Coverage for pygeodesy/ellipsoids.py: 96%

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, e=0.08167337, b=6356256.90923729, f_=299.3249646)

15@var Ellipsoids.AiryModified: Ellipsoid(name='AiryModified', a=6377340.189, f=0.00334085, e=0.08167337, b=6356034.44793853, f_=299.3249646)

16@var Ellipsoids.ATS1977: Ellipsoid(name='ATS1977', a=6378135, f=0.00335281, e=0.08181922, b=6356750.30492159, f_=298.257)

17@var Ellipsoids.Australia1966: Ellipsoid(name='Australia1966', a=6378160, f=0.00335289, e=0.08182018, b=6356774.71919531, f_=298.25)

18@var Ellipsoids.Bessel1841: Ellipsoid(name='Bessel1841', a=6377397.155, f=0.00334277, e=0.08169683, b=6356078.962818, f_=299.1528128)

19@var Ellipsoids.BesselModified: Ellipsoid(name='BesselModified', a=6377492.018, f=0.00334277, e=0.08169683, b=6356173.5087127, f_=299.1528128)

20@var Ellipsoids.CGCS2000: Ellipsoid(name='CGCS2000', a=6378137, f=0.00335281, e=0.08181919, b=6356752.31414036, f_=298.2572221)

21@var Ellipsoids.Clarke1866: Ellipsoid(name='Clarke1866', a=6378206.4, f=0.00339008, e=0.08227185, b=6356583.8, f_=294.97869821)

22@var Ellipsoids.Clarke1880: Ellipsoid(name='Clarke1880', a=6378249.145, f=0.00340756, e=0.0824834, b=6356514.86954978, f_=293.465)

23@var Ellipsoids.Clarke1880IGN: Ellipsoid(name='Clarke1880IGN', a=6378249.2, f=0.00340755, e=0.08248326, b=6356515, f_=293.46602129)

24@var Ellipsoids.Clarke1880Mod: Ellipsoid(name='Clarke1880Mod', a=6378249.145, f=0.00340755, e=0.08248322, b=6356514.96639549, f_=293.46630766)

25@var Ellipsoids.CPM1799: Ellipsoid(name='CPM1799', a=6375738.7, f=0.00299052, e=0.07727934, b=6356671.92557493, f_=334.39)

26@var Ellipsoids.Delambre1810: Ellipsoid(name='Delambre1810', a=6376428, f=0.00321027, e=0.08006397, b=6355957.92616372, f_=311.5)

27@var Ellipsoids.Engelis1985: Ellipsoid(name='Engelis1985', a=6378136.05, f=0.00335282, e=0.08181928, b=6356751.32272154, f_=298.2566)

28@var Ellipsoids.Everest1967: Ellipsoid(name='Everest1967', a=6377298.556, f=0.00332445, e=0.08147298, b=6356097.55030068, f_=300.8017)

29@var Ellipsoids.Everest1969: Ellipsoid(name='Everest1969', a=6377295.664, f=0.00332445, e=0.08147298, b=6356094.667915, f_=300.8017)

30@var Ellipsoids.Everest1975: Ellipsoid(name='Everest1975', a=6377299.151, f=0.00332445, e=0.08147298, b=6356098.14512013, f_=300.8017255)

31@var Ellipsoids.Fisher1968: Ellipsoid(name='Fisher1968', a=6378150, f=0.00335233, e=0.08181333, b=6356768.33724438, f_=298.3)

32@var Ellipsoids.GEM10C: Ellipsoid(name='GEM10C', a=6378137, f=0.00335281, e=0.08181919, b=6356752.31424783, f_=298.2572236)

33@var Ellipsoids.GPES: Ellipsoid(name='GPES', a=6378135, f=0, e=0, b=6378135, f_=0)

34@var Ellipsoids.GRS67: Ellipsoid(name='GRS67', a=6378160, f=0.00335292, e=0.08182057, b=6356774.51609071, f_=298.24716743)

35@var Ellipsoids.GRS80: Ellipsoid(name='GRS80', a=6378137, f=0.00335281, e=0.08181919, b=6356752.31414035, f_=298.2572221)

36@var Ellipsoids.Helmert1906: Ellipsoid(name='Helmert1906', a=6378200, f=0.00335233, e=0.08181333, b=6356818.16962789, f_=298.3)

37@var Ellipsoids.IAU76: Ellipsoid(name='IAU76', a=6378140, f=0.00335281, e=0.08181922, b=6356755.28815753, f_=298.257)

38@var Ellipsoids.IERS1989: Ellipsoid(name='IERS1989', a=6378136, f=0.00335281, e=0.08181922, b=6356751.30156878, f_=298.257)

39@var Ellipsoids.IERS1992TOPEX: Ellipsoid(name='IERS1992TOPEX', a=6378136.3, f=0.00335281, e=0.08181919, b=6356751.61659215, f_=298.25722356)

40@var Ellipsoids.IERS2003: Ellipsoid(name='IERS2003', a=6378136.6, f=0.00335282, e=0.0818193, b=6356751.85797165, f_=298.25642)

41@var Ellipsoids.Intl1924: Ellipsoid(name='Intl1924', a=6378388, f=0.003367, e=0.08199189, b=6356911.94612795, f_=297)

42@var Ellipsoids.Intl1967: Ellipsoid(name='Intl1967', a=6378157.5, f=0.0033529, e=0.08182023, b=6356772.2, f_=298.24961539)

43@var Ellipsoids.Krassovski1940: Ellipsoid(name='Krassovski1940', a=6378245, f=0.00335233, e=0.08181333, b=6356863.01877305, f_=298.3)

44@var Ellipsoids.Krassowsky1940: Ellipsoid(name='Krassowsky1940', a=6378245, f=0.00335233, e=0.08181333, b=6356863.01877305, f_=298.3)

45@var Ellipsoids.Maupertuis1738: Ellipsoid(name='Maupertuis1738', a=6397300, f=0.0052356, e=0.10219488, b=6363806.28272251, f_=191)

46@var Ellipsoids.Mercury1960: Ellipsoid(name='Mercury1960', a=6378166, f=0.00335233, e=0.08181333, b=6356784.28360711, f_=298.3)

47@var Ellipsoids.Mercury1968Mod: Ellipsoid(name='Mercury1968Mod', a=6378150, f=0.00335233, e=0.08181333, b=6356768.33724438, f_=298.3)

48@var Ellipsoids.NWL1965: Ellipsoid(name='NWL1965', a=6378145, f=0.00335289, e=0.08182018, b=6356759.76948868, f_=298.25)

49@var Ellipsoids.OSU86F: Ellipsoid(name='OSU86F', a=6378136.2, f=0.00335281, e=0.08181919, b=6356751.51693008, f_=298.2572236)

50@var Ellipsoids.OSU91A: Ellipsoid(name='OSU91A', a=6378136.3, f=0.00335281, e=0.08181919, b=6356751.6165948, f_=298.2572236)

51@var Ellipsoids.Plessis1817: Ellipsoid(name='Plessis1817', a=6376523, f=0.00324002, e=0.08043347, b=6355862.93325557, f_=308.64)

52@var Ellipsoids.PZ90: Ellipsoid(name='PZ90', a=6378136, f=0.0033528, e=0.08181911, b=6356751.36174571, f_=298.2578393)

53@var Ellipsoids.SGS85: Ellipsoid(name='SGS85', a=6378136, f=0.00335281, e=0.08181922, b=6356751.30156878, f_=298.257)

54@var Ellipsoids.SoAmerican1969: Ellipsoid(name='SoAmerican1969', a=6378160, f=0.00335289, e=0.08182018, b=6356774.71919531, f_=298.25)

55@var Ellipsoids.Sphere: Ellipsoid(name='Sphere', a=6371008.771415, f=0, e=0, b=6371008.771415, f_=0)

56@var Ellipsoids.SphereAuthalic: Ellipsoid(name='SphereAuthalic', a=6371000, f=0, e=0, b=6371000, f_=0)

57@var Ellipsoids.SpherePopular: Ellipsoid(name='SpherePopular', a=6378137, f=0, e=0, b=6378137, f_=0)

58@var Ellipsoids.Struve1860: Ellipsoid(name='Struve1860', a=6378298.3, f=0.00339294, e=0.0823065, b=6356657.14266956, f_=294.73)

59@var Ellipsoids.WGS60: Ellipsoid(name='WGS60', a=6378165, f=0.00335233, e=0.08181333, b=6356783.28695944, f_=298.3)

