Coverage for pygeodesy/ellipsoids.py: 96%
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« prev ^ index » next coverage.py v7.6.1, created at 2025-09-09 13:03 -0400
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 _ # noqa: E702 ;
67# from pygeodesy.albers import AlbersEqualAreaCylindrical # _MODS
68from pygeodesy.basics import copysign0, isbool, _isin, isint, typename
69from pygeodesy.constants import EPS, EPS_2, EPS0, EPS02, EPS1, INF, NINF, \
70 _over, PI_2, PI_3, PI4, R_M, R_MA, R_FM, _EPSqrt, \
71 _EPStol as _TOL, _floatuple as _T, _isfinite, \
72 _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \
73 _0_25, _3_0 # PYCHOK used!
74from pygeodesy.errors import _AssertionError, IntersectionError, _ValueError, _xattr, _xkwds_not
75from pygeodesy.fmath import cbrt, cbrt2, fdot, Fhorner, fpowers, hypot, hypot_, \
76 hypot1, hypot2, sqrt3, Fsum
77# from pygeodesy.fsums import Fsum # from .fmath
78# from pygeodesy.internals import typename # from .basics
79from pygeodesy.interns import NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, _beta_, \
80 _Clarke1866_, _Clarke1880IGN_, _DMAIN_, _DOT_, _f_, _GRS80_, \
81 _height_, _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \
82 _Krassowsky1940_, _lat_, _meridional_, _negative_, _not_finite_, \
83 _prime_vertical_, _radius_, _Sphere_, _SPACE_, _vs_, _WGS72_, _WGS84_
84# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .named
85from pygeodesy.named import _lazyNamedEnumItem as _lazy, _name__, _NamedEnum, \
86 _NamedEnumItem, _NamedTuple, _Pass, _ALL_LAZY, _MODS
87from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple
88from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_, \
89 deprecated_property_RO, property_RO, property_ROver
90from pygeodesy.streprs import Fmt, fstr, instr, strs, unstr
91# from pygeodesy.triaxials import _hartzell3 # _MODS
92from pygeodesy.units import Azimuth, Bearing, Distance, Float, Float_, Height, Lamd, Lat, \
93 Meter, Meter2, Meter3, Phi, Phid, Radius, Radius_, Scalar
94from pygeodesy.utily import atan1, atan1d, atan2b, degrees90, m2radians, radians2m, sincos2d
96from math import asinh, atan, atanh, cos, degrees, exp, fabs, radians, sin, sinh, sqrt, tan # as _tan
98__all__ = _ALL_LAZY.ellipsoids
99__version__ = '25.08.31'
101_f_0_0 = Float(f =_0_0) # zero flattening
102_f__0_0 = Float(f_=_0_0) # zero inverse flattening
103# see U{WGS84_f<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Constants.html>}
104_f__WGS84 = Float(f_=_1_0 / (1000000000 / 298257223563)) # 298.257223562_999_97 vs 298.257223563
107def _aux(lat, inverse, auxLat, clip=90):
108 '''Return a named auxiliary latitude in C{degrees}.
109 '''
110 return Lat(lat, clip=clip, name=_lat_ if inverse else typename(auxLat))
113def _s2_c2(phi):
114 '''(INTERNAL) Return 2-tuple C{(sin(B{phi})**2, cos(B{phi})**2)}.
115 '''
116 if phi:
117 s2 = sin(phi)**2
118 if s2 > EPS:
119 c2 = _1_0 - s2
120 if c2 > EPS:
121 if c2 < EPS1:
122 return s2, c2
123 else:
124 return _1_0, _0_0 # phi == PI_2
125 return _0_0, _1_0 # phi == 0
128class a_f2Tuple(_NamedTuple):
129 '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial}
130 radius C{a} in C{meter} and scalar I{flattening} C{f}.
132 @see: Class L{Ellipsoid2}.
133 '''
134 _Names_ = (_a_, _f_) # name 'f' not 'f_'
135 _Units_ = (_Pass, _Pass)
137 def __new__(cls, a, f, **name):
138 '''New L{a_f2Tuple} ellipsoid specification.
140 @arg a: Equatorial radius (C{scalar} > 0).
141 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
142 @kwarg name: Optional C{B{name}=NN} (C{str}).
144 @return: An L{a_f2Tuple}C{(a, f)}.
146 @raise UnitError: Invalid B{C{a}} or B{C{f}}.
148 @note: C{abs(B{f}) < }L{EPS<pygeodesy.constants.EPS>} is
149 forced to C{B{f}=0}, I{spherical}.
151 @note: Negative C{B{f}} produces a I{prolate} ellipsoid.
152 '''
153 a = Radius_(a=a) # low=EPS, high=None
154 f = Float_( f=f, low=None, high=EPS1)
155 if fabs(f) < EPS: # force spherical
156 f = _f_0_0
157 return _NamedTuple.__new__(cls, a, f, **name)
159 @Property_RO
160 def b(self):
161 '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}.
162 '''
163 return a_f2b(self.a, self.f) # PYCHOK .a and .f
165 def ellipsoid(self, **name):
166 '''Return an L{Ellipsoid} for this 2-tuple C{(a, f)}.
168 @kwarg name: Optional C{B{name}=NN} (C{str}).
170 @raise NameError: A registered C{ellipsoid} with the
171 same B{C{name}} already exists.
172 '''
173 return Ellipsoid(self.a, f=self.f, name=self._name__(name)) # PYCHOK .a and .f
175 @Property_RO
176 def f_(self):
177 '''Get the I{inverse} flattening (C{scalar}), M{1 / f} == M{a / (a - b)}.
178 '''
179 return f2f_(self.f) # PYCHOK .f
182class Circle4Tuple(_NamedTuple):
183 '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height},
184 both conventionally in C{meter} of a parallel I{circle of latitude} at
185 (geodetic) latitude C{lat} and the I{parametric (or reduced) auxiliary
186 latitude} C{beta}, both in C{degrees90}.
188 The C{height} is the (signed) distance along the z-axis between the
189 parallel and the equator. At near-polar C{lat}s, the C{radius} is C{0},
190 the C{height} is the ellipsoid's (signed) polar radius and C{beta}
191 equals C{lat}.
192 '''
193 _Names_ = (_radius_, _height_, _lat_, _beta_)
194 _Units_ = ( Radius, Height, Lat, Lat)
197class Curvature2Tuple(_NamedTuple):
198 '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in
199 C{meter}, conventionally.
200 '''
201 _Names_ = (_meridional_, _prime_vertical_)
202 _Units_ = ( Meter, Meter)
204 @property_RO
205 def transverse(self):
206 '''Get this I{prime_vertical}, aka I{transverse} radius of curvature.
207 '''
208 return self.prime_vertical
211class Ellipsoid(_NamedEnumItem):
212 '''Ellipsoid with I{equatorial} and I{polar} radii, I{flattening}, I{inverse
213 flattening} and other, often used, I{cached} attributes, supporting
214 I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models.
215 '''
216 _a = 0 # equatorial radius, semi-axis (C{meter})
217 _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f
218 _f = 0 # (1st) flattening: (a - b) / a
219 _f_ = 0 # inverse flattening: 1 / f = a / (a - b)
221 _geodsolve = NN # means, use PYGEODESY_GEODSOLVE
222 _KsOrder = 8 # Krüger series order (4, 6 or 8)
223 _rhumbsolve = NN # means, use PYGEODESY_RHUMBSOLVE
225 def __init__(self, a, b=None, f_=None, f=None, **name):
226 '''New L{Ellipsoid} from the I{equatorial} radius I{and} either
227 the I{polar} radius or I{inverse flattening} or I{flattening}.
229 @arg a: Equatorial radius, semi-axis (C{meter}).
230 @arg b: Optional polar radius, semi-axis (C{meter}).
231 @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0).
232 @arg f: Flattening: M{(a - b) / a} (C{scalar}, near zero for
233 spherical).
234 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
236 @raise NameError: Ellipsoid with the same B{C{name}} already exists.
238 @raise ValueError: Invalid B{C{a}}, B{C{b}}, B{C{f_}} or B{C{f}} or
239 B{C{f_}} and B{C{f}} are incompatible.
241 @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced
242 to M{1 / f_ = 0}, spherical.
243 '''
244 ff_ = f, f_ # assertion below
245 n = _name__(**name) if name else NN
246 try:
247 a = Radius_(a=a) # low=EPS
248 if not _isfinite(a):
249 raise ValueError(_SPACE_(_a_, _not_finite_))
251 if b: # not _isin(b, None, _0_0)
252 b = Radius_(b=b) # low=EPS
253 f = a_b2f(a, b) if f is None else Float(f=f)
254 f_ = f2f_(f) if f_ is None else Float(f_=f_)
255 elif f is not None:
256 f = Float(f=f)
257 b = a_f2b(a, f)
258 f_ = f2f_(f) if f_ is None else Float(f_=f_)
259 elif f_:
260 f_ = Float(f_=f_)
261 b = a_f_2b(a, f_) # a * (f_ - 1) / f_
262 f = f_2f(f_)
263 else: # only a, spherical
264 f_ = f = 0
265 b = a # superfluous
267 if not f < _1_0: # sanity check, see .ecef.Ecef.__init__
268 raise ValueError(_SPACE_(_f_, _invalid_))
269 if not _isfinite(b):
270 raise ValueError(_SPACE_(_b_, _not_finite_))
272 if fabs(f) < EPS or a == b or not f_: # spherical
273 b = a
274 f = _f_0_0
275 f_ = _f__0_0
277 except (TypeError, ValueError) as x:
278 d = _xkwds_not(None, b=b, f_=f_, f=f)
279 t = instr(self, a=a, name=n, **d)
280 raise _ValueError(t, cause=x)
282 self._a = a
283 self._b = b
284 self._f = f
285 self._f_ = f_
287 self._register(Ellipsoids, n)
289 if f and f_: # see test/testEllipsoidal
290 d = dict(eps=_TOL)
291 if None in ff_: # both f_ and f given
292 d.update(Error=_ValueError, txt=_incompatible_)
293 self._assert(_1_0 / f, f_=f_, **d)
294 self._assert(_1_0 / f_, f =f, **d)
295 self._assert(self.b2_a2, e21=self.e21, eps=EPS)
297 def __eq__(self, other):
298 '''Compare this and an other ellipsoid.
300 @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}).
302 @return: C{True} if equal, C{False} otherwise.
