCHOK type
2181def f2f_(f):
2182 '''Return C{f_}, the I{inverse flattening} for a given I{flattening}.
2184 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2186 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2188 @note: The result is positive for I{oblate}, negative for I{prolate}
2189 or C{0} for I{near-spherical} ellipsoids.
2190 '''
2191 f_ = 0 if _spherical_f(f) else _1_0 / f
2192 return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) # PYCHOK type
2195def f2f2(f):
2196 '''Return C{f2}, the I{2nd flattening} for a given I{flattening}.
2198 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2200 @return: The I{signed}, 2nd flattening (C{scalar} or C{INF}), M{f / (1 - f)}.
2202 @note: The result is positive for I{oblate}, negative for I{prolate}
2203 or C{0} for I{near-spherical} ellipsoids.
2205 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2206 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2207 <https://WikiPedia.org/wiki/Flattening>}.
2208 '''
2209 t = _1_0 - f
2210 return Float(f2=_0_0 if _spherical_f(f) else (INF if fabs(t) < EPS
2211 else (f / t))) # PYCHOK type
2214def f2n(f):
2215 '''Return C{n}, the I{3rd flattening} for a given I{flattening}.
2217 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2219 @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1),
2220 M{f / (2 - f)}.
2222 @note: The result is positive for I{oblate}, negative for I{prolate}
2223 or C{0} for I{near-spherical} ellipsoids.
2225 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2226 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2227 <https://WikiPedia.org/wiki/Flattening>}.
2228 '''
2229 return Float(n=_0_0 if _spherical_f(f) else (f / float(_2_0 - f)))
2232def n2e2(n):
2233 '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}.
2235 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2237 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF),
2238 M{4 * n / (1 + n)**2}.
2240 @note: The result is positive for I{oblate}, negative for I{prolate}
2241 or C{0} for I{near-spherical} ellipsoids.
2243 @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}.
2244 '''
2245 t = (n + _1_0)**2
2246 return Float(e2=_0_0 if fabs(n) < EPS0 else
2247 (NINF if t < EPS0 else (_4_0 * n / t)))
2250def n2f(n):
2251 '''Return C{f}, the I{flattening} for a given I{3rd flattening}.
2253 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2255 @return: The flattening (C{scalar} or NINF), M{2 * n / (1 + n)}.
2257 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2258 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2259 <https://WikiPedia.org/wiki/Flattening>}.
2260 '''
2261 t = n + _1_0
2262 f = 0 if fabs(n) < EPS0 else (NINF if t < EPS0 else (_2_0 * n / t))
2263 return _f_0_0 if _spherical_f(f) else Float(f=f)
2266def n2f_(n):
2267 '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}.
2269 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2271 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2273 @see: L{n2f} and L{f2f_}.
2274 '''
2275 return f2f_(n2f(n))
2278class Ellipsoids(_NamedEnum):
2279 '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class
2280 to accommodate the L{_LazyNamedEnumItem} properties.
2281 '''
2282 def _Lazy(self, a, b, f_, **kwds):
2283 '''(INTERNAL) Instantiate the L{Ellipsoid}.
2284 '''
2285 return Ellipsoid(a, b=b, f_=f_, **kwds)
2287Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton
2288'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.'''
2289# <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html>
2290# <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS>
2291# <https://kb.OSU.edu/dspace/handle/1811/77986>
2292# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids>
2293# <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html>
2294# <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java>
2295Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid>
2296 Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909
2297 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448
2298# APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965
2299# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid
2300# AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America
2301 Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192
2302 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System"
2303 Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)),
2304 BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)),
2305# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)),
2306 CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC)
2307# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)),
2308 Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)),
2309 Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)),
2310 Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)),
2311 Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc
2312 CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures
2313 Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium
2314 Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)),
2315# Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)),
2316# Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)),
2317# Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)),
2318 Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)),
2319 Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)),
2320 Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)),
2321# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)),
2322 GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)),
2323 GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid"
2324 GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516
2325# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)),
2326 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.25722210088)), # IUGG, ITRS, ETRS89
2327# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297
2328 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)),
2329# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)),
2330 IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union
2331 IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302
2332 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy
2333 IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)),
2334 Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795
2335 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l
2336 Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2337 Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2338# Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961
2339# Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979
2340 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France
2341 Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)),
2342 Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)),
2343# MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983
2344# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab.
2345 NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab.
2346# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965
2347 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)),
2348 OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)),
2349# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect?