60@var Ellipsoids.WGS66: Ellipsoid(name='WGS66', a=6378145, f=0.00335289, e=0.08182018, b=6356759.76948868, f_=298.25)

61@var Ellipsoids.WGS72: Ellipsoid(name='WGS72', a=6378135, f=0.00335278, e=0.08181881, b=6356750.52001609, f_=298.26)

62@var Ellipsoids.WGS84: Ellipsoid(name='WGS84', a=6378137, f=0.00335281, e=0.08181919, b=6356752.31424518, f_=298.25722356)

63@var Ellipsoids.WGS84_NGS: Ellipsoid(name='WGS84_NGS', a=6378137, f=0.00335281, e=0.08181919, b=6356752.31414035, f_=298.2572221)

64'''

65# make sure int/int division yields float quotient, see .basics

66from __future__ import division as _; del _ # noqa: E702 ;

68# from pygeodesy.albers import AlbersEqualAreaCylindrical # _MODS

69from pygeodesy.basics import copysign0, isbool, _isin, isint, typename

70from pygeodesy.constants import EPS, EPS_2, EPS0, EPS02, EPS1, INF, NINF, \

71 _EPSqrt, PI_2, PI_3, PI2, PI4, R_M, R_MA, R_FM, \

72 _EPStol as _TOL, _floatuple as _T, _isfinite, \

73 _over, _SQRT2, _0_0s, _0_0, _0_5, _1_0, _1_EPS, \

74 _2_0, _4_0, _90_0, _0_25, _3_0 # PYCHOK used!

75# from pygeodesy.ellipses import Ellipse # _MODS

76from pygeodesy.errors import _AssertionError, IntersectionError, _ValueError, _xattr, _xkwds_not

77from pygeodesy.fmath import cbrt, cbrt2, fdot, fmean_, Fhorner, fpowers, hypot, hypot_, \

78 hypot1, hypot2, sqrt3, Fsum

79# from pygeodesy.fsums import Fsum # from .fmath

80# from pygeodesy.internals import typename # from .basics

81from pygeodesy.interns import NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, \

82 _Clarke1866_, _Clarke1880IGN_, _DMAIN_, _DOT_, _f_, _GRS80_, \

83 _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \

84 _Krassowsky1940_, _lat_, _meridional_, _negative_, _not_finite_, \

85 _prime_vertical_, _Sphere_, _SPACE_, _vs_, _WGS72_, _WGS84_

86# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .named

87from pygeodesy.named import _lazyNamedEnumItem as _lazy, _name__, _NamedEnum, \

88 _NamedEnumItem, _NamedTuple, _Pass, _ALL_LAZY, _MODS

89from pygeodesy.namedTuples import Circle4Tuple, Distance2Tuple, Vector3Tuple, Vector4Tuple

90from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_, \

91 deprecated_property_RO, property_RO, property_ROver

92from pygeodesy.streprs import Fmt, fstr, instr, strs, unstr

93# from pygeodesy.triaxials.triaxial5 import _hartzell3, _plumbTo3 # _MODS

94from pygeodesy.units import Azimuth, Bearing, Distance, Float, Float_, Lamd, Lat, _Lat0, \

95 Meter, Meter2, Meter3, Phi, Phid, Radius, Radius_, Scalar

96from pygeodesy.utily import atan1, atan1d, atan2b, degrees90, m2radians, radians2m, sincos2d

98from math import asinh, atan, atanh, cos, degrees, exp, fabs, radians, sin, sinh, sqrt, tan # as _tan

100__all__ = _ALL_LAZY.ellipsoids

101__version__ = '26.10.18'

102

103_f_0_0 = Float(f =_0_0) # zero flattening

104_f__0_0 = Float(f_=_0_0) # zero inverse flattening

105# see U{WGS84_f<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Constants.html>}

106# _f_WGS84 = Float(f =_1_0 / (298257223563 / 1000000000)) # 0.003_352_810_664_747_480_5

107_f__WGS84 = Float(f_=_1_0 / (1000000000 / 298257223563)) # 298.257223562_999_97 vs 298.257223563

108

109

110def _aux(lat, inverse, auxLat, clip=90):

111 '''Return a named auxiliary latitude in C{degrees}.

112 '''

113 return Lat(lat, clip=clip, name=_lat_ if inverse else typename(auxLat))

114

115

116def _sin2cos2(rad):

117 '''(INTERNAL) Return 2-tuple C{(sin(B{phi})**2, cos(B{phi})**2)}.

118 '''

119 if rad:

120 s2 = sin(rad)**2

121 if s2 > EPS:

122 c2 = _1_0 - s2

123 if c2 > EPS:

124 if c2 < EPS1:

125 return s2, c2

126 else:

127 return _1_0, _0_0 # rad == PI_2

128 return _0_0, _1_0 # rad == 0

129

130

131class a_f2Tuple(_NamedTuple):

132 '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial}

133 radius C{a} in C{meter} and scalar I{flattening} C{f}.

134

135 @see: Class L{Ellipsoid2}.

136 '''

137 _Names_ = (_a_, _f_) # name 'f' NOT 'f_'

138 _Units_ = (_Pass, _Pass)

139

140 def __new__(cls, a, f, **name):

141 '''New L{a_f2Tuple} ellipsoid specification.

142

143 @arg a: Equatorial radius (C{scalar} > 0).

144 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

145 @kwarg name: Optional C{B{name}=NN} (C{str}).

146

147 @return: An L{a_f2Tuple}C{(a, f)}.

148

149 @raise UnitError: Invalid B{C{a}} or B{C{f}}.

150

151 @note: C{abs(B{f}) < }L{EPS<pygeodesy.constants.EPS>} is

152 forced to C{B{f}=0}, I{spherical}.

153

154 @note: Negative C{B{f}} produces a I{prolate} ellipsoid.

155 '''

156 a = Radius_(a=a) # low=EPS, high=None

157 f = Float_( f=f, low=None, high=EPS1)

158 if fabs(f) < EPS: # force spherical

159 f = _f_0_0

160 return _NamedTuple.__new__(cls, a, f, **name)

161

162 @Property_RO

163 def b(self):

164 '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}.

165 '''

166 return a_f2b(self.a, self.f) # PYCHOK .a and .f

167

168 def ellipsoid(self, **name):

169 '''Return an L{Ellipsoid} for this 2-tuple C{(a, f)}.

170

171 @kwarg name: Optional C{B{name}=NN} (C{str}).

172

173 @raise NameError: A registered C{ellipsoid} with the

174 same B{C{name}} already exists.

175 '''

176 return Ellipsoid(self.a, f=self.f, name=self._name__(name)) # PYCHOK .a and .f

177

178 @Property_RO

179 def f_(self):

180 '''Get the I{inverse} flattening (C{scalar}), M{1 / f} == M{a / (a - b)}.

181 '''

182 return f2f_(self.f) # PYCHOK .f

183

184

185class Curvature2Tuple(_NamedTuple):

186 '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in

187 C{meter}, conventionally.

188 '''

189 _Names_ = (_meridional_, _prime_vertical_)

190 _Units_ = ( Meter, Meter)

191

192 @property_RO

193 def transverse(self):

194 '''Get this I{prime_vertical}, aka I{transverse} radius of curvature.

195 '''

196 return self.prime_vertical

197

198

199class Ellipsoid(_NamedEnumItem):

200 '''Ellipsoid with I{equatorial} and I{polar} radii, I{flattening}, I{inverse

201 flattening} and other, often used, I{cached} attributes, supporting

202 I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models.

203 '''

204 _a = 0 # equatorial radius, semi-axis (C{meter})

205 _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f

206 _f = 0 # (1st) flattening: (a - b) / a

207 _f_ = 0 # inverse flattening: 1 / f = a / (a - b)

208

209 _geodsolve = NN # means, use PYGEODESY_GEODSOLVE

210 _KsOrder = 8 # Krüger series order (4, 6 or 8)

211 _rhumbsolve = NN # means, use PYGEODESY_RHUMBSOLVE

212

213 def __init__(self, a, b=None, f_=None, f=None, **name):

214 '''New L{Ellipsoid} from the I{equatorial} radius I{and} either

215 the I{polar} radius or I{inverse flattening} or I{flattening}.

216

217 @arg a: Equatorial radius, semi-axis (C{meter}).

218 @arg b: Optional polar radius, semi-axis (C{meter}).

219 @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0).

220 @arg f: Flattening: M{(a - b) / a} (C{scalar}, near zero for

221 spherical).

222 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).

