Coverage for pygeodesy/ellipsoids.py: 96%
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« prev ^ index » next coverage.py v7.6.1, created at 2025-04-25 13:15 -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 _ # PYCHOK semicolon
67# from pygeodesy.albers import AlbersEqualAreaCylindrical # _MODS
68from pygeodesy.basics import copysign0, isbool, _isin, isint, typename
69from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, INF, NINF, PI4, PI_2, PI_3, R_M, R_MA, R_FM, \
70 _EPSqrt, _EPStol as _TOL, _floatuple as _T, _isfinite, _over, \
71 _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \
72 _0_25, _3_0 # PYCHOK used!
73from pygeodesy.errors import _AssertionError, IntersectionError, _ValueError, _xattr, _xkwds_not
74from pygeodesy.fmath import cbrt, cbrt2, fdot, Fhorner, fpowers, hypot, hypot_, \
75 hypot1, hypot2, sqrt3, Fsum
76# from pygeodesy.fsums import Fsum # from .fmath
77# from pygeodesy.internals import typename # from .basics
78from pygeodesy.interns import NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, _beta_, \
79 _Clarke1866_, _Clarke1880IGN_, _DMAIN_, _DOT_, _f_, _GRS80_, \
80 _height_, _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \
81 _Krassowsky1940_, _lat_, _meridional_, _negative_, _not_finite_, \
82 _prime_vertical_, _radius_, _Sphere_, _SPACE_, _vs_, _WGS72_, _WGS84_
83# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .named
84from pygeodesy.named import _lazyNamedEnumItem as _lazy, _name__, _NamedEnum, \
85 _NamedEnumItem, _NamedTuple, _Pass, _ALL_LAZY, _MODS
86from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple
87from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_, \
88 deprecated_property_RO, property_RO, property_ROver
89from pygeodesy.streprs import Fmt, fstr, instr, strs, unstr
90# from pygeodesy.triaxials import _hartzell3 # _MODS
91from pygeodesy.units import Azimuth, Bearing, Distance, Float, Float_, Height, Lamd, Lat, \
92 Meter, Meter2, Meter3, Phi, Phid, Radius, Radius_, Scalar
93from pygeodesy.utily import atan1, atan1d, atan2b, degrees90, m2radians, radians2m, sincos2d
95from math import asinh, atan, atanh, cos, degrees, exp, fabs, radians, sin, sinh, sqrt, tan # as _tan
97__all__ = _ALL_LAZY.ellipsoids
98__version__ = '25.04.23'
100_f_0_0 = Float(f =_0_0) # zero flattening
101_f__0_0 = Float(f_=_0_0) # zero inverse flattening
102# see U{WGS84_f<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Constants.html>}
103_f__WGS84 = Float(f_=_1_0 / (1000000000 / 298257223563)) # 298.257223562_999_97 vs 298.257223563
106def _aux(lat, inverse, auxLat, clip=90):
107 '''Return a named auxiliary latitude in C{degrees}.
108 '''
109 return Lat(lat, clip=clip, name=_lat_ if inverse else typename(auxLat))
112def _s2_c2(phi):
113 '''(INTERNAL) Return 2-tuple C{(sin(B{phi})**2, cos(B{phi})**2)}.
114 '''
115 if phi:
116 s2 = sin(phi)**2
117 if s2 > EPS:
118 c2 = _1_0 - s2
119 if c2 > EPS:
120 if c2 < EPS1:
121 return s2, c2
122 else:
123 return _1_0, _0_0 # phi == PI_2
124 return _0_0, _1_0 # phi == 0
127class a_f2Tuple(_NamedTuple):
128 '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial}
129 radius C{a} in C{meter} and scalar I{flattening} C{f}.
131 @see: Class L{Ellipsoid2}.
132 '''
133 _Names_ = (_a_, _f_) # name 'f' not 'f_'
134 _Units_ = (_Pass, _Pass)
136 def __new__(cls, a, f, **name):
137 '''New L{a_f2Tuple} ellipsoid specification.
139 @arg a: Equatorial radius (C{scalar} > 0).
140 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
141 @kwarg name: Optional C{B{name}=NN} (C{str}).
143 @return: An L{a_f2Tuple}C{(a, f)} instance.
145 @raise UnitError: Invalid B{C{a}} or B{C{f}}.
147 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.
148 Negative C{B{f}} produces a I{prolate} ellipsoid.
149 '''
150 a = Radius_(a=a) # low=EPS, high=None
151 f = Float_( f=f, low=None, high=EPS1)
152 if fabs(f) < EPS: # force spherical
153 f = _f_0_0
154 return _NamedTuple.__new__(cls, a, f, **name)
156 @Property_RO
157 def b(self):
158 '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}.
159 '''
160 return a_f2b(self.a, self.f) # PYCHOK .a and .f
162 def ellipsoid(self, **name):
163 '''Return an L{Ellipsoid} for this 2-tuple C{(a, f)}.
165 @kwarg name: Optional C{B{name}=NN} (C{str}).
167 @raise NameError: A registered C{ellipsoid} with the
168 same B{C{name}} already exists.
169 '''
170 return Ellipsoid(self.a, f=self.f, name=self._name__(name)) # PYCHOK .a and .f
172 @Property_RO
173 def f_(self):
174 '''Get the I{inverse} flattening (C{scalar}), M{1 / f} == M{a / (a - b)}.
175 '''
176 return f2f_(self.f) # PYCHOK .f
179class Circle4Tuple(_NamedTuple):
180 '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height},
181 both conventionally in C{meter} of a parallel I{circle of latitude} at
182 (geodetic) latitude C{lat} and the I{parametric (or reduced) auxiliary
183 latitude} C{beta}, both in C{degrees90}.
185 The C{height} is the (signed) distance along the z-axis between the
186 parallel and the equator. At near-polar C{lat}s, the C{radius} is C{0},
187 the C{height} is the ellipsoid's (signed) polar radius and C{beta}
188 equals C{lat}.
189 '''
190 _Names_ = (_radius_, _height_, _lat_, _beta_)
191 _Units_ = ( Radius, Height, Lat, Lat)
194class Curvature2Tuple(_NamedTuple):
195 '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in
196 C{meter}, conventionally.
197 '''
198 _Names_ = (_meridional_, _prime_vertical_)
199 _Units_ = ( Meter, Meter)
201 @property_RO
202 def transverse(self):
203 '''Get this I{prime_vertical}, aka I{transverse} radius of curvature.
204 '''
205 return self.prime_vertical
208class Ellipsoid(_NamedEnumItem):
209 '''Ellipsoid with I{equatorial} and I{polar} radii, I{flattening}, I{inverse
210 flattening} and other, often used, I{cached} attributes, supporting
211 I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models.
212 '''
213 _a = 0 # equatorial radius, semi-axis (C{meter})
214 _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f
215 _f = 0 # (1st) flattening: (a - b) / a
216 _f_ = 0 # inverse flattening: 1 / f = a / (a - b)
218 _geodsolve = NN # means, use PYGEODESY_GEODSOLVE
219 _KsOrder = 8 # Krüger series order (4, 6 or 8)
220 _rhumbsolve = NN # means, use PYGEODESY_RHUMBSOLVE
222 def __init__(self, a, b=None, f_=None, f=None, **name):
223 '''New L{Ellipsoid} from the I{equatorial} radius I{and} either
224 the I{polar} radius or I{inverse flattening} or I{flattening}.
226 @arg a: Equatorial radius, semi-axis (C{meter}).
227 @arg b: Optional polar radius, semi-axis (C{meter}).
228 @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0).
229 @arg f: Flattening: M{(a - b) / a} (C{scalar}, near zero for
230 spherical).
231 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
233 @raise NameError: Ellipsoid with the same B{C{name}} already exists.
235 @raise ValueError: Invalid B{C{a}}, B{C{b}}, B{C{f_}} or B{C{f}} or
236 B{C{f_}} and B{C{f}} are incompatible.
238 @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced
239 to M{1 / f_ = 0}, spherical.
240 '''
241 ff_ = f, f_ # assertion below
242 n = _name__(**name) if name else NN
243 try:
244 a = Radius_(a=a) # low=EPS
245 if not _isfinite(a):
246 raise ValueError(_SPACE_(_a_, _not_finite_))
248 if b: # not _isin(b, None, _0_0)
249 b = Radius_(b=b) # low=EPS
250 f = a_b2f(a, b) if f is None else Float(f=f)
251 f_ = f2f_(f) if f_ is None else Float(f_=f_)
252 elif f is not None:
253 f = Float(f=f)
254 b = a_f2b(a, f)
255 f_ = f2f_(f) if f_ is None else Float(f_=f_)
256 elif f_:
257 f_ = Float(f_=f_)
258 b = a_f_2b(a, f_) # a * (f_ - 1) / f_
259 f = f_2f(f_)
260 else: # only a, spherical
261 f_ = f = 0
262 b = a # superfluous
264 if not f < _1_0: # sanity check, see .ecef.Ecef.__init__
265 raise ValueError(_SPACE_(_f_, _invalid_))
266 if not _isfinite(b):
267 raise ValueError(_SPACE_(_b_, _not_finite_))
269 if fabs(f) < EPS or a == b or not f_: # spherical
270 b = a
271 f = _f_0_0
272 f_ = _f__0_0
274 except (TypeError, ValueError) as x:
275 d = _xkwds_not(None, b=b, f_=f_, f=f)
276 t = instr(self, a=a, name=n, **d)
277 raise _ValueError(t, cause=x)
279 self._a = a
280 self._b = b
281 self._f = f
282 self._f_ = f_
284 self._register(Ellipsoids, n)
286 if f and f_: # see test/testEllipsoidal
287 d = dict(eps=_TOL)
288 if None in ff_: # both f_ and f given
289 d.update(Error=_ValueError, txt=_incompatible_)
290 self._assert(_1_0 / f, f_=f_, **d)
291 self._assert(_1_0 / f_, f =f, **d)
292 self._assert(self.b2_a2, e21=self.e21, eps=EPS)
294 def __eq__(self, other):
295 '''Compare this and an other ellipsoid.
297 @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}).
299 @return: C{True} if equal, C{False} otherwise.