2350 Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972
2351# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)),
2352 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11
2353# SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia
2354 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System
2355 SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American
2356 Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo
2357 SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo
2358 SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)), # EPSG:3857 Spheroid
2359 Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)),
2360# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)),
2361# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)),
2362 WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)),
2363 WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)),
2364 WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52
2365 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # GPS b=6356752.3142451793
2366# U{NOAA/NOS/NGS/inverse<https://GitHub.com/noaa-ngs/inverse/blob/main/invers3d.f>}
2367 WGS84_NGS = _lazy('WGS84_NGS', *_T(R_MA, _0_0, 298.257222100882711243162836600094))
2368)
2370_EWGS84 = Ellipsoids.WGS84 # (INTERNAL) shared
2372if __name__ == '__main__':
2374 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_
2375 from pygeodesy import nameof, printf
2377 for E in (_EWGS84, Ellipsoids.GRS80, # NAD83,
2378 Ellipsoids.Sphere, Ellipsoids.SpherePopular,
2379 Ellipsoid(_EWGS84.b, _EWGS84.a, name='_Prolate')):
2380 e = f2n(E.f) - E.n
2381 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10, terse=0), nl=1)
2382 printf('# e=%s, f_=%s, f=%s, n=%s (%s)', fstr(E.e, prec=13, fmt=Fmt.e),
2383 fstr(E.f_, prec=13, fmt=Fmt.e),
2384 fstr(E.f, prec=13, fmt=Fmt.e),
2385 fstr(E.n, prec=13, fmt=Fmt.e),
2386 fstr(e, prec=9, fmt=Fmt.e))
2387 printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20))
2388 printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20))
2389 printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property
2391 # __doc__ of this file, force all into registry
2392 t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_)
2393 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))
2395# % python3 -m pygeodesy.ellipsoids
2397# Ellipsoids.WGS84: name='WGS84', a=6378137, f=0.0033528107, f_=298.257223563, b=6356752.3142451793, f2=0.0033640898, n=0.0016792204, e=0.0818191908, e2=0.00669438, e21=0.99330562, e22=0.0067394967, e32=0.0033584313, A=6367449.1458234144, L=10001965.7293127235, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.7900091587, Rbiaxial=6367453.6345163295, Rtriaxial=6372797.5559594007
2398# e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00)
2399# AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2400# BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2401# KsOrder 8
2403# Ellipsoids.GRS80: name='GRS80', a=6378137, f=0.0033528107, f_=298.2572221009, b=6356752.3141403468, f2=0.0033640898, n=0.0016792204, e=0.081819191, e2=0.00669438, e21=0.99330562, e22=0.0067394968, e32=0.0033584313, A=6367449.1457710434, L=10001965.7292304561, R1=6371008.7713801153, R2=6371007.1808835147, R3=6371000.7899741363, Rbiaxial=6367453.6344640013, Rtriaxial=6372797.5559332585
2404# e=8.1819191042833e-02, f_=2.9825722210088e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00)
2405# AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2406# BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2407# KsOrder 8
2409# Ellipsoids.Sphere: name='Sphere', a=6371008.7714149999, f=0, f_=0, b=6371008.7714149999, f2=0, n=0, e=0, e2=0, e21=1, e22=0, e32=0, A=6371008.7714149999, L=10007557.1761167478, R1=6371008.7714149999, R2=6371008.7714149999, R3=6371008.7714149999, Rbiaxial=6371008.7714149999, Rtriaxial=6371008.7714149999
2410# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2411# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2412# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2413# KsOrder 8
2415# Ellipsoids.SpherePopular: name='SpherePopular', a=6378137, f=0, f_=0, b=6378137, f2=0, n=0, e=0, e2=0, e21=1, e22=0, e32=0, A=6378137, L=10018754.171394622, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137
2416# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2417# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2418# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2419# KsOrder 8
2421# Ellipsoids._Prolate: name='_Prolate', a=6356752.3142451793, f=-0.0033640898, f_=-297.257223563, b=6378137, f2=-0.0033528107, n=-0.0016792204, e=0.0820944379, e2=-0.0067394967, e21=1.0067394967, e22=-0.00669438, e32=-0.0033584313, A=6367449.1458234144, L=10035500.5204500314, R1=6363880.5428301189, R2=6363878.9413582645, R3=6363872.5644020075, Rbiaxial=6367453.6345163295, Rtriaxial=6362105.2243882557
2422# e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00)
2423# AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0
2424# BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0
2425# KsOrder 8
2427# **) MIT License
2428#
2429# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
2430#
2431# Permission is hereby granted, free of charge, to any person obtaining a
2432# copy of this software and associated documentation files (the "Software"),
2433# to deal in the Software without restriction, including without limitation
2434# the rights to use, copy, modify, merge, publish, distribute, sublicense,
2435# and/or sell copies of the Software, and to permit persons to whom the
2436# Software is furnished to do so, subject to the following conditions:
2437#
2438# The above copyright notice and this permission notice shall be included
2439# in all copies or substantial portions of the Software.
2440#
2441# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
2442# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
2443# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
2444# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
2445# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
2446# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
2447# OTHER DEALINGS IN THE SOFTWARE.