223

224 @raise NameError: Ellipsoid with the same B{C{name}} already exists.

225

226 @raise ValueError: Invalid B{C{a}}, B{C{b}}, B{C{f_}} or B{C{f}} or

227 B{C{f_}} and B{C{f}} are incompatible.

228

229 @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced

230 to M{1 / f_ = 0}, spherical.

231 '''

232 ff_ = f, f_ # assertion below

233 n = _name__(**name) if name else NN

234 try:

235 a = Radius_(a=a) # low=EPS

236 if not _isfinite(a):

237 raise ValueError(_SPACE_(_a_, _not_finite_))

238

239 if b: # not _isin(b, None, _0_0)

240 b = Radius_(b=b) # low=EPS

241 f = a_b2f(a, b) if f is None else Float(f=f)

242 f_ = f2f_(f) if f_ is None else Float(f_=f_)

243 elif f is not None:

244 f = Float(f=f)

245 b = a_f2b(a, f)

246 f_ = f2f_(f) if f_ is None else Float(f_=f_)

247 elif f_:

248 f_ = Float(f_=f_)

249 b = a_f_2b(a, f_) # a * (f_ - 1) / f_

250 f = f_2f(f_)

251 else: # only a, spherical

252 f_ = f = 0

253 b = a # superfluous

254

255 if not (f < _1_0): # sanity check, see .ecef.Ecef.__init__

256 raise ValueError(_SPACE_(_f_, _invalid_))

257 if not _isfinite(b):

258 raise ValueError(_SPACE_(_b_, _not_finite_))

259

260 if fabs(f) < EPS or a == b or not f_: # spherical

261 b = a

262 f = _f_0_0

263 f_ = _f__0_0

264

265 except (TypeError, ValueError) as x:

266 d = _xkwds_not(None, b=b, f_=f_, f=f)

267 t = instr(self, a=a, name=n, **d)

268 raise _ValueError(t, cause=x)

269

270 self._a = a

271 self._b = b

272 self._f = f

273 self._f_ = f_

274

275 self._register(Ellipsoids, n)

276

277 if f and f_: # see test/testEllipsoidal

278 d = dict(eps=_TOL)

279 if None in ff_: # both f_ and f given

280 d.update(Error=_ValueError, txt=_incompatible_)

281 self._assert(_1_0 / f, f_=f_, **d)

282 self._assert(_1_0 / f_, f =f, **d)

283 self._assert(self.b2_a2, e21=self.e21, eps=EPS)

284

285 def __eq__(self, other):

286 '''Compare this and an other ellipsoid.

287

288 @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}).

289

290 @return: C{True} if equal, C{False} otherwise.

291 '''

292 return self is other or (isinstance(other, Ellipsoid) and

293 self.a == other.a and

294 (self.f == other.f or self.b == other.b))

295

296 def __hash__(self):

297 return self._hash # memoized

298

299 @Property_RO

300 def a(self):

301 '''Get the I{equatorial} radius, semi-axis (C{meter}).

302 '''

303 return self._a

304

305 equatoradius = a # = Requatorial

306

307 @Property_RO

308 def a2(self):

309 '''Get the I{equatorial} radius I{squared} (C{meter} I{squared}), M{a**2}.

310 '''

311 return Meter2(a2=self.a**2)

312

313 @Property_RO

314 def a2_(self):

315 '''Get the inverse of the I{equatorial} radius I{squared} (C{meter} I{squared}), M{1 / a**2}.

316 '''

317 return Float(a2_=_1_0 / self.a2)

318

319 @Property_RO

320 def A(self):

321 '''Get the UTM I{meridional (or rectifying)} radius (C{meter}).

322

323 @note: C{A * PI / 2} ≈= L{L<Ellipsoid.L>}, the I{quarter meridian}.

324

325 @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/>},

326 the mean, meridional length I{per radian}.

327 '''

328 A, n = self.a, self.n

329 if n:

330 d = (n + _1_0) * 1048576 / A

331 if d: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8

332 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>

333 # <https://GeographicLib.SourceForge.io/C++/doc/transversemercator.html> and

334 # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> (3)

335 # A *= fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441) / 1048576) / (1 + n)

336 A = Radius(A=Fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441).fover(d))

337 return A

338# # Moritz, H. <https://Geodesy.Geology.Ohio-State.edu/course/refpapers/00740128.pdf>

339# # q = 4 / self.rocPolar

340# # Q = (1 - 3 / 4 * e'2 + 45 / 64 * e'4 - 175 / 256 * e'6 + 11025 / 16384 * e'8) * rocPolar

341# # = (4 + e'2 * (-3 + e'2 * (45 / 16 + e'2 * (-175 / 64 + e'2 * 11025 / 4096)))) / q

342# return Radius(Q=Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(q))

343

344 @Property_RO

345 def a_b(self):

346 '''Get the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b} == M{1 / (1 - f)}.

347 '''

348 return Float(a_b=self.a / self.b if self.f else _1_0)

349

350 @Property_RO

351 def a2_b(self):

352 '''Get the I{polar} meridional (or polar) radius of curvature (C{meter}), M{a**2 / b}.

353

354 @see: U{Radii of Curvature

355 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}

356 and U{Moritz, H. (1980), Geodetic Reference System 1980

357 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}.

358

359 @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature

360 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{c2}

361 and L{R2} or L{Rauthalic}.

362 '''

363 return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar

364

365 @Property_RO

366 def a2_b2(self):

367 '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}),

368 M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e21}.

369 '''

370 return Float(a2_b2=self.a_b**2 if self.f else _1_0)

371

372 @Property_RO

373 def a_f(self):

374 '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}), see method C{toEllipsoid2}.

375 '''

376 return a_f2Tuple(self.a, self.f, name=self.name)

377

378 a_f2 = a_f # synonym

379

380 @Property_RO

381 def _albersCyl(self):

382 '''(INTERNAL) Helper for C{auxAuthalic}.

383 '''

384 return _MODS.albers.AlbersEqualAreaCylindrical(datum=self, name=self.name)

385

386 @Property_RO

387 def AlphaKs(self):

388 '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).

389 '''

390 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # noqa: E702 ;

391 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8

392 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200),

393 _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned

394 _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned

395 _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned

396 _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned

397 _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned

398 _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned

399 _T(1424729850961/743921418240)) # PYCHOK unaligned

400

401 @Property_RO

402 def area(self):

403 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2}.

404

405 @see: Properties L{areax}, L{c2} and L{R2} and functions

406 L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.

407 '''

408 return Meter2(area=self.c2 * PI4)

409

410 @Property_RO

411 def areax(self):

412 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more

413 accurate for very I{oblate} ellipsoids.

414

415 @see: Properties L{area}, L{c2x} and L{R2x}, class L{GeodesicExact} and

416 functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.

417 '''

418 return Meter2(areax=self.c2x * PI4)

419

420 def _assert(self, val, eps=_TOL, f0=_0_0, Error=_AssertionError, txt=NN, **name_value):

421 '''(INTERNAL) Assert a C{name=value} vs C{val}.

422 '''

423 for n, v in name_value.items():

424 if fabs(v - val) > eps: # PYCHOK no cover

425 t = (v, _vs_, val)

426 t = _SPACE_.join(strs(t, prec=12, fmt=Fmt.g))

427 t = Fmt.EQUAL(self._DOT_(n), t)

428 raise Error(t, txt=txt or Fmt.exceeds_eps(eps))

429 return Float(v if self.f else f0, name=n)

430 raise Error(unstr(self._DOT_(typename(self._assert)), val,

431 eps=eps, f0=f0, **name_value))

432

433 def auxAuthalic(self, lat, inverse=False):

434 '''Compute the I{authalic} auxiliary latitude (Xi) or the I{inverse} thereof.

435

436 @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}).

437 @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and

438 return the geodetic latitude (C{bool}).

439

440 @return: The I{authalic} (or geodetic) latitude in C{degrees90}.

441

442 @see: U{Inverse-/AuthalicLatitude<https://GeographicLib.SourceForge.io/

443 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude

444 <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and

445 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16.

446 '''

447 if self.f:

448 _f = self._albersCyl._tanf if inverse else \

449 self._albersCyl._txif # PYCHOK attr

450 lat = atan1d(_f(tan(Phid(lat)))) # PYCHOK attr

451 return _aux(lat, inverse, Ellipsoid.auxAuthalic)

452

453 def auxConformal(self, lat, inverse=False):

454 '''Compute the I{conformal} auxiliary latitude (Chi) or the I{inverse} thereof.