300 '''
301 return self is other or (isinstance(other, Ellipsoid) and
302 self.a == other.a and
303 (self.f == other.f or self.b == other.b))
305 def __hash__(self):
306 return self._hash # memoized
308 @Property_RO
309 def a(self):
310 '''Get the I{equatorial} radius, semi-axis (C{meter}).
311 '''
312 return self._a
314 equatoradius = a # = Requatorial
316 @Property_RO
317 def a2(self):
318 '''Get the I{equatorial} radius I{squared} (C{meter} I{squared}), M{a**2}.
319 '''
320 return Meter2(a2=self.a**2)
322 @Property_RO
323 def a2_(self):
324 '''Get the inverse of the I{equatorial} radius I{squared} (C{meter} I{squared}), M{1 / a**2}.
325 '''
326 return Float(a2_=_1_0 / self.a2)
328 @Property_RO
329 def a_b(self):
330 '''Get the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b} == M{1 / (1 - f)}.
331 '''
332 return Float(a_b=self.a / self.b if self.f else _1_0)
334 @Property_RO
335 def a2_b(self):
336 '''Get the I{polar} meridional (or polar) radius of curvature (C{meter}), M{a**2 / b}.
338 @see: U{Radii of Curvature
339 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
340 and U{Moritz, H. (1980), Geodetic Reference System 1980
341 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}.
343 @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature
344 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{c2}
345 and L{R2} or L{Rauthalic}.
346 '''
347 return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar
349 @Property_RO
350 def a2_b2(self):
351 '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}),
352 M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e21}.
353 '''
354 return Float(a2_b2=self.a_b**2 if self.f else _1_0)
356 @Property_RO
357 def a_f(self):
358 '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}), see method C{toEllipsoid2}.
359 '''
360 return a_f2Tuple(self.a, self.f, name=self.name)
362 @Property_RO
363 def A(self):
364 '''Get the UTM I{meridional (or rectifying)} radius (C{meter}).
366 @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/>}.
367 '''
368 A, n = self.a, self.n
369 if n:
370 d = (n + _1_0) * 1048576 / A
371 if d: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8
372 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>
373 # <https://GeographicLib.SourceForge.io/C++/doc/transversemercator.html> and
374 # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> (3)
375 # A *= fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441) / 1048576) / (1 + n)
376 A = Radius(A=Fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441).fover(d))
377 return A
379 @Property_RO
380 def _albersCyl(self):
381 '''(INTERNAL) Helper for C{auxAuthalic}.
382 '''
383 return _MODS.albers.AlbersEqualAreaCylindrical(datum=self, name=self.name)
385 @Property_RO
386 def AlphaKs(self):
387 '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
388 '''
389 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon
390 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
391 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200),
392 _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned
393 _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned
394 _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned
395 _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned
396 _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned
397 _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned
398 _T(1424729850961/743921418240)) # PYCHOK unaligned
400 @Property_RO
401 def area(self):
402 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2}.
404 @see: Properties L{areax}, L{c2} and L{R2} and functions
405 L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
406 '''
407 return Meter2(area=self.c2 * PI4)
409 @Property_RO
410 def areax(self):
411 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more
412 accurate for very I{oblate} ellipsoids.
414 @see: Properties L{area}, L{c2x} and L{R2x}, class L{GeodesicExact} and
415 functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
416 '''
417 return Meter2(areax=self.c2x * PI4)
419 def _assert(self, val, eps=_TOL, f0=_0_0, Error=_AssertionError, txt=NN, **name_value):
420 '''(INTERNAL) Assert a C{name=value} vs C{val}.
421 '''
422 for n, v in name_value.items():
423 if fabs(v - val) > eps: # PYCHOK no cover
424 t = (v, _vs_, val)
425 t = _SPACE_.join(strs(t, prec=12, fmt=Fmt.g))
426 t = Fmt.EQUAL(self._DOT_(n), t)
427 raise Error(t, txt=txt or Fmt.exceeds_eps(eps))
428 return Float(v if self.f else f0, name=n)
429 raise Error(unstr(self._DOT_(typename(self._assert)), val,
430 eps=eps, f0=f0, **name_value))
432 def auxAuthalic(self, lat, inverse=False):
433 '''Compute the I{authalic} auxiliary latitude or the I{inverse} thereof.
435 @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}).
436 @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and
437 return the geodetic latitude (C{bool}).
439 @return: The I{authalic} (or geodetic) latitude in C{degrees90}.
441 @see: U{Inverse-/AuthalicLatitude<https://GeographicLib.SourceForge.io/
442 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude
443 <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and
444 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16.
445 '''
446 if self.f:
447 f = self._albersCyl._tanf if inverse else \
448 self._albersCyl._txif # PYCHOK attr
449 lat = atan1d(f(tan(Phid(lat)))) # PYCHOK attr
450 return _aux(lat, inverse, Ellipsoid.auxAuthalic)
452 def auxConformal(self, lat, inverse=False):
453 '''Compute the I{conformal} auxiliary latitude or the I{inverse} thereof.
455 @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}).
456 @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and
457 return the geodetic latitude (C{bool}).
459 @return: The I{conformal} (or geodetic) latitude in C{degrees90}.
461 @see: U{Inverse-/ConformalLatitude<https://GeographicLib.SourceForge.io/
462 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude
463 <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and
464 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
465 '''
466 if self.f:
467 f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr
468 lat = atan1d(f(tan(Phid(lat)))) # PYCHOK attr
469 return _aux(lat, inverse, Ellipsoid.auxConformal)
471 def auxGeocentric(self, lat, inverse=False):
472 '''Compute the I{geocentric} auxiliary latitude or the I{inverse} thereof.
474 @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}).
475 @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and
476 return the I{geocentric} latitude (C{bool}).
478 @return: The I{geocentric} (or geodetic) latitude in C{degrees90}.
480 @see: U{Inverse-/GeocentricLatitude<https://GeographicLib.SourceForge.io/
481 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude
482 <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and
483 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18.
484 '''
485 if self.f:
486 f = self.a2_b2 if inverse else self.b2_a2
487 lat = atan1d(tan(Phid(lat)) * f)
488 return _aux(lat, inverse, Ellipsoid.auxGeocentric)
490 def auxIsometric(self, lat, inverse=False):
491 '''Compute the I{isometric} auxiliary latitude or the I{inverse} thereof.
493 @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}).
494 @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and
495 return the geodetic latitude (C{bool}).
497 @return: The I{isometric} (or geodetic) latitude in C{degrees}.
499 @note: The I{isometric} latitude for geodetic C{+/-90} is far
500 outside the C{[-90..+90]} range but the inverse
501 thereof is the original geodetic latitude.
503 @see: U{Inverse-/IsometricLatitude<https://GeographicLib.SourceForge.io/
504 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude
505 <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and
506 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
507 '''
508 if self.f:
509 r = Phid(lat, clip=0)
510 lat = degrees(atan1(self.es_tauf(sinh(r))) if inverse else
511 asinh(self.es_taupf(tan(r))))
512 # clip=0, since auxIsometric(+/-90) is far outside [-90..+90]
513 return _aux(lat, inverse, Ellipsoid.auxIsometric, clip=0)
515 def auxParametric(self, lat, inverse=False):
516 '''Compute the I{parametric} auxiliary latitude or the I{inverse} thereof.
518 @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}).
519 @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and
520 return the geodetic latitude (C{bool}).
522 @return: The I{parametric} (or geodetic) latitude in C{degrees90}.
524 @see: U{Inverse-/ParametricLatitude<https://GeographicLib.SourceForge.io/
525 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude
526 <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>},
527 and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18.
528 '''
529 if self.f:
530 lat = self._beta(Lat(lat), inverse=inverse)
531 return _aux(lat, inverse, Ellipsoid.auxParametric)
533 auxReduced = auxParametric # synonymous
535 def auxRectifying(self, lat, inverse=False):
536 '''Compute the I{rectifying} auxiliary latitude or the I{inverse} thereof.
538 @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}).
539 @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and
540 return the geodetic latitude (C{bool}).
542 @return: The I{rectifying} (or geodetic) latitude in C{degrees90}.
544 @see: U{Inverse-/RectifyingLatitude<https://GeographicLib.SourceForge.io/
545 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude
546 <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and
547 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17.
548 '''
549 if self.f:
550 lat = Lat(lat)
551 if 0 < fabs(lat) < _90_0:
552 if inverse:
553 e = self._elliptic_e22
554 d = degrees90(e.fEinv(e.cE * lat / _90_0))
555 lat = self.auxParametric(d, inverse=True)
556 else:
557 lat = _over(self.Llat(lat), self.L) * _90_0
558 return _aux(lat, inverse, Ellipsoid.auxRectifying)
560 @Property_RO
561 def b(self):
562 '''Get the I{polar} radius, semi-axis (C{meter}).
563 '''
564 return self._b
566 polaradius = b # = Rpolar
568 @Property_RO
569 def b_a(self):
570 '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a == f1 == 1 - f}.
572 @see: Property L{f1}.
573 '''
574 return self._assert(self.b / self.a, b_a=self.f1, f0=_1_0)
576 @Property_RO
577 def b2(self):
578 '''Get the I{polar} radius I{squared} (C{float}), M{b**2}.
579 '''
580 return Meter2(b2=self.b**2)
582 @Property_RO
583 def b2_a(self):
584 '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see C{rocMeridional}C{(0)}.
586 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
587 '''
588 return Radius(b2_a=_over(self.b2, self.a) if self.f else self.b)
590 @Property_RO
591 def b2_a2(self):
592 '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2}
593 == M{(1 - f)**2} == M{1 - e**2} == C{e21}.
594 '''
595 return Float(b2_a2=self.b_a**2 if self.f else _1_0)
597 def _beta(self, lat, inverse=False):
598 '''(INTERNAL) Get the I{parametric (or reduced) auxiliary latitude} or inverse thereof.
599 '''
600 s, c = sincos2d(lat) # like Karney's tand(lat)
601 s *= self.a_b if inverse else self.b_a
602 return atan1d(s, c)
604 @Property_RO
605 def BetaKs(self):
606 '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
607 '''
608 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon
609 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
610 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600),
611 _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned
612 _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned
613 _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned
614 _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned
615 _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne
616 _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned
617 _T(191773887257/3719607091200)) # PYCHOK unaligned
619 @deprecated_Property_RO
620 def c(self): # PYCHOK no cover
621 '''DEPRECATED, use property C{R2} or C{Rauthalic}.'''