303 '''
304 return self is other or (isinstance(other, Ellipsoid) and
305 self.a == other.a and
306 (self.f == other.f or self.b == other.b))
308 def __hash__(self):
309 return self._hash # memoized
311 @Property_RO
312 def a(self):
313 '''Get the I{equatorial} radius, semi-axis (C{meter}).
314 '''
315 return self._a
317 equatoradius = a # = Requatorial
319 @Property_RO
320 def a2(self):
321 '''Get the I{equatorial} radius I{squared} (C{meter} I{squared}), M{a**2}.
322 '''
323 return Meter2(a2=self.a**2)
325 @Property_RO
326 def a2_(self):
327 '''Get the inverse of the I{equatorial} radius I{squared} (C{meter} I{squared}), M{1 / a**2}.
328 '''
329 return Float(a2_=_1_0 / self.a2)
331 @Property_RO
332 def a_b(self):
333 '''Get the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b} == M{1 / (1 - f)}.
334 '''
335 return Float(a_b=self.a / self.b if self.f else _1_0)
337 @Property_RO
338 def a2_b(self):
339 '''Get the I{polar} meridional (or polar) radius of curvature (C{meter}), M{a**2 / b}.
341 @see: U{Radii of Curvature
342 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
343 and U{Moritz, H. (1980), Geodetic Reference System 1980
344 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}.
346 @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature
347 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{c2}
348 and L{R2} or L{Rauthalic}.
349 '''
350 return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar
352 @Property_RO
353 def a2_b2(self):
354 '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}),
355 M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e21}.
356 '''
357 return Float(a2_b2=self.a_b**2 if self.f else _1_0)
359 @Property_RO
360 def a_f(self):
361 '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}), see method C{toEllipsoid2}.
362 '''
363 return a_f2Tuple(self.a, self.f, name=self.name)
365 a_f2 = a_f # synonym
367 @Property_RO
368 def A(self):
369 '''Get the UTM I{meridional (or rectifying)} radius (C{meter}).
371 @note: C{A * PI / 2} ≈= L{L<Ellipsoid.L>}, the I{quarter meridian}.
373 @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/>},
374 the mean, meridional length I{per radian}.
375 '''
376 A, n = self.a, self.n
377 if n:
378 d = (n + _1_0) * 1048576 / A
379 if d: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8
380 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>
381 # <https://GeographicLib.SourceForge.io/C++/doc/transversemercator.html> and
382 # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> (3)
383 # A *= fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441) / 1048576) / (1 + n)
384 A = Radius(A=Fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441).fover(d))
385 return A
386# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
387# # q = 4 / self.rocPolar
388# # Q = (1 - 3 / 4 * e'2 + 45 / 64 * e'4 - 175 / 256 * e'6 + 11025 / 16384 * e'8) * rocPolar
389# # = (4 + e'2 * (-3 + e'2 * (45 / 16 + e'2 * (-175 / 64 + e'2 * 11025 / 4096)))) / q
390# return Radius(Q=Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(q))
392 @Property_RO
393 def _albersCyl(self):
394 '''(INTERNAL) Helper for C{auxAuthalic}.
395 '''
396 return _MODS.albers.AlbersEqualAreaCylindrical(datum=self, name=self.name)
398 @Property_RO
399 def AlphaKs(self):
400 '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
401 '''
402 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # noqa: E702 ;
403 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
404 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200),
405 _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned
406 _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned
407 _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned
408 _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned
409 _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned
410 _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned
411 _T(1424729850961/743921418240)) # PYCHOK unaligned
413 @Property_RO
414 def area(self):
415 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2}.
417 @see: Properties L{areax}, L{c2} and L{R2} and functions
418 L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
419 '''
420 return Meter2(area=self.c2 * PI4)
422 @Property_RO
423 def areax(self):
424 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more
425 accurate for very I{oblate} ellipsoids.
427 @see: Properties L{area}, L{c2x} and L{R2x}, class L{GeodesicExact} and
428 functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
429 '''
430 return Meter2(areax=self.c2x * PI4)
432 def _assert(self, val, eps=_TOL, f0=_0_0, Error=_AssertionError, txt=NN, **name_value):
433 '''(INTERNAL) Assert a C{name=value} vs C{val}.
434 '''
435 for n, v in name_value.items():
436 if fabs(v - val) > eps: # PYCHOK no cover
437 t = (v, _vs_, val)
438 t = _SPACE_.join(strs(t, prec=12, fmt=Fmt.g))
439 t = Fmt.EQUAL(self._DOT_(n), t)
440 raise Error(t, txt=txt or Fmt.exceeds_eps(eps))
441 return Float(v if self.f else f0, name=n)
442 raise Error(unstr(self._DOT_(typename(self._assert)), val,
443 eps=eps, f0=f0, **name_value))
445 def auxAuthalic(self, lat, inverse=False):
446 '''Compute the I{authalic} auxiliary latitude (Xi) or the I{inverse} thereof.
448 @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}).
449 @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and
450 return the geodetic latitude (C{bool}).
452 @return: The I{authalic} (or geodetic) latitude in C{degrees90}.
454 @see: U{Inverse-/AuthalicLatitude<https://GeographicLib.SourceForge.io/
455 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude
456 <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and
457 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16.
458 '''
459 if self.f:
460 f = self._albersCyl._tanf if inverse else \
461 self._albersCyl._txif # PYCHOK attr
462 lat = atan1d(f(tan(Phid(lat)))) # PYCHOK attr
463 return _aux(lat, inverse, Ellipsoid.auxAuthalic)
465 def auxConformal(self, lat, inverse=False):
466 '''Compute the I{conformal} auxiliary latitude (Chi) or the I{inverse} thereof.
468 @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}).
469 @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and
470 return the geodetic latitude (C{bool}).
472 @return: The I{conformal} (or geodetic) latitude in C{degrees90}.
474 @see: U{Inverse-/ConformalLatitude<https://GeographicLib.SourceForge.io/
475 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude
476 <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and
477 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
478 '''
479 if self.f:
480 f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr
481 lat = atan1d(f(tan(Phid(lat)))) # PYCHOK attr
482 return _aux(lat, inverse, Ellipsoid.auxConformal)
484 def auxGeocentric(self, lat, inverse=False, height=_0_0):
485 '''Compute the I{geocentric} auxiliary latitude (Theta) or the I{inverse} thereof.
487 @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}).
488 @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and
489 return the I{geocentric} latitude (C{bool}).
490 @kwarg height: Optional, ellipsoidal height (C{meter}).
492 @return: The I{geocentric} (or geodetic) latitude in C{degrees90}.
494 @see: U{Inverse-/GeocentricLatitude<https://GeographicLib.SourceForge.io/
495 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude
496 <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and
497 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18.
498 '''
499 if self.f: # and lat
500 t = tan(Phid(lat))
501 f = self.b2_a2
502 if height:
503 if inverse:
504 lat = atan1d(t * f) # geodetic n
505 d, f = f, _1_0
506 else:
507 d = _1_0
508 n = self.rocPrimeVertical(lat)
509 f = _over(n * f + height, n * d + height)
510 elif inverse:
511 f = self.a2_b2
512 lat = atan1d(t * f)
513 return _aux(lat, inverse, Ellipsoid.auxGeocentric)
515 def auxIsometric(self, lat, inverse=False):
516 '''Compute the I{isometric} auxiliary latitude (Psi) or the I{inverse} thereof.
518 @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}).
519 @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and
520 return the geodetic latitude (C{bool}).
522 @return: The I{isometric} (or geodetic) latitude in C{degrees}.
524 @note: The I{isometric} latitude for geodetic C{+/-90} is far
525 outside the C{[-90..+90]} range but the inverse thereof
526 is the original geodetic latitude.
528 @see: U{Inverse-/IsometricLatitude<https://GeographicLib.SourceForge.io/
529 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude
530 <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and
531 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
532 '''
533 if self.f:
534 r = Phid(lat, clip=0)
535 lat = degrees(atan1(self.es_tauf(sinh(r))) if inverse else
536 asinh(self.es_taupf(tan(r))))
537 # clip=0, since auxIsometric(+/-90) is far outside [-90..+90]
538 return _aux(lat, inverse, Ellipsoid.auxIsometric, clip=0)
540 def auxParametric(self, lat, inverse=False):
541 '''Compute the I{parametric} auxiliary latitude (Beta) or the I{inverse} thereof.
543 @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}).
544 @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and
545 return the geodetic latitude (C{bool}).
547 @return: The I{parametric} (or geodetic) latitude in C{degrees90}.
549 @see: U{Inverse-/ParametricLatitude<https://GeographicLib.SourceForge.io/
550 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude
551 <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>},
552 and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18.
553 '''
554 if self.f:
555 lat = self._beta(Lat(lat), inverse=inverse)
556 return _aux(lat, inverse, Ellipsoid.auxParametric)
558 auxReduced = auxParametric # synonymous
560 def auxRectifying(self, lat, inverse=False):
561 '''Compute the I{rectifying} auxiliary latitude (Mu) or the I{inverse} thereof.
563 @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}).
564 @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and
565 return the geodetic latitude (C{bool}).
567 @return: The I{rectifying} (or geodetic) latitude in C{degrees90}.
569 @see: U{Inverse-/RectifyingLatitude<https://GeographicLib.SourceForge.io/
570 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude
571 <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and
572 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17.
573 '''
574 if self.f:
575 lat = Lat(lat)
576 if 0 < fabs(lat) < _90_0:
577 if inverse:
578 e = self._elliptic_e22
579 d = degrees90(e.fEinv(e.cE * lat / _90_0))
580 lat = self.auxParametric(d, inverse=True)
581 else:
582 lat = _over(self.Llat(lat), self.L) * _90_0
583 return _aux(lat, inverse, Ellipsoid.auxRectifying)
585 @Property_RO
586 def b(self):
587 '''Get the I{polar} radius, semi-axis (C{meter}).
588 '''
589 return self._b
591 polaradius = b # = Rpolar
593 @Property_RO
594 def b_a(self):
595 '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a == f1 == 1 - f}.
597 @see: Property L{f1}.
598 '''
599 return self._assert(self.b / self.a, b_a=self.f1, f0=_1_0)
601 @Property_RO
602 def b2(self):
603 '''Get the I{polar} radius I{squared} (C{float}), M{b**2}.