455

456 @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}).

457 @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and

458 return the geodetic latitude (C{bool}).

459

460 @return: The I{conformal} (or geodetic) latitude in C{degrees90}.

461

462 @see: U{Inverse-/ConformalLatitude<https://GeographicLib.SourceForge.io/

463 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude

464 <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and

465 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.

466 '''

467 if self.f:

468 _f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr

469 lat = atan1d(_f(tan(Phid(lat)))) # PYCHOK attr

470 return _aux(lat, inverse, Ellipsoid.auxConformal)

471

472 def auxGeocentric(self, lat, inverse=False, height=_0_0):

473 '''Compute the I{geocentric} auxiliary latitude (Theta) or the I{inverse} thereof.

474

475 @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}).

476 @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and

477 return the I{geocentric} latitude (C{bool}).

478 @kwarg height: Optional, ellipsoidal height (C{meter}).

479

480 @return: The I{geocentric} (or geodetic) latitude in C{degrees90}.

481

482 @see: U{Inverse-/GeocentricLatitude<https://GeographicLib.SourceForge.io/

483 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude

484 <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and

485 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18.

486 '''

487 if self.f: # and lat

488 t = tan(Phid(lat))

489 f = self.b2_a2

490 if height:

491 if inverse:

492 lat = atan1d(t * f) # geodetic n

493 d, f = f, _1_0

494 else:

495 d = _1_0

496 n = self.rocPrimeVertical(lat)

497 f = _over(n * f + height, n * d + height)

498 elif inverse:

499 f = self.a2_b2

500 lat = atan1d(t * f)

501 return _aux(lat, inverse, Ellipsoid.auxGeocentric)

502

503 def auxIsometric(self, lat, inverse=False):

504 '''Compute the I{isometric} auxiliary latitude (Psi) or the I{inverse} thereof.

505

506 @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}).

507 @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and

508 return the geodetic latitude (C{bool}).

509

510 @return: The I{isometric} (or geodetic) latitude in C{degrees}.

511

512 @note: The I{isometric} latitude for geodetic C{+/-90} is far

513 outside the C{[-90..+90]} range but the inverse thereof

514 is the original geodetic latitude.

515

516 @see: U{Inverse-/IsometricLatitude<https://GeographicLib.SourceForge.io/

517 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude

518 <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and

519 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.

520 '''

521 if self.f:

522 r = Phid(lat, clip=0)

523 lat = degrees(atan1(self.es_tauf(sinh(r))) if inverse else

524 asinh(self.es_taupf(tan(r))))

525 # clip=0, since auxIsometric(+/-90) is far outside [-90..+90]

526 return _aux(lat, inverse, Ellipsoid.auxIsometric, clip=0)

527

528 def auxParametric(self, lat, inverse=False):

529 '''Compute the I{parametric} auxiliary latitude (Beta) or the I{inverse} thereof.

530

531 @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}).

532 @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and

533 return the geodetic latitude (C{bool}).

534

535 @return: The I{parametric} (or geodetic) latitude in C{degrees90}.

536

537 @see: U{Inverse-/ParametricLatitude<https://GeographicLib.SourceForge.io/

538 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude

539 <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>},

540 and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18.

541 '''

542 if self.f:

543 lat = self._Betad(Lat(lat), inverse=inverse)

544 return _aux(lat, inverse, Ellipsoid.auxParametric)

545

546 auxReduced = auxParametric # synonymous

547

548 def auxRectifying(self, lat, inverse=False):

549 '''Compute the I{rectifying} auxiliary latitude (Mu) or the I{inverse} thereof.

550

551 @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}).

552 @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and

553 return the geodetic latitude (C{bool}).

554

555 @return: The I{rectifying} (or geodetic) latitude in C{degrees90}.

556

557 @see: U{Inverse-/RectifyingLatitude<https://GeographicLib.SourceForge.io/

558 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude

559 <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and

560 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17.

561 '''

562 if self.f:

563 lat = Lat(lat)

564 if 0 < fabs(lat) < _90_0:

565 if inverse:

566 e = self._elliptic_e22

567 d = degrees90(e.fEinv(e.cE * lat / _90_0))

568 lat = self.auxParametric(d, inverse=True)

569 else:

570 lat = _over(self.Llat(lat), self.L) * _90_0

571 return _aux(lat, inverse, Ellipsoid.auxRectifying)

572

573 @Property_RO

574 def b(self):

575 '''Get the I{polar} radius, semi-axis (C{meter}).

576 '''

577 return self._b

578

579 polaradius = b # = Rpolar

580

581 @Property_RO

582 def b_a(self):

583 '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a == f1 == 1 - f}.

584

585 @see: Property L{f1}.

586 '''

587 return self._assert(self.b / self.a, b_a=self.f1, f0=_1_0)

588

589 @Property_RO

590 def b2(self):

591 '''Get the I{polar} radius I{squared} (C{float}), M{b**2}.

592 '''

593 return Meter2(b2=self.b**2)

594

595 @Property_RO

596 def b2_a(self):

597 '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see C{rocMeridional}C{(0)}.

598

599 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.

600 '''

601 return Radius(b2_a=_over(self.b2, self.a) if self.f else self.b)

602

603 @Property_RO

604 def b2_a2(self):

605 '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2}

606 == M{(1 - f)**2} == M{1 - e**2} == C{e21}.

607 '''

608 return Float(b2_a2=self.b_a**2 if self.f else _1_0)

609

610 def _Betad(self, lat, inverse=False):

611 '''(INTERNAL) Get the I{parametric (or reduced) auxiliary latitude} or inverse thereof.

612 '''

613 s, c = sincos2d(lat) # like Karney's tand(lat)

614 s *= self.a_b if inverse else self.b_a

615 return atan1d(s, c) # (s * a, c * b) if inverse else (s * b, c * a)

616

617 @Property_RO

618 def BetaKs(self):

619 '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).

620 '''

621 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # noqa: E702 ;

622 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8

623 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600),

624 _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned

625 _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned

626 _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned

627 _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned

628 _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne

629 _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned

630 _T(191773887257/3719607091200)) # PYCHOK unaligned

631

632 @deprecated_Property_RO

633 def c(self): # PYCHOK no cover

634 '''DEPRECATED, use property C{R2} or C{Rauthalic}.'''

635 return self.R2

636

637 @Property_RO

638 def c2(self):

639 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}).

640

641 @see: Properties L{c2x}, L{area}, L{R2}, L{Rauthalic}, I{Karney's} U{equation (60)

642 <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area

643 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>},

644 U{Authalic radius<https://WikiPedia.org/wiki/Earth_radius#Authalic_radius>}, U{Surface area

645 <https://WikiPedia.org/wiki/Ellipsoid>} and U{surface area

646 <https://www.Numericana.com/answer/geometry.htm#oblate>}.

647 '''

648 return self._c2f(False)

649

650 @Property_RO

651 def c2x(self):

652 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}), more accurate for very I{oblate}

653 ellipsoids.

654

655 @see: Properties L{c2}, L{areax}, L{R2x}, L{Rauthalicx}, class L{GeodesicExact} and I{Karney}'s comments at C++

656 attribute U{GeodesicExact._c2<https://GeographicLib.SourceForge.io/C++/doc/GeodesicExact_8cpp_source.html>}.

657 '''

658 return self._c2f(True)

659

660 def _c2f(self, c2x):

661 '''(INTERNAL) Helper for C{.c2} and C{.c2x}.

662 '''

663 f, c2 = self.f, self.b2

664 if f:

665 e = self.e

666 if e > EPS0:

667 if f > 0: # .isOblate

668 c2 *= (asinh(sqrt(self.e22abs)) if c2x else atanh(e)) / e

669 elif f < 0: # .isProlate

670 c2 *= atan1(e) / e # XXX asin?

671 c2 = Meter2(c2=(self.a2 + c2) * _0_5)

672 return c2

673

674 def circle4(self, lat):

675 '''Get the equatorial or a parallel I{circle of latitude}.

676

677 @arg lat: Geodetic latitude (C{degrees90} or C{str}).

678

679 @return: A L{Circle4Tuple}C{(radius, height, lat, beta)}.

680

681 @raise RangeError: Latitude B{C{lat}} outside valid range, only if

682 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

683

684 @raise TypeError: Invalid B{C{lat}}.

685

686 @raise ValueError: Invalid B{C{lat}}.