622 return self.R2
624 @Property_RO
625 def c2(self):
626 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}).
628 @see: Properties L{c2x}, L{area}, L{R2}, L{Rauthalic}, I{Karney's} U{equation (60)
629 <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area
630 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>},
631 U{Authalic radius<https://WikiPedia.org/wiki/Earth_radius#Authalic_radius>}, U{Surface area
632 <https://WikiPedia.org/wiki/Ellipsoid>} and U{surface area
633 <https://www.Numericana.com/answer/geometry.htm#oblate>}.
634 '''
635 return self._c2f(False)
637 @Property_RO
638 def c2x(self):
639 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}), more accurate for very I{oblate}
640 ellipsoids.
642 @see: Properties L{c2}, L{areax}, L{R2x}, L{Rauthalicx}, class L{GeodesicExact} and I{Karney}'s comments at C++
643 attribute U{GeodesicExact._c2<https://GeographicLib.SourceForge.io/C++/doc/GeodesicExact_8cpp_source.html>}.
644 '''
645 return self._c2f(True)
647 def _c2f(self, c2x):
648 '''(INTERNAL) Helper for C{.c2} and C{.c2x}.
649 '''
650 f, c2 = self.f, self.b2
651 if f:
652 e = self.e
653 if e > EPS0:
654 if f > 0: # .isOblate
655 c2 *= (asinh(sqrt(self.e22abs)) if c2x else atanh(e)) / e
656 elif f < 0: # .isProlate
657 c2 *= atan1(e) / e # XXX asin?
658 c2 = Meter2(c2=(self.a2 + c2) * _0_5)
659 return c2
661 def circle4(self, lat):
662 '''Get the equatorial or a parallel I{circle of latitude}.
664 @arg lat: Geodetic latitude (C{degrees90}, C{str}).
666 @return: A L{Circle4Tuple}C{(radius, height, lat, beta)}.
668 @raise RangeError: Latitude B{C{lat}} outside valid range and
669 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
671 @raise TypeError: Invalid B{C{lat}}.
673 @raise ValueError: Invalid B{C{lat}}.
675 @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude}
676 <https://WikiPedia.org/wiki/Latitude>}, I{Karney's} C++ U{CircleRadius and CircleHeight
677 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>}
678 and method C{Rlat}.
679 '''
680 lat = Lat(lat)
681 if lat:
682 b = lat
683 if fabs(lat) < _90_0:
684 if self.f:
685 b = self._beta(lat)
686 z, r = sincos2d(b)
687 r *= self.a
688 z *= self.b
689 else: # near-polar
690 r, z = _0_0, copysign0(self.b, lat)
691 else: # equator
692 r = self.a
693 z = lat = b = _0_0
694 return Circle4Tuple(r, z, lat, b)
696 def degrees2m(self, deg, lat=0):
697 '''Convert an angle to the distance along the equator or
698 along a parallel of (geodetic) latitude.
700 @arg deg: The angle (C{degrees}).
701 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
703 @return: Distance (C{meter}, same units as the equatorial
704 and polar radii) or C{0} for near-polar B{C{lat}}.
706 @raise RangeError: Latitude B{C{lat}} outside valid range and
707 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
709 @raise ValueError: Invalid B{C{deg}} or B{C{lat}}.
710 '''
711 return self.radians2m(radians(deg), lat=lat)
713 def distance2(self, lat0, lon0, lat1, lon1):
714 '''I{Approximate} the distance and (initial) bearing between
715 two points based on the U{local, flat earth approximation
716 <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny
717 <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
719 I{Suitable only for distances of several hundred Km or Miles
720 and only between points not near-polar}.
722 @arg lat0: From latitude (C{degrees}).
723 @arg lon0: From longitude (C{degrees}).
724 @arg lat1: To latitude (C{degrees}).
725 @arg lon1: To longitude (C{degrees}).
727 @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance}
728 in same units as this ellipsoid's axes.
730 @note: The meridional and prime_vertical radii of curvature are
731 taken and scaled I{at the initial latitude}, see C{roc2}.
733 @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}.
734 '''
735 phi0 = Phid(lat0=lat0)
736 m, n = self.roc2_(phi0, scaled=True)
737 m *= Phid(lat1=lat1) - phi0
738 n *= Lamd(lon1=lon1) - Lamd(lon0=lon0)
739 return Distance2Tuple(hypot(m, n), atan2b(n, m))
741 @Property_RO
742 def e(self):
743 '''Get the I{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}.
745 @see: Property L{es}.
746 '''
747 return Float(e=sqrt(self.e2abs) if self.e2 else _0_0)
749 @deprecated_Property_RO
750 def e12(self): # see property ._e12
751 '''DEPRECATED, use property C{e21}.'''
752 return self.e21
754# @Property_RO
755# def _e12(self): # see property ._elliptic_e12
756# # (INTERNAL) until e12 above can be replaced with e21.
757# return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2)
759 @Property_RO
760 def e2(self):
761 '''Get the I{signed, (1st) eccentricity squared} (C{float}), M{f * (2 - f)
762 == 1 - (b / a)**2}, see C{a_b2e2}.
763 '''
764 return self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f))
766 @Property_RO
767 def e2abs(self):
768 '''Get the I{unsigned, (1st) eccentricity squared} (C{float}).
769 '''
770 return fabs(self.e2)
772 @Property_RO
773 def e21(self):
774 '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2}
775 == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}.
776 '''
777 return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0)
779# _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2
780 _1_e21 = a2_b2 # == M{1 / e21} == M{1 / (1 - e**2)}
782 @Property_RO
783 def e22(self):
784 '''Get the I{signed, 2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)
785 == e2 / (1 - f)**2 == (a / b)**2 - 1}, see C{a_b2e22}.
786 '''
787 return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f))
789 @Property_RO
790 def e22abs(self):
791 '''Get the I{unsigned, 2nd eccentricity squared} (C{float}).
792 '''
793 return fabs(self.e22)
795 @Property_RO
796 def e32(self):
797 '''Get the I{signed, 3rd eccentricity squared} (C{float}), M{e2 / (2 - e2)
798 == (a**2 - b**2) / (a**2 + b**2)}, see C{a_b2e32}.
799 '''
800 return self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f))
802 @Property_RO
803 def e32abs(self):
804 '''Get the I{unsigned, 3rd eccentricity squared} (C{float}).
805 '''
806 return fabs(self.e32)
808 @Property_RO
809 def e4(self):
810 '''Get the I{unsignd, (1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}.
811 '''
812 return Float(e4=self.e2**2 if self.e2 else _0_0)
814 eccentricity = e # eccentricity
815# eccentricity2 = e2 # eccentricity squared
816 eccentricity1st2 = e2 # first eccentricity squared, signed
817 eccentricity2nd2 = e22 # second eccentricity squared, signed
818 eccentricity3rd2 = e32 # third eccentricity squared, signed
820 def ecef(self, Ecef=None):
821 '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter.
823 @kwarg Ecef: ECEF class to use, default L{EcefKarney}.
825 @return: An ECEF converter for this C{ellipsoid}.
827 @raise TypeError: Invalid B{C{Ecef}}.
829 @see: Module L{pygeodesy.ecef}.
830 '''
831 return _MODS.ecef._4Ecef(self, Ecef)
833 @Property_RO
834 def _elliptic_e12(self): # see I{Karney}'s Ellipsoid._e12
835 '''(INTERNAL) Elliptic helper for C{Rhumb}.
836 '''
837 e12 = _over(self.e2, self.e2 - _1_0) # NOT DEPRECATED .e12!
838 return _MODS.elliptic.Elliptic(e12)
840 @Property_RO
841 def _elliptic_e22(self): # aka ._elliptic_ep2
842 '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}.
843 '''
844 return _MODS.elliptic.Elliptic(-self.e22abs) # complex
846 equatoradius = a # Requatorial
848 def e2s(self, s):
849 '''Compute norm M{sqrt(1 - e2 * s**2)}.
851 @arg s: Sine value (C{scalar}).
853 @return: Norm (C{float}).
855 @raise ValueError: Invalid B{C{s}}.
856 '''
857 return sqrt(self.e2s2(s)) if self.e2 else _1_0
859 def e2s2(self, s):
860 '''Compute M{1 - e2 * s**2}.
862 @arg s: Sine value (C{scalar}).
864 @return: Result (C{float}).
866 @raise ValueError: Invalid B{C{s}}.
867 '''
868 r = _1_0
869 if self.e2:
870 try:
871 r -= self.e2 * Scalar(s=s)**2
872 if r < 0:
873 raise ValueError(_negative_)
874 except (TypeError, ValueError) as x:
875 t = self._DOT_(typename(Ellipsoid.e2s2))
876 raise _ValueError(t, s, cause=x)
877 return r
879 @Property_RO
880 def es(self):
881 '''Get the I{signed (1st) eccentricity} (C{float}).
883 @see: Property L{e}.
884 '''
885 # note, self.e is always non-negative
886 return Float(es=copysign0(self.e, self.f)) # see .ups
888 def es_atanh(self, x):
889 '''Compute M{es * atanh(es * x)} or M{-es * atan(es * x)}
890 for I{oblate} respectively I{prolate} ellipsoids where
891 I{es} is the I{signed} (1st) eccentricity.
893 @raise ValueError: Invalid B{C{x}}.
895 @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/
896 C++/doc/classGeographicLib_1_1Math.html>}.
897 '''
898 return self._es_atanh(Scalar(x=x)) if self.f else _0_0
900 def _es_atanh(self, x): # see .albers._atanhee, .AuxLat._atanhee
901 '''(INTERNAL) Helper for .es_atanh, ._es_taupf2 and ._exp_es_atanh.
902 '''
903 es = self.es # signOf(es) == signOf(f)
904 return es * (atanh(es * x) if es > 0 else # .isOblate
905 (-atan(es * x) if es < 0 else # .isProlate
906 _0_0)) # .isSpherical
908 @Property_RO
909 def es_c(self):
910 '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}.