604 '''
605 return Meter2(b2=self.b**2)
607 @Property_RO
608 def b2_a(self):
609 '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see C{rocMeridional}C{(0)}.
611 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
612 '''
613 return Radius(b2_a=_over(self.b2, self.a) if self.f else self.b)
615 @Property_RO
616 def b2_a2(self):
617 '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2}
618 == M{(1 - f)**2} == M{1 - e**2} == C{e21}.
619 '''
620 return Float(b2_a2=self.b_a**2 if self.f else _1_0)
622 def _beta(self, lat, inverse=False):
623 '''(INTERNAL) Get the I{parametric (or reduced) auxiliary latitude} or inverse thereof.
624 '''
625 s, c = sincos2d(lat) # like Karney's tand(lat)
626 s *= self.a_b if inverse else self.b_a
627 return atan1d(s, c)
629 @Property_RO
630 def BetaKs(self):
631 '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
632 '''
633 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # noqa: E702 ;
634 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
635 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600),
636 _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned
637 _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned
638 _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned
639 _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned
640 _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne
641 _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned
642 _T(191773887257/3719607091200)) # PYCHOK unaligned
644 @deprecated_Property_RO
645 def c(self): # PYCHOK no cover
646 '''DEPRECATED, use property C{R2} or C{Rauthalic}.'''
647 return self.R2
649 @Property_RO
650 def c2(self):
651 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}).
653 @see: Properties L{c2x}, L{area}, L{R2}, L{Rauthalic}, I{Karney's} U{equation (60)
654 <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area
655 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>},
656 U{Authalic radius<https://WikiPedia.org/wiki/Earth_radius#Authalic_radius>}, U{Surface area
657 <https://WikiPedia.org/wiki/Ellipsoid>} and U{surface area
658 <https://www.Numericana.com/answer/geometry.htm#oblate>}.
659 '''
660 return self._c2f(False)
662 @Property_RO
663 def c2x(self):
664 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}), more accurate for very I{oblate}
665 ellipsoids.
667 @see: Properties L{c2}, L{areax}, L{R2x}, L{Rauthalicx}, class L{GeodesicExact} and I{Karney}'s comments at C++
668 attribute U{GeodesicExact._c2<https://GeographicLib.SourceForge.io/C++/doc/GeodesicExact_8cpp_source.html>}.
669 '''
670 return self._c2f(True)
672 def _c2f(self, c2x):
673 '''(INTERNAL) Helper for C{.c2} and C{.c2x}.
674 '''
675 f, c2 = self.f, self.b2
676 if f:
677 e = self.e
678 if e > EPS0:
679 if f > 0: # .isOblate
680 c2 *= (asinh(sqrt(self.e22abs)) if c2x else atanh(e)) / e
681 elif f < 0: # .isProlate
682 c2 *= atan1(e) / e # XXX asin?
683 c2 = Meter2(c2=(self.a2 + c2) * _0_5)
684 return c2
686 def circle4(self, lat):
687 '''Get the equatorial or a parallel I{circle of latitude}.
689 @arg lat: Geodetic latitude (C{degrees90}, C{str}).
691 @return: A L{Circle4Tuple}C{(radius, height, lat, beta)}.
693 @raise RangeError: Latitude B{C{lat}} outside valid range and
694 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
696 @raise TypeError: Invalid B{C{lat}}.
698 @raise ValueError: Invalid B{C{lat}}.
700 @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude}
701 <https://WikiPedia.org/wiki/Latitude>}, I{Karney's} C++ U{CircleRadius and CircleHeight
702 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>}
703 and method C{Rlat}.
704 '''
705 lat = Lat(lat)
706 if lat:
707 B = lat # beta
708 if fabs(lat) < _90_0:
709 if self.f:
710 B = self._beta(lat)
711 z, r = sincos2d(B)
712 r *= self.a
713 z *= self.b
714 else: # near-polar
715 r, z = _0_0, copysign0(self.b, lat)
716 else: # equator
717 r = self.a
718 z = lat = B = _0_0
719 return Circle4Tuple(r, z, lat, B)
721 def degrees2m(self, deg, lat=0):
722 '''Convert an angle along the equator or along a parallel
723 of (geodetic) latitude to the distance.
725 @arg deg: The angle (C{degrees}).
726 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
728 @return: Distance (C{meter}, same units as the equatorial
729 and polar radii) or C{0} for near-polar B{C{lat}}.
731 @raise RangeError: Latitude B{C{lat}} outside valid range and
732 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
734 @raise ValueError: Invalid B{C{deg}} or B{C{lat}}.
735 '''
736 return self.radians2m(radians(deg), lat=lat)
738 def distance2(self, lat0, lon0, lat1, lon1):
739 '''I{Approximate} the distance and (initial) bearing between
740 two points based on the U{local, flat earth approximation
741 <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny
742 <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
744 I{Suitable only for distances of several hundred Km or Miles
745 and only between points not near-polar}.
747 @arg lat0: From latitude (C{degrees}).
748 @arg lon0: From longitude (C{degrees}).
749 @arg lat1: To latitude (C{degrees}).
750 @arg lon1: To longitude (C{degrees}).
752 @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance}
753 in same units as this ellipsoid's axes.
755 @note: The meridional and prime_vertical radii of curvature are
756 taken and scaled I{at the initial latitude}, see C{roc2}.
758 @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}.
759 '''
760 phi0 = Phid(lat0=lat0)
761 m, n = self.roc2_(phi0, scaled=True)
762 m *= Phid(lat1=lat1) - phi0
763 n *= Lamd(lon1=lon1) - Lamd(lon0=lon0)
764 return Distance2Tuple(hypot(m, n), atan2b(n, m))
766 @Property_RO
767 def e(self):
768 '''Get the I{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}.
770 @see: Property L{es}.
771 '''
772 return Float(e=sqrt(self.e2abs) if self.e2 else _0_0)
774 @deprecated_Property_RO
775 def e12(self): # see property ._e12
776 '''DEPRECATED, use property C{e21}.'''
777 return self.e21
779# @Property_RO
780# def _e12(self): # see property ._elliptic_e12
781# # (INTERNAL) until e12 above can be replaced with e21.
782# return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2)
784 @Property_RO
785 def e2(self):
786 '''Get the I{signed, (1st) eccentricity squared} (C{float}), M{f * (2 - f)
787 == 1 - (b / a)**2}, see C{a_b2e2}.
788 '''
789 return self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f))
791 @Property_RO
792 def e2abs(self):
793 '''Get the I{unsigned, (1st) eccentricity squared} (C{float}).
794 '''
795 return fabs(self.e2)
797 @Property_RO
798 def e21(self):
799 '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2}
800 == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}.
801 '''
802 return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0)
804# _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2
805 _1_e21 = a2_b2 # == M{1 / e21} == M{1 / (1 - e**2)}
807 @Property_RO
808 def e22(self):
809 '''Get the I{signed, 2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)
810 == e2 / (1 - f)**2 == (a / b)**2 - 1}, see C{a_b2e22}.
811 '''
812 return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f))
814 @Property_RO
815 def e22abs(self):
816 '''Get the I{unsigned, 2nd eccentricity squared} (C{float}).
817 '''
818 return fabs(self.e22)
820 @Property_RO
821 def e32(self):
822 '''Get the I{signed, 3rd eccentricity squared} (C{float}), M{e2 / (2 - e2)
823 == (a**2 - b**2) / (a**2 + b**2)}, see C{a_b2e32}.
824 '''
825 return self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f))
827 @Property_RO
828 def e32abs(self):
829 '''Get the I{unsigned, 3rd eccentricity squared} (C{float}).
830 '''
831 return fabs(self.e32)
833 @Property_RO
834 def e4(self):
835 '''Get the I{unsignd, (1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}.
836 '''
837 return Float(e4=self.e2**2 if self.e2 else _0_0)
839 eccentricity = e # eccentricity
840# eccentricity2 = e2 # eccentricity squared
841 eccentricity1st2 = e2 # first eccentricity squared, signed
842 eccentricity2nd2 = e22 # second eccentricity squared, signed
843 eccentricity3rd2 = e32 # third eccentricity squared, signed
845 def ecef(self, Ecef=None):
846 '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter.
848 @kwarg Ecef: ECEF class to use, default L{EcefKarney}.
850 @return: An ECEF converter for this C{ellipsoid}.
852 @raise TypeError: Invalid B{C{Ecef}}.
854 @see: Module L{pygeodesy.ecef}.
855 '''
856 return _MODS.ecef._4Ecef(self, Ecef)
858 @Property_RO
859 def _elliptic_e12(self): # see I{Karney}'s Ellipsoid._e12
860 '''(INTERNAL) Elliptic helper for C{Rhumb}.
861 '''
862 e12 = _over(self.e2, self.e2 - _1_0) # NOT DEPRECATED .e12!
863 return _MODS.elliptic.Elliptic(e12)
865 @Property_RO
866 def _elliptic_e22(self): # aka ._elliptic_ep2
867 '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}.
868 '''
869 return _MODS.elliptic.Elliptic(-self.e22abs) # complex
871 equatoradius = a # Requatorial
873 def e2s(self, s):
874 '''Compute norm M{sqrt(1 - e2 * s**2)}.
876 @arg s: Sine value (C{scalar}).
878 @return: Norm (C{float}).
880 @raise ValueError: Invalid B{C{s}}.
881 '''
882 return sqrt(self.e2s2(s)) if self.e2 else _1_0
884 def e2s2(self, s):
885 '''Compute M{1 - e2 * s**2}.
887 @arg s: Sine value (C{scalar}).
889 @return: Result (C{float}).
891 @raise ValueError: Invalid B{C{s}}.
892 '''
893 r, e2 = _1_0, self.e2
894 if e2: # and s
895 try:
896 r -= e2 * Scalar(s=s)**2
897 if r < 0:
898 raise ValueError(_negative_)
899 except (TypeError, ValueError) as x:
900 t = self._DOT_(typename(Ellipsoid.e2s2))
901 raise _ValueError(t, s, cause=x)
902 return r
904 @Property_RO
905 def es(self):
906 '''Get the I{signed (1st) eccentricity} (C{float}).
908 @see: Property L{e}.
909 '''
910 # note, self.e is always non-negative
911 return Float(es=copysign0(self.e, self.f)) # see .ups
913 def es_atanh(self, x):
914 '''Compute M{es * atanh(es * x)} or M{-es * atan(es * x)}
915 for I{oblate} respectively I{prolate} ellipsoids where
916 I{es} is the I{signed} (1st) eccentricity.
918 @raise ValueError: Invalid B{C{x}}.