687

688 @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude}

689 <https://WikiPedia.org/wiki/Latitude>}, I{Karney's} C++ U{CircleRadius and CircleHeight

690 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>}

691 and method C{Triaxial.ellipse5}.

692 '''

693 lat = _Lat0(lat)

694 if lat:

695 B = lat # beta

696 if fabs(lat) < _90_0:

697 if self.f:

698 B = self._Betad(lat)

699 z, r = sincos2d(B)

700 r *= self.a

701 z *= self.b

702 else: # near-polar

703 r, z = _0_0, copysign0(self.b, lat)

704 else: # equatorial

705 r = self.a

706 z = lat = B = _0_0

707 return Circle4Tuple(r, z, lat, B)

708

709 def degrees2m(self, deg, lat=0):

710 '''Convert an arc in degrees to a distance along the equator or

711 along a parallel of (geodetic) latitude.

712

713 @arg deg: The angle (C{degrees}).

714 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).

715

716 @return: Distance (C{meter}, same units as the equatorial

717 and polar radii) or C{0} for near-polar B{C{lat}}.

718

719 @raise RangeError: Latitude B{C{lat}} outside valid range and

720 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

721

722 @raise ValueError: Invalid B{C{deg}} or B{C{lat}}.

723 '''

724 return self.radians2m(radians(deg), lat=lat)

725

726 def distance2(self, lat0, lon0, lat1, lon1):

727 '''I{Approximate} the distance and (initial) bearing between

728 two points based on the U{local, flat earth approximation

729 <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny

730 <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.

731

732 I{Suitable only for distances of several hundred Km or Miles

733 and only between points not near-polar}.

734

735 @arg lat0: From latitude (C{degrees}).

736 @arg lon0: From longitude (C{degrees}).

737 @arg lat1: To latitude (C{degrees}).

738 @arg lon1: To longitude (C{degrees}).

739

740 @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance}

741 in same units as this ellipsoid's axes.

742

743 @note: The meridional and prime_vertical radii of curvature are

744 taken and scaled I{at the initial latitude}, see C{roc2}.

745

746 @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}.

747 '''

748 phi0 = Phid(lat0=lat0)

749 m, n = self.roc2_(phi0, scaled=True)

750 m *= Phid(lat1=lat1) - phi0

751 n *= Lamd(lon1=lon1) - Lamd(lon0=lon0)

752 return Distance2Tuple(hypot(m, n), atan2b(n, m))

753

754 @Property_RO

755 def e(self):

756 '''Get the I{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}.

757

758 @see: Property L{es}.

759 '''

760 return Float(e=sqrt(self.e2abs) if self.e2 else _0_0)

761

762 @deprecated_Property_RO

763 def e12(self): # see property ._e12

764 '''DEPRECATED, use property C{e21}.'''

765 return self.e21

766

767# @Property_RO

768# def _e12(self): # see property ._elliptic_e12

769# # (INTERNAL) until e12 above can be replaced with e21.

770# return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2)

771

772 @Property_RO

773 def e2(self):

774 '''Get the I{signed, (1st) eccentricity squared} (C{float}), M{f * (2 - f)

775 == 1 - (b / a)**2}, see C{a_b2e2}.

776 '''

777 return self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f))

778

779 @Property_RO

780 def e2abs(self):

781 '''Get the I{unsigned, (1st) eccentricity squared} (C{float}).

782 '''

783 return fabs(self.e2)

784

785 @Property_RO

786 def e2not(self):

787 '''Get I{self.e**2 and NOT .e2 NOR .e2abs} (C{float}).

788 '''

789 return self.e**2

790

791 @Property_RO

792 def e21(self):

793 '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2}

794 == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}.

795 '''

796 return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0)

797

798# _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2

799 _1_e21 = a2_b2 # == M{1 / e21} == M{1 / (1 - e**2)}

800

801 @Property_RO

802 def e22(self):

803 '''Get the I{signed, 2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)

804 == e2 / (1 - f)**2 == (a / b)**2 - 1}, see C{a_b2e22}.

805 '''

806 return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f))

807

808 @Property_RO

809 def e22abs(self):

810 '''Get the I{unsigned, 2nd eccentricity squared} (C{float}).

811 '''

812 return fabs(self.e22)

813

814 @Property_RO

815 def e32(self):

816 '''Get the I{signed, 3rd eccentricity squared} (C{float}), M{e2 / (2 - e2)

817 == (a**2 - b**2) / (a**2 + b**2)}, see C{a_b2e32}.

818 '''

819 return self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f))

820

821 @Property_RO

822 def e32abs(self):

823 '''Get the I{unsigned, 3rd eccentricity squared} (C{float}).

824 '''

825 return fabs(self.e32)

826

827 @Property_RO

828 def e4(self):

829 '''Get the I{unsignd, (1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}.

830 '''

831 return Float(e4=self.e2**2 if self.e2 else _0_0)

832

833 eccentricity = e # eccentricity

834# eccentricity2 = e2 # eccentricity squared

835 eccentricity1st2 = e2 # first eccentricity squared, signed

836 eccentricity2nd2 = e22 # second eccentricity squared, signed

837 eccentricity3rd2 = e32 # third eccentricity squared, signed

838

839 def ecef(self, Ecef=None):

840 '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter.

841

842 @kwarg Ecef: ECEF class to use, default L{EcefKarney}.

843

844 @return: An ECEF converter for this C{ellipsoid}.

845

846 @raise TypeError: Invalid B{C{Ecef}}.

847

848 @see: Module L{pygeodesy.ecef}.

849 '''

850 return _MODS.ecef._4Ecef(self, Ecef)

851

852 @Property_RO

853 def _elliptic_e12(self): # see I{Karney}'s Ellipsoid._e12

854 '''(INTERNAL) Elliptic helper for C{Rhumb}.

855 '''

856 e12 = _over(self.e2, self.e2 - _1_0) # NOT DEPRECATED .e12!

857 return _MODS.elliptic.Elliptic(e12)

858

859 @Property_RO

860 def _elliptic_e22(self): # aka ._elliptic_ep2

861 '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}.

862 '''

863 # courtesy gaoflow <https://GitHub.com/mrJean1/PyGeodesy/pull/86>

864 return _MODS.elliptic.Elliptic(-self.e22) # signed, like ._elliptic_e12

865

866 equatoradius = a # Requatorial

867

868 @Property_RO

869 def equatorimeter(self):

870 '''Get the ellipsoid's I{equatorial} perimeter (C{meter}).

871 '''

872 return Meter(equatorimeter=self.a * PI2)

873

874 def e2s(self, s):

875 '''Compute norm M{sqrt(1 - e2 * s**2)}.

876

877 @arg s: Sine value (C{scalar}).

878

879 @return: Norm (C{float}).

880

881 @raise ValueError: Invalid B{C{s}}.

882 '''

883 return sqrt(self.e2s2(s)) if self.e2 else _1_0

884

885 def e2s2(self, s):

886 '''Compute M{1 - e2 * s**2}.

887

888 @arg s: Sine value (C{scalar}).

889

890 @return: Result (C{float}).

891

892 @raise ValueError: Invalid B{C{s}}.

893 '''

894 r, e2 = _1_0, self.e2

895 if e2: # and s

896 try:

897 r -= e2 * Scalar(s=s)**2

898 if r < 0:

899 raise ValueError(_negative_)

900 except (TypeError, ValueError) as x:

901 t = self._DOT_(typename(Ellipsoid.e2s2))

902 raise _ValueError(t, s, cause=x)

903 return r

904

905 @Property_RO

906 def es(self):

907 '''Get the I{signed (1st) eccentricity} (C{float}).

908

909 @see: Property L{e}.

910 '''

911 # note, self.e is always non-negative

912 return Float(es=copysign0(self.e, self.f)) # see .ups

913

914 def es_atanh(self, x):

915 '''Compute M{es * atanh(es * x)} or M{-es * atan(es * x)}

916 for I{oblate} respectively I{prolate} ellipsoids where

917 I{es} is the I{signed} (1st) eccentricity.

918

919 @raise ValueError: Invalid B{C{x}}.

920

921 @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/

922 C++/doc/classGeographicLib_1_1Math.html>}.

923 '''

924 return self._es_atanh(Scalar(x=x)) if self.f else _0_0

925

926 def _es_atanh(self, x): # see .albers._atanhee, .AuxLat._atanhee

927 '''(INTERNAL) Helper for .es_atanh, ._es_taupf2 and ._exp_es_atanh.