911 '''
912 return Float(es_c=(self._exp_es_atanh_1 * self.b_a) if self.f else _1_0)
914 def es_tauf(self, taup):
915 '''Compute I{Karney}'s U{equations (19), (20) and (21)
916 <https://ArXiv.org/abs/1002.1417>}.
918 @see: I{Karney}'s C++ method U{Math::tauf<https://GeographicLib.
919 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>} and
920 and I{Veness}' JavaScript method U{toLatLon<https://www.
921 Movable-Type.co.UK/scripts/latlong-utm-mgrs.html>}.
922 '''
923 t = Scalar(taup=taup)
924 if self.f: # .isEllipsoidal
925 a = fabs(t)
926 T = (self._exp_es_atanh_1 if a > 70 else self._1_e21) * t
927 if fabs(T * _EPSqrt) < _2_0: # handles +/- INF and NAN
928 s = (a * _TOL) if a > _1_0 else _TOL
929 for T, _, d in self._es_tauf3(t, T): # max 2
930 if fabs(d) < s:
931 break
932 t = Scalar(tauf=T)
933 return t
935 def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB
936 '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most
937 B{C{N}} Newton iterations, converging rapidly except when C{delta}
938 toggles on +/-1.12e-16 or +/-4.47e-16, see C{.utm.Utm._toLLEB}.
939 '''
940 e = self._1_e21
941 _F2_ = Fsum(T).fsum2f_ # τ0
942 _tf2 = self._es_taupf2
943 for i in range(1, N + 1):
944 a, h = _tf2(T)
945 # = (taup - a) / hypot1(a) / ((e + T**2) / h)
946 d = _over((taup - a) * (T**2 + e), hypot1(a) * h)
947 T, d = _F2_(d) # τi, (τi - τi-1)
948 yield T, i, d
950 def es_taupf(self, tau):
951 '''Compute I{Karney}'s U{equations (7), (8) and (9)
952 <https://ArXiv.org/abs/1002.1417>}.
954 @see: I{Karney}'s C++ method U{Math::taupf<https://GeographicLib.
955 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}.
956 '''
957 t = Scalar(tau=tau)
958 if self.f: # .isEllipsoidal
959 t, _ = self._es_taupf2(t)
960 t = Scalar(taupf=t)
961 return t
963 def _es_taupf2(self, tau):
964 '''(INTERNAL) Return 2-tuple C{(es_taupf(tau), hypot1(tau))}.
965 '''
966 if _isfinite(tau):
967 h = hypot1(tau)
968 s = sinh(self._es_atanh(tau / h))
969 a = hypot1(s) * tau - h * s
970 else:
971 a, h = tau, INF
972 return a, h
974 @Property_RO
975 def _exp_es_atanh_1(self):
976 '''(INTERNAL) Helper for .es_c and .es_tauf.
977 '''
978 return exp(self._es_atanh(_1_0)) if self.es else _1_0
980 @Property_RO
981 def f(self):
982 '''Get the I{flattening} (C{scalar}), M{(a - b) / a}, C{0} for spherical, negative for prolate.
983 '''
984 return self._f
986 @Property_RO
987 def f_(self):
988 '''Get the I{inverse flattening} (C{scalar}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}.
989 '''
990 return self._f_
992 @Property_RO
993 def f1(self):
994 '''Get the I{1 - flattening} (C{float}), M{f1 == 1 - f == b / a}.
996 @see: Property L{b_a}.
997 '''
998 return Float(f1=_1_0 - self.f)
1000 @Property_RO
1001 def f2(self):
1002 '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}.
1003 '''
1004 return self._assert(self.a_b - _1_0, f2=f2f2(self.f))
1006 @deprecated_Property_RO
1007 def geodesic(self):
1008 '''DEPRECATED, use property C{geodesicw}.'''
1009 return self.geodesicw
1011 def geodesic_(self, exact=True):
1012 '''Get the an I{exact} C{Geodesic...} instance for this ellipsoid.
1014 @kwarg exact: If C{bool} return L{GeodesicExact}C{(exact=B{exact}, ...)},
1015 otherwise a L{Geodesic}, L{GeodesicExact} or L{GeodesicSolve}
1016 instance for I{this} ellipsoid.
1018 @return: The C{exact} geodesic (C{Geodesic...}).
1020 @raise TypeError: Invalid B{C{exact}}.
1022 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1023 '''
1024 if isbool(exact): # for consistenccy with C{.rhumb_}
1025 g = _MODS.geodesicx.GeodesicExact(self, C4order=30 if exact else 24,
1026 name=self.name)
1027 else:
1028 g = exact
1029 E = _xattr(g, ellipsoid=None)
1030 if not (E is self and isinstance(g, self._Geodesics)):
1031 raise _ValueError(exact=g, ellipsoid=E, txt_not_=self.name)
1032 return g
1034 @property_ROver
1035 def _Geodesics(self):
1036 '''(INTERNAL) Get all C{Geodesic...} classes, I{once}.
1037 '''
1038 t = (_MODS.geodesicx.GeodesicExact,
1039 _MODS.geodsolve.GeodesicSolve)
1040 try:
1041 t += (_MODS.geodesicw.Geodesic,
1042 _MODS.geodesicw._wrapped.Geodesic)
1043 except ImportError:
1044 pass
1045 return t # overwrite property_ROver
1047 @property_RO
1048 def geodesicw(self):
1049 '''Get this ellipsoid's I{wrapped} U{geodesicw.Geodesic
1050 <https://GeographicLib.SourceForge.io/Python/doc/code.html>}, provided
1051 I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1052 package is installed.
1053 '''
1054 # if not self.isEllipsoidal:
1055 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1056 return _MODS.geodesicw.Geodesic(self)
1058 @property_RO
1059 def geodesicx(self):
1060 '''Get this ellipsoid's I{exact} L{GeodesicExact}.
1061 '''
1062 # if not self.isEllipsoidal:
1063 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1064 return _MODS.geodesicx.GeodesicExact(self, name=self.name)
1066 @property
1067 def geodsolve(self):
1068 '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility
1069 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1070 provided the path to the C{GeodSolve} executable is specified with env
1071 variable C{PYGEODESY_GEODSOLVE} or re-/set with this property..
1072 '''
1073 # if not self.isEllipsoidal:
1074 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1075 return _MODS.geodsolve.GeodesicSolve(self, path=self._geodsolve, name=self.name)
1077 @geodsolve.setter # PYCHOK setter!
1078 def geodsolve(self, path):
1079 '''Re-/set the (fully qualified) path to the U{GeodSolve
1080 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1081 overriding env variable C{PYGEODESY_GEODSOLVE} (C{str}).
1082 '''
1083 self._geodsolve = path
1085 def hartzell4(self, pov, los=None):
1086 '''Compute the intersection of this ellipsoid's surface and a Line-Of-Sight
1087 from a Point-Of-View in space.
1089 @arg pov: Point-Of-View outside this ellipsoid (C{Cartesian}, L{Ecef9Tuple}
1090 or L{Vector3d}).
1091 @kwarg los: Line-Of-Sight, I{direction} to this ellipsoid (L{Los}, L{Vector3d})
1092 or C{True} for the I{normal, perpendicular, plumb} to the surface
1093 of this ellipsoid or C{False} or C{None} to point to its center.
1095 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1096 C{y} and C{z} of the projection on or the intersection with this
1097 ellipsoid and the I{distance} C{h} from B{C{pov}} to C{(x, y, z)}
1098 along B{C{los}}, all in C{meter}, conventionally.
1100 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, or B{C{pov}}
1101 is inside this ellipsoid or B{C{los}} points
1102 outside this ellipsoid or in opposite direction.
1104 @raise TypeError: Invalid B{C{pov}} or B{C{los}}.
1106 @see: U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.
1107 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and
1108 methods L{Ellipsoid.height4} and L{Triaxial.hartzell4}.
1109 '''
1110 try:
1111 v, d, i = _MODS.triaxials._hartzell3(pov, los, self._triaxial)
1112 except Exception as x:
1113 raise IntersectionError(pov=pov, los=los, cause=x)
1114 return Vector4Tuple(v.x, v.y, v.z, d, iteration=i, name__=self.hartzell4)
1116 @Property_RO
1117 def _hash(self):
1118 return hash((self.a, self.f))
1120 def height4(self, xyz, normal=True):
1121 '''Compute the projection on and the height of a cartesian above or below
1122 this ellipsoid's surface.
1124 @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d},
1125 L{Vector3Tuple} or L{Vector4Tuple}).
1126 @kwarg normal: If C{True}, the projection is perpendicular to (the nearest
1127 point on) this ellipsoid's surface, otherwise the C{radial}
1128 line to this ellipsoid's center (C{bool}).
1130 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1131 C{y} and C{z} of the projection on and the height C{h} above or
1132 below this ellipsoid's surface, all in C{meter}, conventionally.
1134 @raise ValueError: Null B{C{xyz}}.
1136 @raise TypeError: Non-cartesian B{C{xyz}}.
1138 @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>}
1139 and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse and
1140 methods L{Ellipsoid.hartzell4} and L{Triaxial.height4}.
1141 '''
1142 v = _MODS.vector3d._otherV3d(xyz=xyz)
1143 r = v.length
1145 a, b, i = self.a, self.b, None
1146 if r < EPS0: # EPS
1147 v = v.times(_0_0)
1148 h = -a
1150 elif self.isSpherical:
1151 v = v.times(a / r)
1152 h = r - a
1154 elif normal: # perpendicular to ellipsoid
1155 x, y = hypot(v.x, v.y), fabs(v.z)
1156 if x < EPS0: # PYCHOK no cover
1157 z = copysign0(b, v.z)
1158 v = Vector3Tuple(v.x, v.y, z)
1159 h = y - b # polar
1160 elif y < EPS0: # PYCHOK no cover
1161 t = a / r
1162 v = v.times_(t, t, 0) # force z=0.0
1163 h = x - a # equatorial
1164 else: # normal in 1st quadrant
1165 x, y, i = _MODS.triaxials._plumbTo3(x, y, self)
1166 t, v = v, v.times_(x, x, y)
1167 h = t.minus(v).length
1169 else: # radial to ellipsoid's center
1170 h = hypot_(a * v.z, b * v.x, b * v.y)
1171 t = (a * b / h) if h > EPS0 else _0_0 # EPS
1172 v = v.times(t)
1173 h = r * (_1_0 - t)
1175 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name__=self.height4)
1177 def _hubeny_2(self, phi2, phi1, lam21, scaled=True, squared=True):
1178 '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_},
1179 returning the I{angular} distance in C{radians squared} or C{radians}
1180 '''
1181 m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=scaled)
1182 h, r = (hypot2, self.a2_) if squared else (hypot, _1_0 / self.a)
1183 return h(m * (phi2 - phi1), n * lam21) * r
1185 @Property_RO
1186 def isEllipsoidal(self):
1187 '''Is this model I{ellipsoidal} (C{bool})?