920 @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/
921 C++/doc/classGeographicLib_1_1Math.html>}.
922 '''
923 return self._es_atanh(Scalar(x=x)) if self.f else _0_0
925 def _es_atanh(self, x): # see .albers._atanhee, .AuxLat._atanhee
926 '''(INTERNAL) Helper for .es_atanh, ._es_taupf2 and ._exp_es_atanh.
927 '''
928 es = self.es # signOf(es) == signOf(f)
929 return es * (atanh(es * x) if es > 0 else # .isOblate
930 (-atan(es * x) if es < 0 else # .isProlate
931 _0_0)) # .isSpherical
933 @Property_RO
934 def es_c(self):
935 '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}.
936 '''
937 return Float(es_c=(self._exp_es_atanh_1 * self.b_a) if self.f else _1_0)
939 def es_tauf(self, taup):
940 '''Compute I{Karney}'s U{equations (19), (20) and (21)
941 <https://ArXiv.org/abs/1002.1417>}.
943 @see: I{Karney}'s C++ method U{Math::tauf<https://GeographicLib.
944 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>} and
945 and I{Veness}' JavaScript method U{toLatLon<https://www.
946 Movable-Type.co.UK/scripts/latlong-utm-mgrs.html>}.
947 '''
948 t = Scalar(taup=taup)
949 if self.f: # .isEllipsoidal
950 a = fabs(t)
951 T = (self._exp_es_atanh_1 if a > 70 else self._1_e21) * t
952 if fabs(T * _EPSqrt) < _2_0: # handles +/- INF and NAN
953 s = (a * _TOL) if a > _1_0 else _TOL
954 for T, _, d in self._es_tauf3(t, T): # max 2
955 if fabs(d) < s:
956 break
957 t = Scalar(tauf=T)
958 return t
960 def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB
961 '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most
962 B{C{N}} Newton iterations, converging rapidly except when C{delta}
963 toggles on +/-1.12e-16 or +/-4.47e-16, see C{.utm.Utm._toLLEB}.
964 '''
965 e = self._1_e21
966 _F2_ = Fsum(T).fsum2f_ # τ0
967 _tf2 = self._es_taupf2
968 for i in range(1, N + 1):
969 a, h = _tf2(T)
970 # = (taup - a) / hypot1(a) / ((e + T**2) / h)
971 d = _over((taup - a) * (T**2 + e), hypot1(a) * h)
972 T, d = _F2_(d) # τi, (τi - τi-1)
973 yield T, i, d
975 def es_taupf(self, tau):
976 '''Compute I{Karney}'s U{equations (7), (8) and (9)
977 <https://ArXiv.org/abs/1002.1417>}.
979 @see: I{Karney}'s C++ method U{Math::taupf<https://GeographicLib.
980 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}.
981 '''
982 t = Scalar(tau=tau)
983 if self.f: # .isEllipsoidal
984 t, _ = self._es_taupf2(t)
985 t = Scalar(taupf=t)
986 return t
988 def _es_taupf2(self, tau):
989 '''(INTERNAL) Return 2-tuple C{(es_taupf(tau), hypot1(tau))}.
990 '''
991 if _isfinite(tau):
992 h = hypot1(tau)
993 s = sinh(self._es_atanh(tau / h))
994 a = hypot1(s) * tau - h * s
995 else:
996 a, h = tau, INF
997 return a, h
999 @Property_RO
1000 def _exp_es_atanh_1(self):
1001 '''(INTERNAL) Helper for .es_c and .es_tauf.
1002 '''
1003 return exp(self._es_atanh(_1_0)) if self.es else _1_0
1005 @Property_RO
1006 def f(self):
1007 '''Get the I{flattening} (C{scalar}), M{(a - b) / a}, C{0} for spherical, negative for prolate.
1008 '''
1009 return self._f
1011 @Property_RO
1012 def f_(self):
1013 '''Get the I{inverse flattening} (C{scalar}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}.
1014 '''
1015 return self._f_
1017 @Property_RO
1018 def f1(self):
1019 '''Get the I{1 - flattening} (C{float}), M{f1 == 1 - f == b / a}.
1021 @see: Property L{b_a}.
1022 '''
1023 return Float(f1=_1_0 - self.f)
1025 @Property_RO
1026 def f2(self):
1027 '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}.
1028 '''
1029 return self._assert(self.a_b - _1_0, f2=f2f2(self.f))
1031 @deprecated_Property_RO
1032 def geodesic(self):
1033 '''DEPRECATED, use property C{geodesicw}.'''
1034 return self.geodesicw
1036 def geodesic_(self, exact=True):
1037 '''Get the an I{exact} C{Geodesic...} instance for this ellipsoid.
1039 @kwarg exact: If C{bool} return L{GeodesicExact}C{(exact=B{exact}, ...)},
1040 otherwise a L{Geodesic}, L{GeodesicExact} or L{GeodesicSolve}
1041 instance for I{this} ellipsoid.
1043 @return: The C{exact} geodesic (C{Geodesic...}).
1045 @raise TypeError: Invalid B{C{exact}}.
1047 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1048 '''
1049 if isbool(exact): # for consistenccy with C{.rhumb_}
1050 g = _MODS.geodesicx.GeodesicExact(self, C4order=30 if exact else 24,
1051 name=self.name)
1052 else:
1053 g = exact
1054 E = _xattr(g, ellipsoid=None)
1055 if not (E is self and isinstance(g, self._Geodesics)):
1056 raise _ValueError(exact=g, ellipsoid=E, txt_not_=self.name)
1057 return g
1059 @property_ROver
1060 def _Geodesics(self):
1061 '''(INTERNAL) Get all C{Geodesic...} classes, I{once}.
1062 '''
1063 t = (_MODS.geodesicx.GeodesicExact,
1064 _MODS.geodsolve.GeodesicSolve)
1065 try:
1066 t += (_MODS.geodesicw.Geodesic,
1067 _MODS.geodesicw._wrapped.Geodesic)
1068 except ImportError:
1069 pass
1070 return t # overwrite property_ROver
1072 @property_RO
1073 def geodesicw(self):
1074 '''Get this ellipsoid's I{wrapped} U{geodesicw.Geodesic
1075 <https://GeographicLib.SourceForge.io/Python/doc/code.html>}, provided
1076 I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1077 package is installed.
1078 '''
1079 # if not self.isEllipsoidal:
1080 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1081 return _MODS.geodesicw.Geodesic(self)
1083 @property_RO
1084 def geodesicx(self):
1085 '''Get this ellipsoid's I{exact} L{GeodesicExact}.
1086 '''
1087 # if not self.isEllipsoidal:
1088 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1089 return _MODS.geodesicx.GeodesicExact(self, name=self.name)
1091 @property
1092 def geodsolve(self):
1093 '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility
1094 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1095 provided the path to the C{GeodSolve} executable is specified with env
1096 variable C{PYGEODESY_GEODSOLVE} or re-/set with this property..
1097 '''
1098 # if not self.isEllipsoidal:
1099 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1100 return _MODS.geodsolve.GeodesicSolve(self, path=self._geodsolve, name=self.name)
1102 @geodsolve.setter # PYCHOK setter!
1103 def geodsolve(self, path):
1104 '''Re-/set the (fully qualified) path to the U{GeodSolve
1105 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1106 overriding env variable C{PYGEODESY_GEODSOLVE} (C{str}).
1107 '''
1108 self._geodsolve = path
1110 def hartzell4(self, pov, los=None):
1111 '''Compute the intersection of this ellipsoid's surface and a Line-Of-Sight
1112 from a Point-Of-View in space.
1114 @arg pov: Point-Of-View outside this ellipsoid (C{Cartesian}, L{Ecef9Tuple}
1115 or L{Vector3d}).
1116 @kwarg los: Line-Of-Sight, I{direction} to this ellipsoid (L{Los}, L{Vector3d})
1117 or C{True} for the I{normal, perpendicular, plumb} to the surface
1118 of this ellipsoid or C{False} or C{None} to point to its center.
1120 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1121 C{y} and C{z} of the projection on or the intersection with this
1122 ellipsoid and the I{distance} C{h} from B{C{pov}} to C{(x, y, z)}
1123 along B{C{los}}, all in C{meter}, conventionally.
1125 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, or B{C{pov}}
1126 is inside this ellipsoid or B{C{los}} points
1127 outside this ellipsoid or in opposite direction.
1129 @raise TypeError: Invalid B{C{pov}} or B{C{los}}.
1131 @see: U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.
1132 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and
1133 methods L{Ellipsoid.height4} and L{Triaxial.hartzell4}.
1134 '''
1135 try:
1136 v, d, i = _MODS.triaxials._hartzell3(pov, los, self._triaxial)
1137 except Exception as x:
1138 raise IntersectionError(pov=pov, los=los, cause=x)
1139 return Vector4Tuple(v.x, v.y, v.z, d, iteration=i, name__=self.hartzell4)
1141 @Property_RO
1142 def _hash(self):
1143 return hash((self.a, self.f))
1145 def height4(self, xyz, normal=True):
1146 '''Compute the projection on and the height of a cartesian above or below
1147 this ellipsoid's surface.
1149 @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d},
1150 L{Vector3Tuple} or L{Vector4Tuple}).
1151 @kwarg normal: If C{True}, the projection is perpendicular to (the nearest
1152 point on) this ellipsoid's surface, otherwise the C{radial}
1153 line to this ellipsoid's center (C{bool}).
1155 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1156 C{y} and C{z} of the projection on and the height C{h} above or
1157 below this ellipsoid's surface, all in C{meter}, conventionally.
1159 @raise ValueError: Null B{C{xyz}}.
1161 @raise TypeError: Non-cartesian B{C{xyz}}.
1163 @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>}
1164 and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse and
1165 methods L{Ellipsoid.hartzell4} and L{Triaxial.height4}.