928 '''

929 es = self.es # signOf(es) == signOf(f)

930 return es * (atanh(es * x) if es > 0 else # .isOblate

931 (-atan(es * x) if es < 0 else # .isProlate

932 _0_0)) # .isSpherical

933

934 @Property_RO

935 def es_c(self):

936 '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}.

937 '''

938 return Float(es_c=(self._exp_es_atanh_1 * self.b_a) if self.f else _1_0)

939

940 def es_tauf(self, taup):

941 '''Compute I{Karney}'s U{equations (19), (20) and (21)

942 <https://ArXiv.org/abs/1002.1417>}.

943

944 @see: I{Karney}'s C++ method U{Math::tauf<https://GeographicLib.

945 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>} and

946 and I{Veness}' JavaScript method U{toLatLon<https://www.

947 Movable-Type.co.UK/scripts/latlong-utm-mgrs.html>}.

948 '''

949 t = Scalar(taup=taup)

950 if self.f: # .isEllipsoidal

951 a = fabs(t)

952 T = (self._exp_es_atanh_1 if a > 70 else self._1_e21) * t

953 if fabs(T * _EPSqrt) < _2_0: # handles +/- INF and NAN

954 s = (a * _TOL) if a > _1_0 else _TOL

955 for T, _, d in self._es_tauf3(t, T): # max 2

956 if fabs(d) < s:

957 break

958 t = Scalar(tauf=T)

959 return t

960

961 def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB

962 '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most

963 B{C{N}} Newton iterations, converging rapidly except when C{delta}

964 toggles on +/-1.12e-16 or +/-4.47e-16, see C{.utm.Utm._toLLEB}.

965 '''

966 e = self._1_e21

967 _F2_ = Fsum(T).fsum2f_ # τ0

968 _tf2 = self._es_taupf2

969 for i in range(1, N + 1):

970 a, h = _tf2(T)

971 # = (taup - a) / hypot1(a) / ((e + T**2) / h)

972 d = _over((taup - a) * (T**2 + e), hypot1(a) * h)

973 T, d = _F2_(d) # τi, (τi - τi-1)

974 yield T, i, d

975

976 def es_taupf(self, tau):

977 '''Compute I{Karney}'s U{equations (7), (8) and (9)

978 <https://ArXiv.org/abs/1002.1417>}.

979

980 @see: I{Karney}'s C++ method U{Math::taupf<https://GeographicLib.

981 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}.

982 '''

983 t = Scalar(tau=tau)

984 if self.f: # .isEllipsoidal

985 t, _ = self._es_taupf2(t)

986 t = Scalar(taupf=t)

987 return t

988

989 def _es_taupf2(self, tau):

990 '''(INTERNAL) Return 2-tuple C{(es_taupf(tau), hypot1(tau))}.

991 '''

992 if _isfinite(tau):

993 h = hypot1(tau)

994 s = sinh(self._es_atanh(tau / h))

995 a = hypot1(s) * tau - h * s

996 else:

997 a, h = tau, INF

998 return a, h

999

1000 @Property_RO

1001 def _exp_es_atanh_1(self):

1002 '''(INTERNAL) Helper for .es_c and .es_tauf.

1003 '''

1004 return exp(self._es_atanh(_1_0)) if self.es else _1_0

1005

1006 @Property_RO

1007 def f(self):

1008 '''Get the I{flattening} (C{scalar}), M{(a - b) / a}, C{0} for spherical, negative for prolate.

1009 '''

1010 return self._f

1011

1012 @Property_RO

1013 def f_(self):

1014 '''Get the I{inverse flattening} (C{scalar}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}.

1015 '''

1016 return self._f_

1017

1018 @Property_RO

1019 def f1(self):

1020 '''Get the I{1 - flattening} (C{float}), M{f1 == 1 - f == b / a}.

1021

1022 @see: Property L{b_a}.

1023 '''

1024 return Float(f1=_1_0 - self.f)

1025

1026 @Property_RO

1027 def f2(self):

1028 '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}.

1029 '''

1030 return self._assert(self.a_b - _1_0, f2=f2f2(self.f))

1031

1032 @deprecated_Property_RO

1033 def geodesic(self):

1034 '''DEPRECATED, use property C{geodesicw}.'''

1035 return self.geodesicw

1036

1037 def geodesic_(self, exact=True):

1038 '''Get the an I{exact} C{Geodesic...} instance for this ellipsoid.

1039

1040 @kwarg exact: If C{bool} return L{GeodesicExact}C{(exact=B{exact}, ...)},

1041 otherwise a L{Geodesic}, L{GeodesicExact} or L{GeodesicSolve}

1042 instance for I{this} ellipsoid.

1043

1044 @return: The C{exact} geodesic (C{Geodesic...}).

1045

1046 @raise TypeError: Invalid B{C{exact}}.

1047

1048 @raise ValueError: Incompatible B{C{exact}} ellipsoid.

1049 '''

1050 if isbool(exact): # for consistenccy with C{.rhumb_}

1051 g = _MODS.geodesicx.GeodesicExact(self, C4order=30 if exact else 24,

1052 name=self.name)

1053 else:

1054 g = exact

1055 E = _xattr(g, ellipsoid=None)

1056 if not (E is self and isinstance(g, self._Geodesics)):

1057 raise _ValueError(exact=g, ellipsoid=E, txt_not_=self.name)

1058 return g

1059

1060 @property_ROver

1061 def _Geodesics(self):

1062 '''(INTERNAL) Get all C{Geodesic...} classes, I{once}.

1063 '''

1064 t = (_MODS.geodesicx.GeodesicExact,

1065 _MODS.geodsolve.GeodesicSolve)

1066 try:

1067 t += (_MODS.geodesicw.Geodesic,

1068 _MODS.geodesicw._wrapped.Geodesic)

1069 except ImportError:

1070 pass

1071 return t # overwrite property_ROver

1072

1073 @property_RO

1074 def geodesicw(self):

1075 '''Get this ellipsoid's I{wrapped} U{geodesicw.Geodesic

1076 <https://GeographicLib.SourceForge.io/Python/doc/code.html>}, provided

1077 I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}

1078 package is installed.

1079 '''

1080 # if not self.isEllipsoidal:

1081 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1082 return _MODS.geodesicw.Geodesic(self)

1083

1084 @property_RO

1085 def geodesicx(self):

1086 '''Get this ellipsoid's I{exact} L{GeodesicExact}.

1087 '''

1088 # if not self.isEllipsoidal:

1089 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1090 return _MODS.geodesicx.GeodesicExact(self, name=self.name)

1091

1092 @property

1093 def geodsolve(self):

1094 '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility

1095 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},

1096 provided the path to the C{GeodSolve} executable is specified with env

1097 variable C{PYGEODESY_GEODSOLVE} or re-/set with this property..

1098 '''

1099 # if not self.isEllipsoidal:

1100 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1101 return _MODS.geodsolve.GeodesicSolve(self, path=self._geodsolve, name=self.name)

1102

1103 @geodsolve.setter # PYCHOK setter!

1104 def geodsolve(self, path):

1105 '''Re-/set the (fully qualified) path to the U{GeodSolve

1106 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,

1107 overriding env variable C{PYGEODESY_GEODSOLVE} (C{str}).

1108 '''

1109 self._geodsolve = path

1110

1111 def hartzell4(self, pov, los=None):

1112 '''Compute the intersection of this ellipsoid's surface and a Line-Of-Sight

1113 from a Point-Of-View in space.

1114

1115 @arg pov: Point-Of-View outside this ellipsoid (C{Cartesian}, L{Ecef9Tuple}

1116 or L{Vector3d}).

1117 @kwarg los: Line-Of-Sight, I{direction} to this ellipsoid (L{Los}, L{Vector3d})

1118 or C{True} for the I{normal, perpendicular, plumb} to the surface

1119 of this ellipsoid or C{False} or C{None} to point to its center.

1120

1121 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},

1122 C{y} and C{z} of the projection on or the intersection with this

1123 ellipsoid and the I{distance} C{h} from B{C{pov}} to C{(x, y, z)}

1124 along B{C{los}}, all in C{meter}, conventionally.

1125

1126 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, or B{C{pov}}

1127 is inside this ellipsoid or B{C{los}} points

1128 outside this ellipsoid or in opposite direction.

1129

1130 @raise TypeError: Invalid B{C{pov}} or B{C{los}}.

1131

1132 @see: U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.

1133 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and

1134 methods L{Ellipsoid.height4} and L{Triaxial.hartzell4}.