1188 '''
1189 return self.f != 0
1191 @Property_RO
1192 def isOblate(self):
1193 '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or
1194 spherical otherwise.
1195 '''
1196 return self.f > 0
1198 @Property_RO
1199 def isProlate(self):
1200 '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or
1201 spherical otherwise.
1202 '''
1203 return self.f < 0
1205 @Property_RO
1206 def isSpherical(self):
1207 '''Is this ellipsoid I{spherical} (C{bool})?
1208 '''
1209 return self.f == 0
1211 def _Kseries(self, *AB8Ks):
1212 '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha
1213 or Beta series coefficients per I{Karney}'s U{equations (35)
1214 and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}.
1216 @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series
1217 coefficient tuples.
1219 @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}).
1221 @see: I{Karney}'s 30-th order U{TMseries30
1222 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}.
1223 '''
1224 k = self.KsOrder
1225 if self.n:
1226 ns = fpowers(self.n, k)
1227 ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k))
1228 else:
1229 ks = _0_0s(k)
1230 return ks
1232 @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''')
1233 def KsOrder(self):
1234 '''Get the I{Krüger} series' order (C{int} 4, 6 or 8).
1235 '''
1236 return self._KsOrder
1238 @KsOrder.setter # PYCHOK setter!
1239 def KsOrder(self, order):
1240 '''Set the I{Krüger} series' order (C{int} 4, 6 or 8).
1242 @raise ValueError: Invalid B{C{order}}.
1243 '''
1244 if not (isint(order) and _isin(order, 4, 6, 8)):
1245 raise _ValueError(order=order)
1246 if self._KsOrder != order:
1247 Ellipsoid.AlphaKs._update(self)
1248 Ellipsoid.BetaKs._update(self)
1249 self._KsOrder = order
1251 @Property_RO
1252 def L(self):
1253 '''Get the I{quarter meridian} C{L}, aka the C{polar distance}
1254 along a meridian between the equator and a pole (C{meter}),
1255 M{b * Elliptic(-e2 / (1 - e2)).cE} or M{b * PI / 2}.
1256 '''
1257 r = self._elliptic_e22.cE if self.f else PI_2
1258 return Distance(L=self.b * r)
1260 def Llat(self, lat):
1261 '''Return the I{meridional length}, the distance along a meridian
1262 between the equator and a (geodetic) latitude, see C{L}.
1264 @arg lat: Geodetic latitude (C{degrees90}).
1266 @return: The meridional length at B{C{lat}}, negative on southern
1267 hemisphere (C{meter}).
1268 '''
1269 r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phid(lat)
1270 return Distance(Llat=self.b * r)
1272 Lmeridian = Llat # meridional distance
1274 @property_RO
1275 def _Lpd(self):
1276 '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}.
1277 '''
1278 return Meter(_Lpd=self.L / _90_0)
1280 @property_RO
1281 def _Lpr(self):
1282 '''Get the I{quarter meridian} per radian (C{meter}), M{self.L / PI_2}.
1283 '''
1284 return Meter(_Lpr=self.L / PI_2)
1286 @deprecated_Property_RO
1287 def majoradius(self): # PYCHOK no cover
1288 '''DEPRECATED, use property C{a} or C{Requatorial}.'''
1289 return self.a
1291 def m2degrees(self, distance, lat=0):
1292 '''Convert a distance to an angle along the equator or along
1293 a parallel of (geodetic) latitude.
1295 @arg distance: Distance (C{meter}).
1296 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1298 @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}.
1300 @raise RangeError: Latitude B{C{lat}} outside valid range and
1301 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1303 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1304 '''
1305 return degrees(self.m2radians(distance, lat=lat))
1307 def m2radians(self, distance, lat=0):
1308 '''Convert a distance to an angle along the equator or along
1309 a parallel of (geodetic) latitude.
1311 @arg distance: Distance (C{meter}).
1312 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1314 @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}.
1316 @raise RangeError: Latitude B{C{lat}} outside valid range and
1317 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1319 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1320 '''
1321 r = self.circle4(lat).radius if lat else self.a
1322 return m2radians(distance, radius=r, lat=0)
1324 @deprecated_Property_RO
1325 def minoradius(self): # PYCHOK no cover
1326 '''DEPRECATED, use property C{b}, C{polaradius} or C{Rpolar}.'''
1327 return self.b
1329 @Property_RO
1330 def n(self):
1331 '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}.
1332 '''
1333 return self._assert(a_b2n(self.a, self.b), n=f2n(self.f))
1335 flattening = f
1336 flattening1st = f
1337 flattening2nd = f2
1338 flattening3rd = n
1340 polaradius = b # Rpolar
1342# @Property_RO
1343# def Q(self):
1344# '''Get the I{meridian arc unit} C{Q}, the mean, meridional length I{per radian} C({float}).
1345#
1346# @note: C{Q * PI / 2} ≈ C{L}, the I{quarter meridian}.
1347#
1348# @see: Property C{A} and U{Engsager, K., Poder, K.<https://StudyLib.net/doc/7443565/
1349# a-highly-accurate-world-wide-algorithm-for-the-transverse...>}.
1350# '''
1351# n = self.n
1352# d = (n + _1_0) / self.a
1353# return Float(Q=Fhorner(n**2, _1_0, _0_25, _1_16th, _0_25).fover(d) if d else self.b)
1355# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1356# # Q = (1 - 3/4 * e'2 + 45/64 * e'4 - 175/256 * e'6 + 11025/16384 * e'8) * rocPolar
1357# # = (4 + e'2 * (-3 + e'2 * (45/16 + e'2 * (-175/64 + e'2 * 11025/4096)))) * rocPolar / 4
1358# return Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(4 / self.rocPolar)
1360 @deprecated_Property_RO
1361 def quarteradius(self): # PYCHOK no cover
1362 '''DEPRECATED, use property C{L} or method C{Llat}.'''
1363 return self.L
1365 @Property_RO
1366 def R1(self):
1367 '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}.
1369 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}
1370 and method C{Rgeometric}.
1371 '''
1372 r = Fsum(self.a, self.a, self.b).fover(_3_0) if self.f else self.a
1373 return Radius(R1=r)
1375 Rmean = R1
1377 @Property_RO
1378 def R2(self):
1379 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}.
1381 @see: C{R2x}, C{c2}, C{area} and U{Earth radius
1382 <https://WikiPedia.org/wiki/Earth_radius>}.
1383 '''
1384 return Radius(R2=sqrt(self.c2) if self.f else self.a)
1386 Rauthalic = R2
1388# @Property_RO
1389# def R2(self):
1390# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1391# # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar
1392# # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3
1393# return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar)
1395 @Property_RO
1396 def R2x(self):
1397 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}.
1399 @see: C{R2}, C{c2x} and C{areax}.
1400 '''
1401 return Radius(R2x=sqrt(self.c2x) if self.f else self.a)
1403 Rauthalicx = R2x
1405 @Property_RO
1406 def R3(self):
1407 '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}.
1409 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}.
1410 '''
1411 r = (cbrt(self.b_a) * self.a) if self.f else self.a
1412 return Radius(R3=r)
1414 Rvolumetric = R3
1416 def radians2m(self, rad, lat=0):
1417 '''Convert an angle to the distance along the equator or along
1418 a parallel of (geodetic) latitude.
1420 @arg rad: The angle (C{radians}).
1421 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1423 @return: Distance (C{meter}, same units as the equatorial
1424 and polar radii) or C{0} for near-polar B{C{lat}}.
1426 @raise RangeError: Latitude B{C{lat}} outside valid range and
1427 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1429 @raise ValueError: Invalid B{C{rad}} or B{C{lat}}.
1430 '''
1431 r = self.circle4(lat).radius if lat else self.a
1432 return radians2m(rad, radius=r, lat=0)
1434 @Property_RO
1435 def Rbiaxial(self):
1436 '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}.
1438 @see: C{Rtriaxial}
1439 '''
1440 a, b = self.a, self.b
1441 if b < a:
1442 b = sqrt(_0_5 + self.b2_a2 * _0_5) * a
1443 elif b > a:
1444 b *= sqrt(_0_5 + self.a2_b2 * _0_5)
1445 return Radius(Rbiaxial=b)
1447 Requatorial = a # for consistent naming
1449 def Rgeocentric(self, lat):
1450 '''Compute the I{geocentric} earth radius of (geodetic) latitude.
1452 @arg lat: Latitude (C{degrees90}).
1454 @return: Geocentric earth radius (C{meter}).
1456 @raise ValueError: Invalid B{C{lat}}.
1458 @see: U{Geocentric Radius
1459 <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>}
1460 '''
1461 r, a = self.a, Phid(lat)
1462 if a and self.f:
1463 if fabs(a) < PI_2:
1464 s2, c2 = _s2_c2(a)
1465 b2_a2_s2 = self.b2_a2 * s2
1466 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2))
1467 # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2)))
1468 # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2))
1469 # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2))
1470 # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2))
1471 r *= sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2))
1472 else:
1473 r = self.b
1474 return Radius(Rgeocentric=r)
1476 @Property_RO
1477 def Rgeometric(self):
1478 '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}.
1480 @see: C{R1}.
1481 '''
1482 g = sqrt(self.a * self.b) if self.f else self.a
1483 return Radius(Rgeometric=g)
1485 def rhumb_(self, exact=True):
1486 '''Get the an I{exact} C{Rhumb...} instance for this ellipsoid.