1166 '''
1167 v = _MODS.vector3d._otherV3d(xyz=xyz)
1168 r = v.length
1170 a, b, i = self.a, self.b, None
1171 if r < EPS0: # EPS
1172 v = v.times(_0_0)
1173 h = -a
1175 elif self.isSpherical:
1176 v = v.times(a / r)
1177 h = r - a
1179 elif normal: # perpendicular to ellipsoid
1180 x, y = hypot(v.x, v.y), fabs(v.z)
1181 if x < EPS0: # PYCHOK no cover
1182 z = copysign0(b, v.z)
1183 v = Vector3Tuple(v.x, v.y, z)
1184 h = y - b # polar
1185 elif y < EPS0: # PYCHOK no cover
1186 t = a / r
1187 v = v.times_(t, t, 0) # force z=0.0
1188 h = x - a # equatorial
1189 else: # normal in 1st quadrant
1190 x, y, i = _MODS.triaxials._plumbTo3(x, y, self)
1191 t, v = v, v.times_(x, x, y)
1192 h = t.minus(v).length
1194 else: # radial to ellipsoid's center
1195 h = hypot_(a * v.z, b * v.x, b * v.y)
1196 t = (a * b / h) if h > EPS0 else _0_0 # EPS
1197 v = v.times(t)
1198 h = r * (_1_0 - t)
1200 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name__=self.height4)
1202 def _heightB(self, sa, ca, z, p): # in ecef.EcefSudano, ecec.EcefVeness
1203 '''(INTERNAL) Height above ellipsoid (Bowring eqn 7) at C{lat}.
1204 '''
1205 # sa, ca = sincos2d(lat)
1206 # p = hypot(x, y) # distance to polar axis
1208 # r = a / self.e2s(sa) # length of normal terminated by polar axis
1209 # h = p * ca + z * sa - (a * a / r)
1210 return (p * ca + fabs(z * sa) - self.a * self.e2s(sa)) if sa else (p - self.a)
1212 @Property_RO
1213 def _heightMax(self):
1214 '''(INTERNAL) Get the height limit (C{meter}, conventionally).
1215 '''
1216 return self.a / EPS_2 # self.a * _2_EPS, about 12M lightyears
1218 def _hubeny_2(self, phi2, phi1, lam21, scaled=True, squared=True):
1219 '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_},
1220 returning the I{angular} distance in C{radians squared} or C{radians}
1221 '''
1222 m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=scaled)
1223 h, r = (hypot2, self.a2_) if squared else (hypot, _1_0 / self.a)
1224 return h(m * (phi2 - phi1), n * lam21) * r
1226 @Property_RO
1227 def isEllipsoidal(self):
1228 '''Is this model I{ellipsoidal} (C{bool})?
1229 '''
1230 return self.f != 0
1232 @Property_RO
1233 def isOblate(self):
1234 '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or
1235 spherical otherwise.
1236 '''
1237 return self.f > 0
1239 @Property_RO
1240 def isProlate(self):
1241 '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or
1242 spherical otherwise.
1243 '''
1244 return self.f < 0
1246 @Property_RO
1247 def isSpherical(self):
1248 '''Is this ellipsoid I{spherical} (C{bool})?
1249 '''
1250 return self.f == 0
1252 def _Kseries(self, *AB8Ks):
1253 '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha
1254 or Beta series coefficients per I{Karney}'s U{equations (35)
1255 and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}.
1257 @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series
1258 coefficient tuples.
1260 @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}).
1262 @see: I{Karney}'s 30-th order U{TMseries30
1263 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}.
1264 '''
1265 k = self.KsOrder
1266 if self.n:
1267 ns = fpowers(self.n, k)
1268 ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k))
1269 else:
1270 ks = _0_0s(k)
1271 return ks
1273 @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''')
1274 def KsOrder(self):
1275 '''Get the I{Krüger} series' order (C{int} 4, 6 or 8).
1276 '''
1277 return self._KsOrder
1279 @KsOrder.setter # PYCHOK setter!
1280 def KsOrder(self, order):
1281 '''Set the I{Krüger} series' order (C{int} 4, 6 or 8).
1283 @raise ValueError: Invalid B{C{order}}.
1284 '''
1285 if not (isint(order) and _isin(order, 4, 6, 8)):
1286 raise _ValueError(order=order)
1287 if self._KsOrder != order:
1288 Ellipsoid.AlphaKs._update(self)
1289 Ellipsoid.BetaKs._update(self)
1290 self._KsOrder = order
1292 @Property_RO
1293 def L(self):
1294 '''Get the I{quarter meridian} C{L}, aka the C{polar distance}
1295 along a meridian between the equator and a pole (C{meter}),
1296 M{b * Elliptic(-e2 / (1 - e2)).cE} or M{b * PI / 2}.
1297 '''
1298 r = self._elliptic_e22.cE if self.f else PI_2
1299 return Distance(L=self.b * r)
1301 def Llat(self, lat):
1302 '''Return the I{meridional length}, the distance along a meridian
1303 between the equator and a (geodetic) latitude, see C{L}.
1305 @arg lat: Geodetic latitude (C{degrees90}).
1307 @return: The meridional length at B{C{lat}}, negative on southern
1308 hemisphere (C{meter}).
1309 '''
1310 r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phid(lat)
1311 return Distance(Llat=self.b * r)
1313 Lmeridian = Llat # meridional distance
1315 @property_RO
1316 def _Lpd(self):
1317 '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}.
1318 '''
1319 return Meter(_Lpd=self.L / _90_0)
1321 @property_RO
1322 def _Lpr(self):
1323 '''Get the I{quarter meridian} per radian (C{meter}), M{self.L / PI_2}.
1324 '''
1325 return Meter(_Lpr=self.L / PI_2)
1327 @deprecated_Property_RO
1328 def majoradius(self): # PYCHOK no cover
1329 '''DEPRECATED, use property C{a} or C{Requatorial}.'''
1330 return self.a
1332 def m2degrees(self, distance, lat=0):
1333 '''Convert a distance to an angle along the equator or along
1334 a parallel of (geodetic) latitude.
1336 @arg distance: Distance (C{meter}).
1337 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1339 @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}.
1341 @raise RangeError: Latitude B{C{lat}} outside valid range and
1342 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1344 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1345 '''
1346 return degrees(self.m2radians(distance, lat=lat))
1348 def m2radians(self, distance, lat=0):
1349 '''Convert a distance to an angle along the equator or along
1350 a parallel of (geodetic) latitude.
1352 @arg distance: Distance (C{meter}).
1353 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1355 @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}.
1357 @raise RangeError: Latitude B{C{lat}} outside valid range and
1358 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1360 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1361 '''
1362 r = self.circle4(lat).radius if lat else self.a
1363 return m2radians(distance, radius=r, lat=0)
1365 @deprecated_Property_RO
1366 def minoradius(self): # PYCHOK no cover
1367 '''DEPRECATED, use property C{b}, C{polaradius} or C{Rpolar}.'''
1368 return self.b
1370 @Property_RO
1371 def n(self):
1372 '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}.
1373 '''
1374 return self._assert(a_b2n(self.a, self.b), n=f2n(self.f))
1376 flattening = f
1377 flattening1st = f
1378 flattening2nd = f2
1379 flattening3rd = n
1381 polaradius = b # Rpolar
1383# Q = A # I{meridian arc unit} C{Q}, the mean, meridional length I{per radian}
1385 @deprecated_Property_RO
1386 def quarteradius(self): # PYCHOK no cover
1387 '''DEPRECATED, use property C{L} or method C{Llat}.'''
1388 return self.L
1390 @Property_RO
1391 def R1(self):
1392 '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}.
1394 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}
1395 and method C{Rgeometric}.
1396 '''
1397 r = Fsum(self.a, self.a, self.b).fover(_3_0) if self.f else self.a
1398 return Radius(R1=r)
1400 Rmean = R1
1402 @Property_RO
1403 def R2(self):
1404 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}.
1406 @see: C{R2x}, C{c2}, C{area} and U{Earth radius
1407 <https://WikiPedia.org/wiki/Earth_radius>}.
1408 '''
1409 return Radius(R2=sqrt(self.c2) if self.f else self.a)
1410# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1411# # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar
1412# # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3
1413# return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar)
1415 Rauthalic = R2
1417 @Property_RO
1418 def R2x(self):
1419 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}.
1421 @see: C{R2}, C{c2x} and C{areax}.
1422 '''
1423 return Radius(R2x=sqrt(self.c2x) if self.f else self.a)
1425 Rauthalicx = R2x
1427 @Property_RO
1428 def R3(self):
1429 '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}.
1431 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}.
1432 '''
1433 r = (cbrt(self.b_a) * self.a) if self.f else self.a
1434 return Radius(R3=r)
1436 Rvolumetric = R3
1438 def radians2m(self, rad, lat=0):
1439 '''Convert an angle to the distance along the equator or along
1440 a parallel of (geodetic) latitude.
1442 @arg rad: The angle (C{radians}).
1443 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1445 @return: Distance (C{meter}, same units as the equatorial
1446 and polar radii) or C{0} for near-polar B{C{lat}}.
1448 @raise RangeError: Latitude B{C{lat}} outside valid range and
1449 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1451 @raise ValueError: Invalid B{C{rad}} or B{C{lat}}.
1452 '''
1453 r = self.circle4(lat).radius if lat else self.a
1454 return radians2m(rad, radius=r, lat=0)
1456 @Property_RO
1457 def Rbiaxial(self):
1458 '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}.
1460 @see: C{Rtriaxial}
1461 '''
1462 a, b = self.a, self.b
1463 if b < a:
1464 b = sqrt(_0_5 + self.b2_a2 * _0_5) * a
1465 elif b > a:
1466 b *= sqrt(_0_5 + self.a2_b2 * _0_5)
1467 return Radius(Rbiaxial=b)
1469 Requatorial = a # for consistent naming
1471 def Rgeocentric(self, lat):
1472 '''Compute the I{geocentric} earth radius of (geodetic) latitude.
1474 @arg lat: Latitude (C{degrees90}).
1476 @return: Geocentric earth radius (C{meter}).
1478 @raise ValueError: Invalid B{C{lat}}.
1480 @see: U{Geocentric Radius
1481 <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>}
1482 '''
1483 r, p = self.a, Phid(lat)
1484 if p and self.f:
1485 if fabs(p) < PI_2:
1486 s2, c2 = _s2_c2(p)
1487 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2))
1488 # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2)))
1489 # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2))
1490 # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2))
1491 s2 *= self.b2_a2
1492 r *= sqrt((c2 + self.b2_a2 * s2) / (c2 + s2))
1493 else:
1494 r = self.b
1495 return Radius(Rgeocentric=r)
1497 @Property_RO
1498 def Rgeometric(self):
1499 '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}.