1135 '''

1136 try:

1137 m = _MODS._triaxials_triaxial5

1138 v, d, i = m._hartzell3(pov, los, self._triaxial) # PYLINT E1101

1139 except Exception as x:

1140 raise IntersectionError(pov=pov, los=los, cause=x)

1141 return Vector4Tuple(v.x, v.y, v.z, d, iteration=i, name__=self.hartzell4)

1142

1143 @Property_RO

1144 def _hash(self):

1145 return hash((self.a, self.f))

1146

1147 def height4(self, xyz, normal=True):

1148 '''Compute the projection on and the height of a cartesian above or below

1149 this ellipsoid's surface.

1150

1151 @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d},

1152 L{Vector3Tuple} or L{Vector4Tuple}).

1153 @kwarg normal: If C{True}, the projection is perpendicular to (the nearest

1154 point on) this ellipsoid's surface, otherwise the C{radial}

1155 line to this ellipsoid's center (C{bool}).

1156

1157 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},

1158 C{y} and C{z} of the projection on and the height C{h} above or

1159 below this ellipsoid's surface, all in C{meter}, conventionally.

1160

1161 @raise ValueError: Null B{C{xyz}}.

1162

1163 @raise TypeError: Non-cartesian B{C{xyz}}.

1164

1165 @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>}

1166 and U{intersection with<https://MathWorld.Wolfram.com/Ellipse-LineIntersection.html>} an ellipse and

1167 methods L{Ellipsoid.hartzell4} and L{Triaxial.height4}.

1168 '''

1169 v = _MODS.vector3d._otherV3d(xyz=xyz)

1170 r = v.length

1171

1172 a, b, i = self.a, self.b, None

1173 if r < EPS0: # EPS

1174 v = v.times(_0_0)

1175 h = -a

1176

1177 elif self.isSpherical:

1178 v = v.times(a / r)

1179 h = r - a

1180

1181 elif normal: # perpendicular to ellipsoid

1182 x, y = hypot(v.x, v.y), fabs(v.z)

1183 if x < EPS0: # PYCHOK no cover

1184 z = copysign0(b, v.z)

1185 v = Vector3Tuple(v.x, v.y, z)

1186 h = y - b # polar

1187 elif y < EPS0: # PYCHOK no cover

1188 t = a / r

1189 v = v.times_(t, t, 0) # force z=0.0

1190 h = x - a # equatorial

1191 else: # normal in 1st quadrant

1192 m = _MODS._triaxials_triaxial5

1193 x, y, i = m._plumbTo3(x, y, self) # PYLINT E1101

1194 t, v = v, v.times_(x, x, y)

1195 h = t.minus(v).length

1196

1197 else: # radial to ellipsoid's center

1198 h = hypot_(a * v.z, b * v.x, b * v.y)

1199 t = (a * b / h) if h > EPS0 else _0_0 # EPS

1200 v = v.times(t)

1201 h = r * (_1_0 - t)

1202

1203 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name__=self.height4)

1204

1205 @Property_RO

1206 def heightMax(self):

1207 '''Get the height limit (C{meter}, conventionally).

1208 '''

1209 return self.a / EPS_2 # self.a * _2_EPS, about 12M lightyears

1210

1211 def _hubeny_2(self, phi2, phi1, lam21, scaled=True, squared=True):

1212 '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_},

1213 returning the I{angular} distance in C{radians squared} or C{radians}

1214 '''

1215 m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=scaled)

1216 h, r = (hypot2, self.a2_) if squared else (hypot, _1_0 / self.a)

1217 return h(m * (phi2 - phi1), n * lam21) * r

1218

1219 @Property_RO

1220 def isEllipsoidal(self):

1221 '''Is this model I{ellipsoidal} (C{bool})?

1222 '''

1223 return self.f != 0

1224

1225 @Property_RO

1226 def isOblate(self):

1227 '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or

1228 spherical otherwise.

1229 '''

1230 return self.f > 0

1231

1232 @Property_RO

1233 def isProlate(self):

1234 '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or

1235 spherical otherwise.

1236 '''

1237 return self.f < 0

1238

1239 @Property_RO

1240 def isSpherical(self):

1241 '''Is this ellipsoid I{spherical} (C{bool})?

1242 '''

1243 return self.f == 0

1244

1245 def _Kseries(self, *AB8Ks):

1246 '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha

1247 or Beta series coefficients per I{Karney}'s U{equations (35)

1248 and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}.

1249

1250 @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series

1251 coefficient tuples.

1252

1253 @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}).

1254

1255 @see: I{Karney}'s 30-th order U{TMseries30

1256 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}.

1257 '''

1258 k = self.KsOrder

1259 if self.n:

1260 ns = fpowers(self.n, k)

1261 ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k))

1262 else:

1263 ks = _0_0s(k)

1264 return ks

1265

1266 @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''')

1267 def KsOrder(self):

1268 '''Get the I{Krüger} series' order (C{int} 4, 6 or 8).

1269 '''

1270 return self._KsOrder

1271

1272 @KsOrder.setter # PYCHOK setter!

1273 def KsOrder(self, order):

1274 '''Set the I{Krüger} series' order (C{int} 4, 6 or 8).

1275

1276 @raise ValueError: Invalid B{C{order}}.

1277 '''

1278 if not (isint(order) and _isin(order, 4, 6, 8)):

1279 raise _ValueError(order=order)

1280 if self._KsOrder != order:

1281 Ellipsoid.AlphaKs._update(self)

1282 Ellipsoid.BetaKs._update(self)

1283 self._KsOrder = order

1284

1285 @Property_RO

1286 def L(self):

1287 '''Get the I{quarter meridian} C{L}, aka the C{polar distance}

1288 along a meridian between the equator and a pole (C{meter}),

1289 M{b * Elliptic(-e2 / (1 - e2)).cE} or M{b * PI / 2}.

1290 '''

1291 r = self._elliptic_e22.cE if self.f else PI_2

1292 return Distance(L=self.b * r)

1293

1294 def Llat(self, lat):

1295 '''Return the I{meridional length}, the distance along a meridian

1296 between the equator and a (geodetic) latitude, see C{L}.

1297

1298 @arg lat: Geodetic latitude (C{degrees90}).

1299

1300 @return: The meridional length at B{C{lat}}, negative on southern

1301 hemisphere (C{meter}).

1302 '''

1303 r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phid(lat)

1304 return Distance(Llat=self.b * r)

1305

1306 Lmeridian = Llat # meridional distance

1307

1308 @property_RO

1309 def _Lpd(self):

1310 '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}.

1311 '''

1312 return Meter(_Lpd=self.L / _90_0)

1313

1314 @property_RO

1315 def _Lpr(self):

1316 '''Get the I{quarter meridian} per radian (C{meter}), M{self.L / PI_2}.

1317 '''

1318 return Meter(_Lpr=self.L / PI_2)

1319

1320 @deprecated_Property_RO

1321 def majoradius(self): # PYCHOK no cover

1322 '''DEPRECATED, use property C{a} or C{Requatorial}.'''

1323 return self.a

1324

1325 def m2degrees(self, distance, lat=0):

1326 '''Convert a distance to an arc in degrees along the equator or

1327 along a parallel of (geodetic) latitude.

1328

1329 @arg distance: Distance (C{meter}).

1330 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).

1331

1332 @return: Angle (C{degrees}) or C{0} for near-polar B{C{lat}}.

1333

1334 @raise RangeError: Latitude B{C{lat}} outside valid range and

1335 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1336

1337 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.

1338 '''

1339 return degrees(self.m2radians(distance, lat=lat))

1340

1341 def m2radians(self, distance, lat=0):

1342 '''Convert a distance to an angle along the equator or along

1343 a parallel of (geodetic) latitude.

1344

1345 @arg distance: Distance (C{meter}).

1346 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).

1347

1348 @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}.

1349

1350 @raise RangeError: Latitude B{C{lat}} outside valid range and

1351 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1352

1353 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.

1354 '''

1355 r = self.circle4(lat).radius if lat else self.a

1356 return m2radians(distance, radius=r, lat=0)

1357

1358 @deprecated_Property_RO

1359 def minoradius(self): # PYCHOK no cover

1360 '''DEPRECATED, use property C{b}, C{polaradius} or C{Rpolar}.'''

1361 return self.b

1362

1363 @Property_RO

1364 def n(self):

1365 '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}.