1488 @kwarg exact: If C{bool} or C{None} return L{Rhumb}C{(exact=B{exact}, ...)},
1489 otherwise a L{Rhumb}, L{RhumbAux} or L{RhumbSolve} instance
1490 for I{this} ellipsoid.
1492 @return: The C{exact} rhumb (C{Rhumb...}).
1494 @raise TypeError: Invalid B{C{exact}}.
1496 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1497 '''
1498 if isbool(exact): # use Rhumb for backward compatibility
1499 r = _MODS.rhumb.ekx.Rhumb(self, exact=exact, name=self.name)
1500 else:
1501 r = exact
1502 E = _xattr(r, ellipsoid=None)
1503 if not (E is self and isinstance(r, self._Rhumbs)):
1504 raise _ValueError(exact=r, ellipsosid=E, txt_not_=self.name)
1505 return r
1507 @property_RO
1508 def rhumbaux(self):
1509 '''Get this ellipsoid's I{Auxiliary} C{rhumb.RhumbAux}.
1510 '''
1511 # if not self.isEllipsoidal:
1512 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1513 return _MODS.rhumb.aux_.RhumbAux(self, name=self.name)
1515 @property_RO
1516 def rhumbekx(self):
1517 '''Get this ellipsoid's I{Elliptic, Krüger} C{rhumb.Rhumb}.
1518 '''
1519 # if not self.isEllipsoidal:
1520 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1521 return _MODS.rhumb.ekx.Rhumb(self, name=self.name)
1523 @property_ROver
1524 def _Rhumbs(self):
1525 '''(INTERNAL) Get all C{Rhumb...} classes, I{once}.
1526 '''
1527 r = _MODS.rhumb
1528 return (r.aux_.RhumbAux, # overwrite property_ROver
1529 r.ekx.Rhumb, r.solve.RhumbSolve)
1531 @property
1532 def rhumbsolve(self):
1533 '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility
1534 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1535 provided the path to the C{RhumbSolve} executable is specified with env
1536 variable C{PYGEODESY_RHUMBSOLVE} or re-/set with this property.
1537 '''
1538 # if not self.isEllipsoidal:
1539 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1540 return _MODS.rhumb.solve.RhumbSolve(self, path=self._rhumbsolve, name=self.name)
1542 @rhumbsolve.setter # PYCHOK setter!
1543 def rhumbsolve(self, path):
1544 '''Re-/set the (fully qualified) path to the U{RhumbSolve
1545 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1546 overriding env variable C{PYGEODESY_RHUMBSOLVE} (C{str}).
1547 '''
1548 self._rhumbsolve = path
1550 @deprecated_property_RO
1551 def rhumbx(self):
1552 '''DEPRECATED on 2023.11.28, use property C{rhumbekx}. '''
1553 return self.rhumbekx
1555 def Rlat(self, lat):
1556 '''I{Approximate} the earth radius of (geodetic) latitude.
1558 @arg lat: Latitude (C{degrees90}).
1560 @return: Approximate earth radius (C{meter}).
1562 @raise RangeError: Latitude B{C{lat}} outside valid range and
1563 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1565 @raise TypeError: Invalid B{C{lat}}.
1567 @raise ValueError: Invalid B{C{lat}}.
1569 @note: C{Rlat(B{90})} equals C{Rpolar}.
1571 @see: Method C{circle4}.
1572 '''
1573 # r = a - (a - b) * |lat| / 90
1574 r = self.a
1575 if self.f and lat: # .isEllipsoidal
1576 r -= (r - self.b) * fabs(Lat(lat)) / _90_0
1577 r = Radius(Rlat=r)
1578 return r
1580 Rpolar = b # for consistent naming
1582 def roc1_(self, sa, ca=None):
1583 '''Compute the I{prime-vertical}, I{normal} radius of curvature
1584 of (geodetic) latitude, I{unscaled}.
1586 @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]).
1587 @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0])
1588 to use an alternate formula.
1590 @return: The prime-vertical radius of curvature (C{float}).
1592 @note: The delta between both formulae with C{Ellipsoids.WGS84}
1593 is less than 2 nanometer over the entire latitude range.
1595 @see: Method L{roc2_} and class L{EcefYou}.
1596 '''
1597 if not self.f: # .isSpherical
1598 n = self.a
1599 elif ca is None:
1600 r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward
1601 n = sqrt(self.a2 / r) if r > EPS02 else _0_0
1602 elif ca: # derived from EcefYou.forward
1603 h = hypot(ca, self.b_a * sa) if sa else fabs(ca)
1604 n = self.a / h
1605 elif sa:
1606 n = self.a2_b / fabs(sa)
1607 else:
1608 n = self.a
1609 return n
1611 def roc2(self, lat, scaled=False):
1612 '''Compute the I{meridional} and I{prime-vertical}, I{normal}
1613 radii of curvature of (geodetic) latitude.
1615 @arg lat: Latitude (C{degrees90}).
1616 @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}).
1618 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with
1619 the radii of curvature.
1621 @raise ValueError: Invalid B{C{lat}}.
1623 @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation
1624 <https://www.EdWilliams.org/avform.htm#flat>} and meridional and
1625 prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/
1626 Earth_radius#Radii_of_curvature>}.
1627 '''
1628 return self.roc2_(Phid(lat), scaled=scaled)
1630 def roc2_(self, phi, scaled=False):
1631 '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of
1632 curvature of (geodetic) latitude.
1634 @arg phi: Latitude (C{radians}).
1635 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}).
1637 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the
1638 radii of curvature.
1640 @raise ValueError: Invalid B{C{phi}}.
1642 @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local,
1643 flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>}
1644 and the meridional and prime vertical U{Radii of Curvature
1645 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1646 '''
1647 a = fabs(Phi(phi))
1648 if self.f:
1649 r = self.e2s2(sin(a))
1650 if r > EPS02:
1651 n = self.a / sqrt(r)
1652 m = n * self.e21 / r
1653 else:
1654 m = n = _0_0
1655 else:
1656 m = n = self.a
1657 if scaled and a:
1658 n *= cos(a) if a < PI_2 else _0_0
1659 return Curvature2Tuple(m, n)
1661 def rocAzimuth(self, lat, azimuth):
1662 '''Compute the I{directional} radius of curvature of (geodetic) latitude
1663 and C{azimuth} compass direction.
1665 @see: Method L{rocBearing<Ellipsoid.rocBearing>} for details, using C{azimuth} for C{bearing}.
1666 '''
1667 return Radius(rocAzimuth=self._rocDirectional(lat, Azimuth(azimuth)))
1669 def rocBearing(self, lat, bearing):
1670 '''Compute the I{directional} radius of curvature of (geodetic) latitude
1671 and C{bearing} compass direction.
1673 @arg lat: Latitude (C{degrees90}).
1674 @arg bearing: Direction (compass C{degrees360}).
1676 @return: Directional radius of curvature (C{meter}).
1678 @raise RangeError: Latitude B{C{lat}} outside valid range and
1679 L{rangerrors<pygeodesy.rangerrors>} is C{True}.
1681 @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}.
1683 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
1684 '''
1685 return Radius(rocBearing=self._rocDirectional(lat, Bearing(bearing)))
1687 def _rocDirectional(self, lat, deg):
1688 '''(INTERNAL) Helper for C{rocAzimuth} and C{rocBearing}.
1689 '''
1690 if self.f:
1691 s2, c2 = _s2_c2(radians(deg))
1692 m, n = self.roc2_(Phid(lat))
1693 if n < m: # == n / (c2 * n / m + s2)
1694 c2 *= n / m
1695 elif m < n: # == m / (c2 + s2 * m / n)
1696 s2 *= m / n
1697 n = m
1698 r = _over(n, c2 + s2) # == 1 / (c2 / m + s2 / n)
1699 else:
1700 r = self.b # == self.a
1701 return r
1703 @Property_RO
1704 def rocEquatorial2(self):
1705 '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature
1706 at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}.
1708 @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a},
1709 L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature
1710 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1711 '''
1712 m = self.b2_a if self.f else self.a
1713 return Curvature2Tuple(m, self.a)
1715 def rocGauss(self, lat):
1716 '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude.
1718 @arg lat: Latitude (C{degrees90}).
1720 @return: Gaussian radius of curvature (C{meter}).
1722 @raise ValueError: Invalid B{C{lat}}.
1724 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1725 Earth_radius#Radii_of_curvature>}
1726 '''
1727 # using ...
1728 # m, n = self.roc2_(Phid(lat))
1729 # return sqrt(m * n)
1730 # ... requires 1 or 2 sqrt
1731 g = self.b
1732 if self.f:
1733 s2, c2 = _s2_c2(Phid(lat))
1734 g = _over(g, c2 + self.b2_a2 * s2)
1735 return Radius(rocGauss=g)
1737 def rocMean(self, lat):
1738 '''Compute the I{mean} radius of curvature of (geodetic) latitude.
1740 @arg lat: Latitude (C{degrees90}).
1742 @return: Mean radius of curvature (C{meter}).
1744 @raise ValueError: Invalid B{C{lat}}.
1746 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1747 Earth_radius#Radii_of_curvature>}
1748 '''
1749 if self.f:
1750 m, n = self.roc2_(Phid(lat))
1751 m *= _over(n * _2_0, m + n) # == 2 / (1 / m + 1 / n)
1752 else:
1753 m = self.a
1754 return Radius(rocMean=m)
1756 def rocMeridional(self, lat):
1757 '''Compute the I{meridional} radius of curvature of (geodetic) latitude.
1759 @arg lat: Latitude (C{degrees90}).
1761 @return: Meridional radius of curvature (C{meter}).
1763 @raise ValueError: Invalid B{C{lat}}.
1765 @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation
1766 <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of
1767 Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1768 '''
1769 r = self.roc2_(Phid(lat)) if lat else self.rocEquatorial2
1770 return Radius(rocMeridional=r.meridional)
1772 rocPolar = a2_b # synonymous
1774 def rocPrimeVertical(self, lat):
1775 '''Compute the I{prime-vertical}, I{normal} radius of curvature of
1776 (geodetic) latitude, aka the I{transverse} radius of curvature.