1501 @see: C{R1}.
1502 '''
1503 g = sqrt(self.a * self.b) if self.f else self.a
1504 return Radius(Rgeometric=g)
1506 def rhumb_(self, exact=True):
1507 '''Get the an I{exact} C{Rhumb...} instance for this ellipsoid.
1509 @kwarg exact: If C{bool} or C{None} return L{Rhumb}C{(exact=B{exact}, ...)},
1510 otherwise a L{Rhumb}, L{RhumbAux} or L{RhumbSolve} instance
1511 for I{this} ellipsoid.
1513 @return: The C{exact} rhumb (C{Rhumb...}).
1515 @raise TypeError: Invalid B{C{exact}}.
1517 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1518 '''
1519 if isbool(exact): # use Rhumb for backward compatibility
1520 r = _MODS.rhumb.ekx.Rhumb(self, exact=exact, name=self.name)
1521 else:
1522 r = exact
1523 E = _xattr(r, ellipsoid=None)
1524 if not (E is self and isinstance(r, self._Rhumbs)):
1525 raise _ValueError(exact=r, ellipsosid=E, txt_not_=self.name)
1526 return r
1528 @property_RO
1529 def rhumbaux(self):
1530 '''Get this ellipsoid's I{Auxiliary} C{rhumb.RhumbAux}.
1531 '''
1532 # if not self.isEllipsoidal:
1533 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1534 return _MODS.rhumb.aux_.RhumbAux(self, name=self.name)
1536 @property_RO
1537 def rhumbekx(self):
1538 '''Get this ellipsoid's I{Elliptic, Krüger} C{rhumb.Rhumb}.
1539 '''
1540 # if not self.isEllipsoidal:
1541 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1542 return _MODS.rhumb.ekx.Rhumb(self, name=self.name)
1544 @property_ROver
1545 def _Rhumbs(self):
1546 '''(INTERNAL) Get all C{Rhumb...} classes, I{once}.
1547 '''
1548 r = _MODS.rhumb
1549 return (r.aux_.RhumbAux, # overwrite property_ROver
1550 r.ekx.Rhumb, r.solve.RhumbSolve)
1552 @property
1553 def rhumbsolve(self):
1554 '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility
1555 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1556 provided the path to the C{RhumbSolve} executable is specified with env
1557 variable C{PYGEODESY_RHUMBSOLVE} or re-/set with this property.
1558 '''
1559 # if not self.isEllipsoidal:
1560 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1561 return _MODS.rhumb.solve.RhumbSolve(self, path=self._rhumbsolve, name=self.name)
1563 @rhumbsolve.setter # PYCHOK setter!
1564 def rhumbsolve(self, path):
1565 '''Re-/set the (fully qualified) path to the U{RhumbSolve
1566 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1567 overriding env variable C{PYGEODESY_RHUMBSOLVE} (C{str}).
1568 '''
1569 self._rhumbsolve = path
1571 @deprecated_property_RO
1572 def rhumbx(self):
1573 '''DEPRECATED on 2023.11.28, use property C{rhumbekx}.'''
1574 return self.rhumbekx
1576 def Rlat(self, lat):
1577 '''I{Approximate} the earth radius of (geodetic) latitude.
1579 @arg lat: Latitude (C{degrees90}).
1581 @return: Approximate earth radius (C{meter}).
1583 @raise RangeError: Latitude B{C{lat}} outside valid range and
1584 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1586 @raise TypeError: Invalid B{C{lat}}.
1588 @raise ValueError: Invalid B{C{lat}}.
1590 @note: C{Rlat(B{90})} equals C{Rpolar}.
1592 @see: Method C{circle4}.
1593 '''
1594 # r = a - (a - b) * |lat| / 90
1595 r = self.a
1596 if self.f and lat: # .isEllipsoidal
1597 r -= (r - self.b) * fabs(Lat(lat)) / _90_0
1598 r = Radius(Rlat=r)
1599 return r
1601 Rpolar = b # for consistent naming
1603 def roc1_(self, sa, ca=None):
1604 '''Compute the I{prime-vertical}, I{normal} radius of curvature
1605 of (geodetic) latitude, I{unscaled}.
1607 @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]).
1608 @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0])
1609 to use an alternate formula.
1611 @return: The prime-vertical radius of curvature (C{float}).
1613 @note: The delta between both formulae with C{Ellipsoids.WGS84}
1614 is less than 2 nanometer over the entire latitude range.
1616 @see: Method L{roc2_} and class L{EcefYou}.
1617 '''
1618 if sa and self.f: # .isEllipsoidal
1619 if ca is None:
1620 r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward
1621 n = sqrt(self.a2 / r) if r > EPS02 else _0_0
1622 elif ca: # derived from EcefYou.forward
1623 h = hypot(ca, self.b_a * sa)
1624 n = self.a / h
1625 else:
1626 n = self.a2_b / fabs(sa)
1627 else:
1628 n = self.a
1629 return n
1631 def roc2(self, lat, scaled=False):
1632 '''Compute the I{meridional} and I{prime-vertical}, I{normal}
1633 radii of curvature of (geodetic) latitude.
1635 @arg lat: Latitude (C{degrees90}).
1636 @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}).
1638 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with
1639 the radii of curvature.
1641 @raise ValueError: Invalid B{C{lat}}.
1643 @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation
1644 <https://www.EdWilliams.org/avform.htm#flat>} and meridional and
1645 prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/
1646 Earth_radius#Radii_of_curvature>}.
1647 '''
1648 return self.roc2_(Phid(lat), scaled=scaled)
1650 def roc2_(self, phi, scaled=False):
1651 '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of
1652 curvature of (geodetic) latitude.
1654 @arg phi: Latitude (C{radians}).
1655 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}).
1657 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the
1658 radii of curvature.
1660 @raise ValueError: Invalid B{C{phi}}.
1662 @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local,
1663 flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>}
1664 and the meridional and prime vertical U{Radii of Curvature
1665 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1666 '''
1667 p = fabs(Phi(phi))
1668 if self.f:
1669 r = self.e2s2(sin(p))
1670 if r > EPS02:
1671 n = sqrt(self.a2 / r)
1672 m = n * self.e21 / r
1673 else:
1674 m = n = _0_0
1675 else:
1676 m = n = self.a
1677 if scaled and p:
1678 n *= cos(p) if p < PI_2 else _0_0
1679 return Curvature2Tuple(m, n)
1681 def rocAzimuth(self, lat, azimuth):
1682 '''Compute the I{directional} radius of curvature of (geodetic) latitude
1683 and C{azimuth} compass direction.
1685 @see: Method L{rocBearing<Ellipsoid.rocBearing>} for details, using C{azimuth} for C{bearing}.
1686 '''
1687 return Radius(rocAzimuth=self._rocDirectional(lat, Azimuth(azimuth)))
1689 def rocBearing(self, lat, bearing):
1690 '''Compute the I{directional} radius of curvature of (geodetic) latitude
1691 and C{bearing} compass direction.
1693 @arg lat: Latitude (C{degrees90}).
1694 @arg bearing: Direction (compass C{degrees360}).
1696 @return: Directional radius of curvature (C{meter}).
1698 @raise RangeError: Latitude B{C{lat}} outside valid range and
1699 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1701 @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}.
1703 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
1704 '''
1705 return Radius(rocBearing=self._rocDirectional(lat, Bearing(bearing)))
1707 def _rocDirectional(self, lat, deg):
1708 '''(INTERNAL) Helper for C{rocAzimuth} and C{rocBearing}.
1709 '''
1710 if self.f:
1711 s2, c2 = _s2_c2(radians(deg))
1712 m, n = self.roc2_(Phid(lat))
1713 if n < m: # == n / (c2 * n / m + s2)
1714 c2 *= n / m
1715 elif m < n: # == m / (c2 + s2 * m / n)
1716 s2 *= m / n
1717 n = m
1718 r = _over(n, c2 + s2) # == 1 / (c2 / m + s2 / n)
1719 else:
1720 r = self.b # == self.a
1721 return r
1723 @Property_RO
1724 def rocEquatorial2(self):
1725 '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature
1726 at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}.
1728 @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a},
1729 L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature
1730 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1731 '''
1732 m = self.b2_a if self.f else self.a
1733 return Curvature2Tuple(m, self.a)
1735 def rocGauss(self, lat):
1736 '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude.
1738 @arg lat: Latitude (C{degrees90}).
1740 @return: Gaussian radius of curvature (C{meter}).
1742 @raise ValueError: Invalid B{C{lat}}.
1744 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1745 Earth_radius#Radii_of_curvature>}
1746 '''
1747 # using ...
1748 # m, n = self.roc2_(Phid(lat))
1749 # return sqrt(m * n)
1750 # ... requires 1 or 2 sqrt
1751 g = self.b
1752 if self.f:
1753 s2, c2 = _s2_c2(Phid(lat))
1754 g = _over(g, c2 + self.b2_a2 * s2)
1755 return Radius(rocGauss=g)
1757 def rocMean(self, lat):
1758 '''Compute the I{mean} radius of curvature of (geodetic) latitude.
1760 @arg lat: Latitude (C{degrees90}).
1762 @return: Mean radius of curvature (C{meter}).
1764 @raise ValueError: Invalid B{C{lat}}.
1766 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1767 Earth_radius#Radii_of_curvature>}
1768 '''
1769 if self.f:
1770 m, n = self.roc2_(Phid(lat))
1771 m *= _over(n * _2_0, m + n) # == 2 / (1 / m + 1 / n)
1772 else:
1773 m = self.a
1774 return Radius(rocMean=m)
1776 def rocMeridional(self, lat):
1777 '''Compute the I{meridional} radius of curvature of (geodetic) latitude.
1779 @arg lat: Latitude (C{degrees90}).
1781 @return: Meridional radius of curvature (C{meter}).
1783 @raise ValueError: Invalid B{C{lat}}.
1785 @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation
1786 <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of
1787 Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1788 '''
1789 r = self.roc2_(Phid(lat)) if lat else self.rocEquatorial2
1790 return Radius(rocMeridional=r.meridional)
1792 rocPolar = a2_b # synonymous
1794 def rocPrimeVertical(self, lat):
1795 '''Compute the I{prime-vertical}, I{normal} radius of curvature of
1796 (geodetic) latitude, aka the I{transverse} radius of curvature.
1798 @arg lat: Latitude (C{degrees90}).
1800 @return: Prime-vertical radius of curvature (C{meter}).
1802 @raise ValueError: Invalid B{C{lat}}.
1804 @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth
1805 approximation<https://www.EdWilliams.org/avform.htm#flat>} and
1806 U{Radii of Curvature<https://WikiPedia.org/wiki/
1807 Earth_radius#Radii_of_curvature>}.
1808 '''
1809 r = self.roc1_(sin(Phid(lat))) if lat else self.a
1810 return Radius(rocPrimeVertical=r)
1812 rocTransverse = rocPrimeVertical # synonymous
1814 @deprecated_Property_RO
1815 def Rquadratic(self): # PYCHOK no cover
1816 '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.'''
1817 return self.Rbiaxial
1819 @deprecated_Property_RO
1820 def Rr(self): # PYCHOK no cover
1821 '''DEPRECATED, use property C{Rrectifying}.'''
1822 return self.Rrectifying
1824 @Property_RO
1825 def Rrectifying(self):
1826 '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}.
1828 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}.
1829 '''
1830 r = self.a
1831 if self.f:
1832 r *= cbrt2((sqrt3(self.b_a) + _1_0) * _0_5)
1833 return Radius(Rrectifying=r)
1835 @deprecated_Property_RO
1836 def Rs(self): # PYCHOK no cover
1837 '''DEPRECATED, use property C{Rgeometric}.'''
1838 return self.Rgeometric
1840 @Property_RO
1841 def Rtriaxial(self):
1842 '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}.
1844 @see: C{Rbiaxial}
1845 '''
1846 q, b = self.a, self.b
1847 if b < q:
1848 q *= sqrt((self.b2_a2 + _3_0) * _0_25)
1849 elif b > q:
1850 q = sqrt((self.a2_b2 * _3_0 + _1_0) * _0_25) * b
1851 return Radius(Rtriaxial=q)
1853 def toEllipsoid2(self, **name):
1854 '''Get a copy of this ellipsoid as an L{Ellipsoid2}.
1856 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
1858 @see: Property C{a_f}.
1859 '''
1860 return Ellipsoid2(self, None, **name)
1862 def toStr(self, prec=8, terse=4, **sep_name): # PYCHOK expected
1863 '''Return this ellipsoid as a text string.
1865 @kwarg prec: Number of decimal digits, unstripped (C{int}).
1866 @kwarg terse: Limit the number of items (C{int}, 0...18),
1867 use C{B{terse}=0} or C{=None} for all.
1868 @kwarg sep_name: Optional C{B{name}=NN} (C{str}) or C{None}
1869 to exclude this ellipsoid's name and separator
1870 C{B{sep}=", "} to join the items (C{str}).
1872 @return: This C{Ellipsoid}'s attributes (C{str}).
1873 '''
1874 E = Ellipsoid
1875 t = (E.a, E.f, E.f_, E.b, E.f2, E.n, E.e,
1876 E.e2, E.e21, E.e22, E.e32,
1877 E.A, E.L, E.R1, E.R2, E.R3,
1878 E.Rbiaxial, E.Rtriaxial)
1879 if terse:
1880 t = t[:terse]
1881 return self._instr(prec=prec, props=t, **sep_name)
1883 def toTriaxial(self, **name):
1884 '''Convert this ellipsoid to a L{Triaxial_}.
1886 @kwarg name: Optional C{B{name}=NN} (C{str}).
1888 @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis
1889 pointing east and C{Z} pointing north.
1891 @see: Method L{Triaxial_.toEllipsoid}.
1892 '''
1893 T = self._triaxial
1894 return T.copy(**name) if name else T
1896 @property_RO
1897 def _triaxial(self):
1898 '''(INTERNAL) Get this ellipsoid's un-/ordered C{Triaxial/_}.
1899 '''
1900 a, b, m = self.a, self.b, _MODS.triaxials
1901 T = m.Triaxial if a > b else m.Triaxial_
1902 return T(a, a, b, name=self.name)
1904 @Property_RO
1905 def volume(self):
1906 '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}.
1908 @see: C{R3}.
1909 '''
1910 return Meter3(volume=self.a2 * self.b * PI_3 * _4_0)
1913class Ellipsoid2(Ellipsoid):
1914 '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}.
1915 '''
1916 def __init__(self, a, f=None, **name):
1917 '''New L{Ellipsoid2}.
1919 @arg a: Equatorial radius, semi-axis (C{meter}) or a previous
1920 L{Ellipsoid} instance.
1921 @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}),
1922 if B{C{a}} is in C{meter}.
1923 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
1925 @raise NameError: Ellipsoid with that B{C{name}} already exists.
1927 @raise ValueError: Invalid B{C{a}} or B{C{f}}.
1929 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.
1930 Negative C{B{f}} produces a I{prolate} ellipsoid.
1931 '''
1932 if f is None and isinstance(a, Ellipsoid):
1933 Ellipsoid.__init__(self, a.a, f =a.f,
1934 b=a.b, f_=a.f_, **name)
1935 else:
1936 Ellipsoid.__init__(self, a, f=f, **name)
1939def _ispherical_a_b(a, b):
1940 '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}.
1941 '''
1942 return a < EPS0 or b < EPS0 or fabs(a - b) < EPS0
1945def _ispherical_f(f):
1946 '''(INTERNAL) C{True} for spherical or invalid C{f}.
1947 '''
1948 return f > EPS1 or fabs(f) < EPS
1951def _ispherical_f_(f_):
1952 '''(INTERNAL) C{True} for spherical or invalid C{f_}.
1953 '''
1954 f_ = fabs(f_)
1955 return f_ < EPS or f_ > _1_EPS
1958def a_b2e(a, b):
1959 '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius.
1961 @arg a: Equatorial radius (C{scalar} > 0).
1962 @arg b: Polar radius (C{scalar} > 0).
1964 @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}), M{sqrt(1 - (b / a)**2)}.
1966 @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids.
1967 '''
1968 e2 = _a2b2e2(a, b, b2=False)
1969 return Float(e=sqrt(fabs(e2)) if e2 else _0_0) # == sqrt(fabs((a - b) * (a + b))) / a
1972def a_b2e2(a, b):
1973 '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius.
1975 @arg a: Equatorial radius (C{scalar} > 0).
1976 @arg b: Polar radius (C{scalar} > 0).
1978 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}), M{1 - (b / a)**2}.
1980 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1981 for I{near-spherical} ellipsoids.
1982 '''
1983 return Float(e2=_a2b2e2(a, b, b2=False))
1986def a_b2e22(a, b):
1987 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1989 @arg a: Equatorial radius (C{scalar} > 0).
1990 @arg b: Polar radius (C{scalar} > 0).
1992 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}), M{(a / b)**2 - 1}.
1994 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1995 for I{near-spherical} ellipsoids.
1996 '''
1997 return Float(e22=_a2b2e2(a, b, a2=False))
2000def a_b2e32(a, b):
2001 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius.
2003 @arg a: Equatorial radius (C{scalar} > 0).
2004 @arg b: Polar radius (C{scalar} > 0).
2006 @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}),
2007 M{(a**2 - b**2) / (a**2 + b**2)}.
2009 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2010 for I{near-spherical} ellipsoids.
2011 '''
2012 return Float(e32=_a2b2e2(a, b))
2015def _a2b2e2(a, b, a2=True, b2=True):
2016 '''(INTERNAL) Helper for C{a_b2e}, C{a_b2e2}, C{a_b2e22} and C{a_b2e32}.
2017 '''
2018 if _ispherical_a_b(a, b):
2019 e2 = _0_0
2020 else: # a > 0, b > 0
2021 a, b = (_1_0, b / a) if a > b else (a / b, _1_0)
2022 a2b2 = float(a - b) * (a + b)
2023 e2 = _over(a2b2, (a**2 if a2 else _0_0) +
2024 (b**2 if b2 else _0_0)) if a2b2 else _0_0
2025 return e2
2028def a_b2f(a, b):
2029 '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius.
2031 @arg a: Equatorial radius (C{scalar} > 0).
2032 @arg b: Polar radius (C{scalar} > 0).
2034 @return: The flattening (C{scalar} or C{0}), M{(a - b) / a}.
2036 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2037 for I{near-spherical} ellipsoids.
2038 '''
2039 f = 0 if _ispherical_a_b(a, b) else _over(float(a - b), a)
2040 return _f_0_0 if _ispherical_f(f) else Float(f=f)
2043def a_b2f_(a, b):
2044 '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius.
2046 @arg a: Equatorial radius (C{scalar} > 0).
2047 @arg b: Polar radius (C{scalar} > 0).
2049 @return: The inverse flattening (C{scalar} or C{0}), M{a / (a - b)}.
2051 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2052 for I{near-spherical} ellipsoids.
2053 '''
2054 f_ = 0 if _ispherical_a_b(a, b) else _over(a, float(a - b))
2055 return _f__0_0 if _ispherical_f_(f_) else Float(f_=f_)
2058def a_b2f2(a, b):
2059 '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius.
2061 @arg a: Equatorial radius (C{scalar} > 0).
2062 @arg b: Polar radius (C{scalar} > 0).
2064 @return: The I{signed}, 2nd flattening (C{scalar} or C{0}), M{(a - b) / b}.
2066 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2067 for I{near-spherical} ellipsoids.
2068 '''
2069 t = 0 if _ispherical_a_b(a, b) else float(a - b)
2070 return Float(f2=_0_0 if fabs(t) < EPS0 else _over(t, b))
2073def a_b2n(a, b):
2074 '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius.
2076 @arg a: Equatorial radius (C{scalar} > 0).
2077 @arg b: Polar radius (C{scalar} > 0).
2079 @return: The I{signed}, 3rd flattening (C{scalar} or C{0}), M{(a - b) / (a + b)}.
2081 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2082 for I{near-spherical} ellipsoids.
2083 '''
2084 t = 0 if _ispherical_a_b(a, b) else float(a - b)
2085 return Float(n=_0_0 if fabs(t) < EPS0 else _over(t, a + b))
2088def a_f2b(a, f):
2089 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}.
2091 @arg a: Equatorial radius (C{scalar} > 0).
2092 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2094 @return: The polar radius (C{float}), M{a * (1 - f)}.
2095 '''
2096 b = a if _ispherical_f(f) else (a * (_1_0 - f))
2097 return Radius_(b=a if _ispherical_a_b(a, b) else b)
2100def a_f_2b(a, f_):
2101 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}.