1778 @arg lat: Latitude (C{degrees90}).
1780 @return: Prime-vertical radius of curvature (C{meter}).
1782 @raise ValueError: Invalid B{C{lat}}.
1784 @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth
1785 approximation<https://www.EdWilliams.org/avform.htm#flat>} and
1786 U{Radii of Curvature<https://WikiPedia.org/wiki/
1787 Earth_radius#Radii_of_curvature>}.
1788 '''
1789 r = self.roc2_(Phid(lat)) if lat else self.rocEquatorial2
1790 return Radius(rocPrimeVertical=r.prime_vertical)
1792 rocTransverse = rocPrimeVertical # synonymous
1794 @deprecated_Property_RO
1795 def Rquadratic(self): # PYCHOK no cover
1796 '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.'''
1797 return self.Rbiaxial
1799 @deprecated_Property_RO
1800 def Rr(self): # PYCHOK no cover
1801 '''DEPRECATED, use property C{Rrectifying}.'''
1802 return self.Rrectifying
1804 @Property_RO
1805 def Rrectifying(self):
1806 '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}.
1808 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}.
1809 '''
1810 r = self.a
1811 if self.f:
1812 r *= cbrt2((sqrt3(self.b_a) + _1_0) * _0_5)
1813 return Radius(Rrectifying=r)
1815 @deprecated_Property_RO
1816 def Rs(self): # PYCHOK no cover
1817 '''DEPRECATED, use property C{Rgeometric}.'''
1818 return self.Rgeometric
1820 @Property_RO
1821 def Rtriaxial(self):
1822 '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}.
1824 @see: C{Rbiaxial}
1825 '''
1826 q, b = self.a, self.b
1827 if b < q:
1828 q *= sqrt((self.b2_a2 + _3_0) * _0_25)
1829 elif b > q:
1830 q = sqrt((self.a2_b2 * _3_0 + _1_0) * _0_25) * b
1831 return Radius(Rtriaxial=q)
1833 def toEllipsoid2(self, **name):
1834 '''Get a copy of this ellipsoid as an L{Ellipsoid2}.
1836 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
1838 @see: Property C{a_f}.
1839 '''
1840 return Ellipsoid2(self, None, **name)
1842 def toStr(self, prec=8, terse=4, **sep_name): # PYCHOK expected
1843 '''Return this ellipsoid as a text string.
1845 @kwarg prec: Number of decimal digits, unstripped (C{int}).
1846 @kwarg terse: Limit the number of items (C{int}, 0...18),
1847 use C{B{terse}=0} or C{=None} for all.
1848 @kwarg sep_name: Optional C{B{name}=NN} (C{str}) or C{None}
1849 to exclude this ellipsoid's name and separator
1850 C{B{sep}=", "} to join the items (C{str}).
1852 @return: This C{Ellipsoid}'s attributes (C{str}).
1853 '''
1854 E = Ellipsoid
1855 t = (E.a, E.f, E.f_, E.b, E.f2, E.n, E.e,
1856 E.e2, E.e21, E.e22, E.e32,
1857 E.A, E.L, E.R1, E.R2, E.R3,
1858 E.Rbiaxial, E.Rtriaxial)
1859 if terse:
1860 t = t[:terse]
1861 return self._instr(prec=prec, props=t, **sep_name)
1863 def toTriaxial(self, **name):
1864 '''Convert this ellipsoid to a L{Triaxial_}.
1866 @kwarg name: Optional C{B{name}=NN} (C{str}).
1868 @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis
1869 pointing east and C{Z} pointing north.
1871 @see: Method L{Triaxial_.toEllipsoid}.
1872 '''
1873 T = self._triaxial
1874 return T.copy(**name) if name else T
1876 @property_RO
1877 def _triaxial(self):
1878 '''(INTERNAL) Get this ellipsoid's un-/ordered C{Triaxial/_}.
1879 '''
1880 a, b, m = self.a, self.b, _MODS.triaxials
1881 T = m.Triaxial if a > b else m.Triaxial_
1882 return T(a, a, b, name=self.name)
1884 @Property_RO
1885 def volume(self):
1886 '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}.
1888 @see: C{R3}.
1889 '''
1890 return Meter3(volume=self.a2 * self.b * PI_3 * _4_0)
1893class Ellipsoid2(Ellipsoid):
1894 '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}.
1895 '''
1896 def __init__(self, a, f=None, **name):
1897 '''New L{Ellipsoid2}.
1899 @arg a: Equatorial radius, semi-axis (C{meter}) or a previous
1900 L{Ellipsoid} instance.
1901 @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}),
1902 if B{C{a}} is in C{meter}.
1903 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
1905 @raise NameError: Ellipsoid with that B{C{name}} already exists.
1907 @raise ValueError: Invalid B{C{a}} or B{C{f}}.
1909 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.
1910 Negative C{B{f}} produces a I{prolate} ellipsoid.
1911 '''
1912 if f is None and isinstance(a, Ellipsoid):
1913 Ellipsoid.__init__(self, a.a, f =a.f,
1914 b=a.b, f_=a.f_, **name)
1915 else:
1916 Ellipsoid.__init__(self, a, f=f, **name)
1919def _ispherical_a_b(a, b):
1920 '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}.
1921 '''
1922 return a < EPS0 or b < EPS0 or fabs(a - b) < EPS0
1925def _ispherical_f(f):
1926 '''(INTERNAL) C{True} for spherical or invalid C{f}.
1927 '''
1928 return f > EPS1 or fabs(f) < EPS
1931def _ispherical_f_(f_):
1932 '''(INTERNAL) C{True} for spherical or invalid C{f_}.
1933 '''
1934 f_ = fabs(f_)
1935 return f_ < EPS or f_ > _1_EPS
1938def a_b2e(a, b):
1939 '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius.
1941 @arg a: Equatorial radius (C{scalar} > 0).
1942 @arg b: Polar radius (C{scalar} > 0).
1944 @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}), M{sqrt(1 - (b / a)**2)}.
1946 @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids.
1947 '''
1948 e2 = _a2b2e2(a, b, b2=False)
1949 return Float(e=sqrt(fabs(e2)) if e2 else _0_0) # == sqrt(fabs((a - b) * (a + b))) / a
1952def a_b2e2(a, b):
1953 '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius.
1955 @arg a: Equatorial radius (C{scalar} > 0).
1956 @arg b: Polar radius (C{scalar} > 0).
1958 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}), M{1 - (b / a)**2}.
1960 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1961 for I{near-spherical} ellipsoids.
1962 '''
1963 return Float(e2=_a2b2e2(a, b, b2=False))
1966def a_b2e22(a, b):
1967 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1969 @arg a: Equatorial radius (C{scalar} > 0).
1970 @arg b: Polar radius (C{scalar} > 0).
1972 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}), M{(a / b)**2 - 1}.
1974 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1975 for I{near-spherical} ellipsoids.
1976 '''
1977 return Float(e22=_a2b2e2(a, b, a2=False))
1980def a_b2e32(a, b):
1981 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1983 @arg a: Equatorial radius (C{scalar} > 0).
1984 @arg b: Polar radius (C{scalar} > 0).
1986 @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}),
1987 M{(a**2 - b**2) / (a**2 + b**2)}.
1989 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1990 for I{near-spherical} ellipsoids.
1991 '''
1992 return Float(e32=_a2b2e2(a, b))
1995def _a2b2e2(a, b, a2=True, b2=True):
1996 '''(INTERNAL) Helper for C{a_b2e}, C{a_b2e2}, C{a_b2e22} and C{a_b2e32}.
1997 '''
1998 if _ispherical_a_b(a, b):
1999 e2 = _0_0
2000 else: # a > 0, b > 0
2001 a, b = (_1_0, b / a) if a > b else (a / b, _1_0)
2002 a2b2 = float(a - b) * (a + b)
2003 e2 = _over(a2b2, (a**2 if a2 else _0_0) +
2004 (b**2 if b2 else _0_0)) if a2b2 else _0_0
2005 return e2
2008def a_b2f(a, b):
2009 '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius.
2011 @arg a: Equatorial radius (C{scalar} > 0).
2012 @arg b: Polar radius (C{scalar} > 0).
2014 @return: The flattening (C{scalar} or C{0}), M{(a - b) / a}.
2016 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2017 for I{near-spherical} ellipsoids.
2018 '''
2019 f = 0 if _ispherical_a_b(a, b) else _over(float(a - b), a)
2020 return _f_0_0 if _ispherical_f(f) else Float(f=f)
2023def a_b2f_(a, b):
2024 '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius.
2026 @arg a: Equatorial radius (C{scalar} > 0).
2027 @arg b: Polar radius (C{scalar} > 0).
2029 @return: The inverse flattening (C{scalar} or C{0}), M{a / (a - b)}.
2031 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2032 for I{near-spherical} ellipsoids.
2033 '''
2034 f_ = 0 if _ispherical_a_b(a, b) else _over(a, float(a - b))
2035 return _f__0_0 if _ispherical_f_(f_) else Float(f_=f_)
2038def a_b2f2(a, b):
2039 '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius.
2041 @arg a: Equatorial radius (C{scalar} > 0).
2042 @arg b: Polar radius (C{scalar} > 0).
2044 @return: The I{signed}, 2nd flattening (C{scalar} or C{0}), M{(a - b) / b}.
2046 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2047 for I{near-spherical} ellipsoids.
2048 '''
2049 t = 0 if _ispherical_a_b(a, b) else float(a - b)
2050 return Float(f2=_0_0 if fabs(t) < EPS0 else _over(t, b))
2053def a_b2n(a, b):
2054 '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius.
2056 @arg a: Equatorial radius (C{scalar} > 0).
2057 @arg b: Polar radius (C{scalar} > 0).
2059 @return: The I{signed}, 3rd flattening (C{scalar} or C{0}), M{(a - b) / (a + b)}.
2061 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2062 for I{near-spherical} ellipsoids.
2063 '''
2064 t = 0 if _ispherical_a_b(a, b) else float(a - b)
2065 return Float(n=_0_0 if fabs(t) < EPS0 else _over(t, a + b))
2068def a_f2b(a, f):
2069 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}.
2071 @arg a: Equatorial radius (C{scalar} > 0).