2103 @arg a: Equatorial radius (C{scalar} > 0).
2104 @arg f_: Inverse flattening (C{scalar} >>> 1).
2106 @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}.
2107 '''
2108 b = a if _ispherical_f_(f_) else _over(a * (f_ - _1_0), f_)
2109 return Radius_(b=a if _ispherical_a_b(a, b) else b)
2112def b_f2a(b, f):
2113 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}.
2115 @arg b: Polar radius (C{scalar} > 0).
2116 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2118 @return: The equatorial radius (C{float}), M{b / (1 - f)}.
2119 '''
2120 t = _1_0 - f
2121 a = b if fabs(t) < EPS0 else _over(b, t)
2122 return Radius_(a=b if _ispherical_a_b(a, b) else a)
2125def b_f_2a(b, f_):
2126 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}.
2128 @arg b: Polar radius (C{scalar} > 0).
2129 @arg f_: Inverse flattening (C{scalar} >>> 1).
2131 @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}.
2132 '''
2133 t = f_ - _1_0
2134 a = b if _ispherical_f_(f_) or fabs(t) < EPS0 \
2135 or fabs(t - f_) < EPS0 else _over(b * f_, t)
2136 return Radius_(a=b if _ispherical_a_b(a, b) else a)
2139def e2f(e):
2140 '''Return C{f}, the I{flattening} for a given I{1st eccentricity}.
2142 @arg e: The (1st) eccentricity (0 <= C{float} < 1)
2144 @return: The flattening (C{scalar} or C{0}).
2146 @see: Function L{e22f}.
2147 '''
2148 return e22f(e**2)
2151def e22f(e2):
2152 '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}.
2154 @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1)
2156 @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(1 - e2) + 1)}.
2157 '''
2158 return Float(f=_over(e2, sqrt(_1_0 - e2) + _1_0)) if e2 and e2 < _1_0 else _f_0_0
2161def f2e2(f):
2162 '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}.
2164 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2166 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1), M{f * (2 - f)}.
2168 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2169 for I{near-spherical} ellipsoids.
2171 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2172 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2173 <https://WikiPedia.org/wiki/Flattening>}.
2174 '''
2175 return Float(e2=_0_0 if _ispherical_f(f) else (f * (_2_0 - f)))
2178def f2e22(f):
2179 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}.
2181 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2183 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or C{INF}),
2184 M{f * (2 - f) / (1 - f)**2}.
2186 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2187 for near-spherical ellipsoids.
2189 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2190 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.
2191 '''
2192 # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2
2193 t = (_1_0 - f)**2
2194 return Float(e22=INF if t < EPS0 else _over(f2e2(f), t)) # PYCHOK type
2197def f2e32(f):
2198 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}.
2200 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2202 @return: The I{signed}, 3rd eccentricity I{squared} (C{float}),
2203 M{f * (2 - f) / (1 + (1 - f)**2)}.
2205 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2206 for I{near-spherical} ellipsoids.
2208 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2209 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.
2210 '''
2211 # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2)
2212 e2 = f2e2(f)
2213 return Float(e32=_over(e2, _2_0 - e2))
2216def f_2f(f_):
2217 '''Return C{f}, the I{flattening} for a given I{inverse flattening}.
2219 @arg f_: Inverse flattening (C{scalar} >>> 1).
2221 @return: The flattening (C{scalar} or C{0}), M{1 / f_}.
2223 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2224 for I{near-spherical} ellipsoids.
2225 '''
2226 f = 0 if _ispherical_f_(f_) else _over(_1_0, f_)
2227 return _f_0_0 if _ispherical_f(f) else Float(f=f) # PYCHOK type
2230def f2f_(f):
2231 '''Return C{f_}, the I{inverse flattening} for a given I{flattening}.
2233 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2235 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2237 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2238 for I{near-spherical} ellipsoids.
2239 '''
2240 f_ = 0 if _ispherical_f(f) else _over(_1_0, f)
2241 return _f__0_0 if _ispherical_f_(f_) else Float(f_=f_) # PYCHOK type
2244def f2f2(f):
2245 '''Return C{f2}, the I{2nd flattening} for a given I{flattening}.
2247 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2249 @return: The I{signed}, 2nd flattening (C{scalar} or C{INF}), M{f / (1 - f)}.
2251 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2252 for I{near-spherical} ellipsoids.
2254 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2255 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2256 <https://WikiPedia.org/wiki/Flattening>}.
2257 '''
2258 t = _1_0 - f
2259 return Float(f2=_0_0 if _ispherical_f(f) else
2260 (INF if fabs(t) < EPS else _over(f, t))) # PYCHOK type
2263def f2n(f):
2264 '''Return C{n}, the I{3rd flattening} for a given I{flattening}.
2266 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2268 @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1),
2269 M{f / (2 - f)}.
2271 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2272 for I{near-spherical} ellipsoids.
2274 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2275 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2276 <https://WikiPedia.org/wiki/Flattening>}.
2277 '''
2278 return Float(n=_0_0 if _ispherical_f(f) else _over(f, float(_2_0 - f)))
2281def n2e2(n):
2282 '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}.
2284 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2286 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF),
2287 M{4 * n / (1 + n)**2}.
2289 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2290 for I{near-spherical} ellipsoids.
2292 @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}.
2293 '''
2294 t = (n + _1_0)**2
2295 return Float(e2=_0_0 if fabs(n) < EPS0 else
2296 (NINF if t < EPS0 else _over(_4_0 * n, t)))
2299def n2f(n):
2300 '''Return C{f}, the I{flattening} for a given I{3rd flattening}.
2302 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2304 @return: The flattening (C{scalar} or NINF), M{2 * n / (1 + n)}.
2306 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2307 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2308 <https://WikiPedia.org/wiki/Flattening>}.
2309 '''
2310 t = n + _1_0
2311 f = 0 if fabs(n) < EPS0 else (NINF if t < EPS0 else _over(_2_0 * n, t))
2312 return _f_0_0 if _ispherical_f(f) else Float(f=f)
2315def n2f_(n):
2316 '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}.
2318 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2320 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2322 @see: L{n2f} and L{f2f_}.
2323 '''
2324 return f2f_(n2f(n))
2327class Ellipsoids(_NamedEnum):
2328 '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class
2329 to accommodate the L{_LazyNamedEnumItem} properties.
2330 '''
2331 def _Lazy(self, a, b, f_, **kwds):
2332 '''(INTERNAL) Instantiate the L{Ellipsoid}.
2333 '''
2334 return Ellipsoid(a, b=b, f_=f_, **kwds)
2336Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton
2337'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.'''
2338# <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html>
2339# <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS>
2340# <https://kb.OSU.edu/dspace/handle/1811/77986>
2341# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids>
2342# <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html>
2343# <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java>
2344Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid>
2345 Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909
2346 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448
2347# APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965
2348# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid
2349# AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America
2350 Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192
2351 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System"
2352 Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)),
2353 BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)),
2354# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)),
2355 CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC)
2356# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)),
2357 Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)),
2358 Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)),
2359 Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)),
2360 Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc
2361 CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures
2362 Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium
2363 Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)),
2364# Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)),
2365# Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)),
2366# Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)),
2367 Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)),
2368 Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)),
2369 Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)),
2370# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)),
2371 GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)),
2372 GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid"
2373 GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516
2374# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)),
2375 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.25722210088)), # IUGG, ITRS, ETRS89
2376# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297
2377 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)),
2378# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)),
2379 IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union
2380 IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302
2381 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy
2382 IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)),
2383 Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795
2384 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l
2385 Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2386 Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2387# Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961
2388# Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979
2389 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France
2390 Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)),
2391 Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)),
2392# MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983
2393# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab.
2394 NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab.
2395# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965
2396 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)),
2397 OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)),
2398# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect?
2399 Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972
2400# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)),
2401 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11
2402# SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia
2403 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System
2404 SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American
2405 Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo
2406 SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo
2407 SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)), # EPSG:3857 Spheroid
2408 Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)),
2409# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)),
2410# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)),
2411 WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)),
2412 WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)),
2413 WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52
2414 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # GPS b=6356752.3142451793
2415# U{NOAA/NOS/NGS/inverse<https://GitHub.com/noaa-ngs/inverse/blob/main/invers3d.f>}
2416 WGS84_NGS = _lazy('WGS84_NGS', *_T(R_MA, _0_0, 298.257222100882711243162836600094))
2417)
2419_EWGS84 = Ellipsoids.WGS84 # (INTERNAL) shared
2421if __name__ == _DMAIN_:
2423 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_
2424 from pygeodesy import nameof, printf
2426 for E in (_EWGS84, Ellipsoids.GRS80, # NAD83,
2427 Ellipsoids.Sphere, Ellipsoids.SpherePopular,
2428 Ellipsoid(_EWGS84.b, _EWGS84.a, name='_Prolate')):
2429 e = f2n(E.f) - E.n
2430 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10, terse=0), nl=1)
2431 printf('# e=%s, f_=%s, f=%s, n=%s (%s)', fstr(E.e, prec=13, fmt=Fmt.e),
2432 fstr(E.f_, prec=13, fmt=Fmt.e),
2433 fstr(E.f, prec=13, fmt=Fmt.e),
2434 fstr(E.n, prec=13, fmt=Fmt.e),
2435 fstr(e, prec=9, fmt=Fmt.e))
2436 printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20))
2437 printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20))
2438 printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property
2440 # __doc__ of this file, force all into registry
2441 t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_)
2442 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))
2444# % python3.13 -m pygeodesy.ellipsoids
2446# 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
2447# e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00)
2448# AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2449# BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2450# KsOrder 8
2452# 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
2453# e=8.1819191042833e-02, f_=2.9825722210088e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00)
2454# AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2455# BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2456# KsOrder 8
2458# 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
2459# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2460# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2461# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2462# KsOrder 8
2464# 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
2465# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2466# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2467# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2468# KsOrder 8
2470# 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=10035500.5204500332, R1=6363880.5428301189, R2=6363878.9413582645, R3=6363872.5644020075, Rbiaxial=6367453.6345163295, Rtriaxial=6362105.2243882557
2471# e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00)
2472# AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0
2473# BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0
2474# KsOrder 8
2476# **) MIT License
2477#
2478# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
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2480# Permission is hereby granted, free of charge, to any person obtaining a
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2485# Software is furnished to do so, subject to the following conditions:
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