2072 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2074 @return: The polar radius (C{float}), M{a * (1 - f)}.
2075 '''
2076 b = a if _ispherical_f(f) else (a * (_1_0 - f))
2077 return Radius_(b=a if _ispherical_a_b(a, b) else b)
2080def a_f_2b(a, f_):
2081 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}.
2083 @arg a: Equatorial radius (C{scalar} > 0).
2084 @arg f_: Inverse flattening (C{scalar} >>> 1).
2086 @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}.
2087 '''
2088 b = a if _ispherical_f_(f_) else _over(a * (f_ - _1_0), f_)
2089 return Radius_(b=a if _ispherical_a_b(a, b) else b)
2092def b_f2a(b, f):
2093 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}.
2095 @arg b: Polar radius (C{scalar} > 0).
2096 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2098 @return: The equatorial radius (C{float}), M{b / (1 - f)}.
2099 '''
2100 t = _1_0 - f
2101 a = b if fabs(t) < EPS0 else _over(b, t)
2102 return Radius_(a=b if _ispherical_a_b(a, b) else a)
2105def b_f_2a(b, f_):
2106 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}.
2108 @arg b: Polar radius (C{scalar} > 0).
2109 @arg f_: Inverse flattening (C{scalar} >>> 1).
2111 @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}.
2112 '''
2113 t = f_ - _1_0
2114 a = b if _ispherical_f_(f_) or fabs(t) < EPS0 \
2115 or fabs(t - f_) < EPS0 else _over(b * f_, t)
2116 return Radius_(a=b if _ispherical_a_b(a, b) else a)
2119def e2f(e):
2120 '''Return C{f}, the I{flattening} for a given I{1st eccentricity}.
2122 @arg e: The (1st) eccentricity (0 <= C{float} < 1)
2124 @return: The flattening (C{scalar} or C{0}).
2126 @see: Function L{e22f}.
2127 '''
2128 return e22f(e**2)
2131def e22f(e2):
2132 '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}.
2134 @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1)
2136 @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(1 - e2) + 1)}.
2137 '''
2138 return Float(f=_over(e2, sqrt(_1_0 - e2) + _1_0)) if e2 and e2 < _1_0 else _f_0_0
2141def f2e2(f):
2142 '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}.
2144 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2146 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1), M{f * (2 - f)}.
2148 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2149 for I{near-spherical} ellipsoids.
2151 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2152 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2153 <https://WikiPedia.org/wiki/Flattening>}.
2154 '''
2155 return Float(e2=_0_0 if _ispherical_f(f) else (f * (_2_0 - f)))
2158def f2e22(f):
2159 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}.
2161 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2163 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or C{INF}),
2164 M{f * (2 - f) / (1 - f)**2}.
2166 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2167 for near-spherical ellipsoids.
2169 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2170 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.
2171 '''
2172 # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2
2173 t = (_1_0 - f)**2
2174 return Float(e22=INF if t < EPS0 else _over(f2e2(f), t)) # PYCHOK type
2177def f2e32(f):
2178 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}.
2180 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2182 @return: The I{signed}, 3rd eccentricity I{squared} (C{float}),
2183 M{f * (2 - f) / (1 + (1 - f)**2)}.
2185 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2186 for I{near-spherical} ellipsoids.
2188 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2189 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.
2190 '''
2191 # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2)
2192 e2 = f2e2(f)
2193 return Float(e32=_over(e2, _2_0 - e2))
2196def f_2f(f_):
2197 '''Return C{f}, the I{flattening} for a given I{inverse flattening}.
2199 @arg f_: Inverse flattening (C{scalar} >>> 1).
2201 @return: The flattening (C{scalar} or C{0}), M{1 / f_}.
2203 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2204 for I{near-spherical} ellipsoids.
2205 '''
2206 f = 0 if _ispherical_f_(f_) else _over(_1_0, f_)
2207 return _f_0_0 if _ispherical_f(f) else Float(f=f) # PYCHOK type
2210def f2f_(f):
2211 '''Return C{f_}, the I{inverse flattening} for a given I{flattening}.
2213 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2215 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2217 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2218 for I{near-spherical} ellipsoids.
2219 '''
2220 f_ = 0 if _ispherical_f(f) else _over(_1_0, f)
2221 return _f__0_0 if _ispherical_f_(f_) else Float(f_=f_) # PYCHOK type
2224def f2f2(f):
2225 '''Return C{f2}, the I{2nd flattening} for a given I{flattening}.
2227 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2229 @return: The I{signed}, 2nd flattening (C{scalar} or C{INF}), M{f / (1 - f)}.
2231 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2232 for I{near-spherical} ellipsoids.
2234 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2235 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2236 <https://WikiPedia.org/wiki/Flattening>}.
2237 '''
2238 t = _1_0 - f
2239 return Float(f2=_0_0 if _ispherical_f(f) else
2240 (INF if fabs(t) < EPS else _over(f, t))) # PYCHOK type
2243def f2n(f):
2244 '''Return C{n}, the I{3rd flattening} for a given I{flattening}.
2246 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2248 @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1),
2249 M{f / (2 - 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 return Float(n=_0_0 if _ispherical_f(f) else _over(f, float(_2_0 - f)))
2261def n2e2(n):
2262 '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}.
2264 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2266 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF),
2267 M{4 * n / (1 + n)**2}.
2269 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2270 for I{near-spherical} ellipsoids.
2272 @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}.
2273 '''
2274 t = (n + _1_0)**2
2275 return Float(e2=_0_0 if fabs(n) < EPS0 else
2276 (NINF if t < EPS0 else _over(_4_0 * n, t)))
2279def n2f(n):
2280 '''Return C{f}, the I{flattening} for a given I{3rd flattening}.
2282 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2284 @return: The flattening (C{scalar} or NINF), M{2 * n / (1 + n)}.
2286 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2287 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2288 <https://WikiPedia.org/wiki/Flattening>}.
2289 '''
2290 t = n + _1_0
2291 f = 0 if fabs(n) < EPS0 else (NINF if t < EPS0 else _over(_2_0 * n, t))
2292 return _f_0_0 if _ispherical_f(f) else Float(f=f)
2295def n2f_(n):
2296 '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}.
2298 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2300 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2302 @see: L{n2f} and L{f2f_}.
2303 '''
2304 return f2f_(n2f(n))
2307class Ellipsoids(_NamedEnum):
2308 '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class
2309 to accommodate the L{_LazyNamedEnumItem} properties.
2310 '''
2311 def _Lazy(self, a, b, f_, **kwds):
2312 '''(INTERNAL) Instantiate the L{Ellipsoid}.
2313 '''
2314 return Ellipsoid(a, b=b, f_=f_, **kwds)
2316Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton
2317'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.'''
2318# <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html>
2319# <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS>
2320# <https://kb.OSU.edu/dspace/handle/1811/77986>
2321# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids>
2322# <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html>
2323# <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java>
2324Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid>
2325 Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909
2326 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448
2327# APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965
2328# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid
2329# AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America
2330 Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192
2331 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System"
2332 Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)),
2333 BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)),
2334# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)),
2335 CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC)
2336# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)),
2337 Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)),
2338 Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)),
2339 Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)),
2340 Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc
2341 CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures
2342 Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium
2343 Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)),
2344# Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)),
2345# Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)),
2346# Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)),
2347 Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)),
2348 Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)),
2349 Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)),
2350# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)),
2351 GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)),
2352 GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid"
2353 GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516
2354# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)),
2355 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.25722210088)), # IUGG, ITRS, ETRS89
2356# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297
2357 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)),
2358# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)),
2359 IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union
2360 IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302
2361 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy
2362 IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)),
2363 Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795
2364 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l
2365 Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2366 Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2367# Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961
2368# Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979
2369 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France
2370 Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)),
2371 Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)),
2372# MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983
2373# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab.
2374 NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab.
2375# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965
2376 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)),
2377 OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)),
2378# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect?
2379 Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972
2380# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)),
2381 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11
2382# SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia
2383 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System
2384 SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American
2385 Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo
2386 SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo
2387 SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)), # EPSG:3857 Spheroid
2388 Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)),
2389# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)),
2390# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)),
2391 WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)),
2392 WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)),
2393 WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52
2394 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # GPS b=6356752.3142451793
2395# U{NOAA/NOS/NGS/inverse<https://GitHub.com/noaa-ngs/inverse/blob/main/invers3d.f>}
2396 WGS84_NGS = _lazy('WGS84_NGS', *_T(R_MA, _0_0, 298.257222100882711243162836600094))
2397)
2399_EWGS84 = Ellipsoids.WGS84 # (INTERNAL) shared
2401if __name__ == _DMAIN_:
2403 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_
2404 from pygeodesy import nameof, printf
2406 for E in (_EWGS84, Ellipsoids.GRS80, # NAD83,
2407 Ellipsoids.Sphere, Ellipsoids.SpherePopular,
2408 Ellipsoid(_EWGS84.b, _EWGS84.a, name='_Prolate')):
2409 e = f2n(E.f) - E.n
2410 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10, terse=0), nl=1)
2411 printf('# e=%s, f_=%s, f=%s, n=%s (%s)', fstr(E.e, prec=13, fmt=Fmt.e),
2412 fstr(E.f_, prec=13, fmt=Fmt.e),
2413 fstr(E.f, prec=13, fmt=Fmt.e),
2414 fstr(E.n, prec=13, fmt=Fmt.e),
2415 fstr(e, prec=9, fmt=Fmt.e))
2416 printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20))
2417 printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20))
2418 printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property
2420 # __doc__ of this file, force all into registry
2421 t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_)
2422 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))
2424# % python3.13 -m pygeodesy.ellipsoids
2426# 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
2427# e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00)
2428# AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2429# BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2430# KsOrder 8
2432# 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
2433# e=8.1819191042833e-02, f_=2.9825722210088e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00)
2434# AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2435# BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2436# KsOrder 8
2438# 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
2439# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2440# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2441# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2442# KsOrder 8
2444# 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
2445# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2446# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2447# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2448# KsOrder 8
2450# 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
2451# e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00)
2452# AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0
2453# BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0
2454# KsOrder 8
2456# **) MIT License
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2458# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.
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2460# Permission is hereby granted, free of charge, to any person obtaining a
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