1366 '''

1367 return self._assert(a_b2n(self.a, self.b), n=f2n(self.f))

1368

1369 flattening = f

1370 flattening1st = f

1371 flattening2nd = f2

1372 flattening3rd = n

1373

1374 polaradius = b # Rpolar

1375

1376 @property_RO

1377 def polarimeter(self):

1378 '''Get the ellipsoid's I{polar}, meridional perimeter (C{meter}).

1379 '''

1380 return Meter(polarimeter=self.L * _4_0)

1381

1382# Q = A # I{meridian arc unit} C{Q}, the mean, meridional length I{per radian}

1383

1384 @deprecated_Property_RO

1385 def quarteradius(self): # PYCHOK no cover

1386 '''DEPRECATED, use property C{L} or method C{Llat}.'''

1387 return self.L

1388

1389 @Property_RO

1390 def R1(self):

1391 '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}.

1392

1393 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}

1394 and method C{Rgeometric}.

1395 '''

1396 return Radius(R1=fmean_(self.a, self.a, self.b) if self.f else self.a)

1397

1398 Rmean = R1

1399

1400 @Property_RO

1401 def R2(self):

1402 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}.

1403

1404 @see: C{R2x}, C{c2}, C{area} and U{Earth radius

1405 <https://WikiPedia.org/wiki/Earth_radius>}.

1406 '''

1407 return Radius(R2=sqrt(self.c2) if self.f else self.a)

1408# # Moritz, H. <https://Geodesy.Geology.Ohio-State.edu/course/refpapers/00740128.pdf>

1409# # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar

1410# # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3

1411# return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar)

1412

1413 Rauthalic = R2

1414

1415 @Property_RO

1416 def R2x(self):

1417 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}.

1418

1419 @see: C{R2}, C{c2x} and C{areax}.

1420 '''

1421 return Radius(R2x=sqrt(self.c2x) if self.f else self.a)

1422

1423 Rauthalicx = R2x

1424

1425 @Property_RO

1426 def R3(self):

1427 '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}.

1428

1429 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}.

1430 '''

1431 r = (cbrt(self.b_a) * self.a) if self.f else self.a

1432 return Radius(R3=r)

1433

1434 Rvolumetric = R3

1435

1436 def radians2m(self, rad, lat=0):

1437 '''Convert an angle to the distance along the equator or along

1438 a parallel of (geodetic) latitude.

1439

1440 @arg rad: The angle (C{radians}).

1441 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).

1442

1443 @return: Distance (C{meter}, same units as the equatorial

1444 and polar radii) or C{0} for near-polar B{C{lat}}.

1445

1446 @raise RangeError: Latitude B{C{lat}} outside valid range and

1447 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1448

1449 @raise ValueError: Invalid B{C{rad}} or B{C{lat}}.

1450 '''

1451 r = self.circle4(lat).radius if lat else self.a

1452 return radians2m(rad, radius=r, lat=0)

1453

1454 @Property_RO

1455 def Rbiaxial(self):

1456 '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}.

1457

1458 @see: C{Rtriaxial}

1459 '''

1460 a, b = self.a, self.b

1461 if b != a:

1462 b = hypot(a, b) / _SQRT2

1463 return Radius(Rbiaxial=b)

1464

1465 Requatorial = a # for consistent naming

1466

1467 def Rgeocentric(self, lat):

1468 '''Compute the I{geocentric} earth radius of (geodetic) latitude.

1469

1470 @arg lat: Latitude (C{degrees90}).

1471

1472 @return: Geocentric earth radius (C{meter}).

1473

1474 @raise ValueError: Invalid B{C{lat}}.

1475

1476 @see: U{Geocentric Radius

1477 <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>}

1478 '''

1479 r, p = self.a, Phid(lat)

1480 if p and self.f:

1481 if fabs(p) < PI_2:

1482 s2, c2 = _sin2cos2(p)

1483 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2))

1484 # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2)))

1485 # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2))

1486 # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2))

1487 s2 *= self.b2_a2

1488 r *= sqrt((c2 + self.b2_a2 * s2) / (c2 + s2))

1489 else:

1490 r = self.b

1491 return Radius(Rgeocentric=r)

1492

1493 @Property_RO

1494 def Rgeometric(self):

1495 '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}.

1496

1497 @see: C{R1}.

1498 '''

1499 g = sqrt(self.a * self.b) if self.f else self.a

1500 return Radius(Rgeometric=g)

1501

1502 def rhumb_(self, exact=True):

1503 '''Get the an I{exact} C{Rhumb...} instance for this ellipsoid.

1504

1505 @kwarg exact: If C{bool} or C{None} return L{Rhumb}C{(exact=B{exact}, ...)},

1506 otherwise a L{Rhumb}, L{RhumbAux} or L{RhumbSolve} instance

1507 for I{this} ellipsoid.

1508

1509 @return: The C{exact} rhumb (C{Rhumb...}).

1510

1511 @raise TypeError: Invalid B{C{exact}}.

1512

1513 @raise ValueError: Incompatible B{C{exact}} ellipsoid.

1514 '''

1515 if isbool(exact): # use Rhumb for backward compatibility

1516 r = _MODS.rhumb.ekx.Rhumb(self, exact=exact, name=self.name)

1517 else:

1518 r = exact

1519 E = _xattr(r, ellipsoid=None)

1520 if not (E is self and isinstance(r, self._Rhumbs)):

1521 raise _ValueError(exact=r, ellipsosid=E, txt_not_=self.name)

1522 return r

1523

1524 @property_RO

1525 def rhumbaux(self):

1526 '''Get this ellipsoid's I{Auxiliary} C{rhumb.RhumbAux}.

1527 '''

1528 # if not self.isEllipsoidal:

1529 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1530 return _MODS.rhumb.aux_.RhumbAux(self, name=self.name)

1531

1532 @property_RO

1533 def rhumbekx(self):

1534 '''Get this ellipsoid's I{Elliptic, Krüger} C{rhumb.Rhumb}.

1535 '''

1536 # if not self.isEllipsoidal:

1537 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1538 return _MODS.rhumb.ekx.Rhumb(self, name=self.name)

1539

1540 @property_ROver

1541 def _Rhumbs(self):

1542 '''(INTERNAL) Get all C{Rhumb...} classes, I{once}.

1543 '''

1544 r = _MODS.rhumb

1545 return (r.aux_.RhumbAux, # overwrite property_ROver

1546 r.ekx.Rhumb, r.solve.RhumbSolve)

1547

1548 @property

1549 def rhumbsolve(self):

1550 '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility

1551 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},

1552 provided the path to the C{RhumbSolve} executable is specified with env

1553 variable C{PYGEODESY_RHUMBSOLVE} or re-/set with this property.

1554 '''

1555 # if not self.isEllipsoidal:

1556 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1557 return _MODS.rhumb.solve.RhumbSolve(self, path=self._rhumbsolve, name=self.name)

1558

1559 @rhumbsolve.setter # PYCHOK setter!

1560 def rhumbsolve(self, path):

1561 '''Re-/set the (fully qualified) path to the U{RhumbSolve

1562 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,

1563 overriding env variable C{PYGEODESY_RHUMBSOLVE} (C{str}).

1564 '''

1565 self._rhumbsolve = path

1566

1567 @deprecated_property_RO

1568 def rhumbx(self):

1569 '''DEPRECATED on 2023.11.28, use property C{rhumbekx}.'''

1570 return self.rhumbekx

1571

1572 def Rlat(self, lat):

1573 '''I{Average} the earth radius between C{equatoradius} at C{0} and

1574 C{polaradius} at C{+/-90} degrees latitude.

1575

1576 @arg lat: Latitude (C{degrees90}, C{str} or C{Ang}).

1577

1578 @return: Averaged earth radius (C{meter}).

1579

1580 @raise RangeError: Latitude B{C{lat}} outside valid range, only if

1581 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1582

1583 @raise TypeError: Invalid B{C{lat}}.

1584

1585 @raise ValueError: Invalid B{C{lat}}.

1586 '''

1587 # r = a - (a - b) * |lat| / 90

1588 r = self.a

1589 if self.f: # .isEllipsoidal

1590 lat = _Lat0(lat)

1591 if lat:

1592 r -= (r - self.b) * fabs(lat) / _90_0

1593 r = Radius(Rlat=r)

1594 return r

1595

1596 Rpolar = b # for consistent naming

1597

1598 def roc1_(self, sa, ca=None):

1599 '''Compute the I{prime-vertical}, I{normal} radius of curvature

1600 of (geodetic) latitude, I{unscaled}.

1601

1602 @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]).

1603 @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0])

1604 to use an alternate formula.

1605

1606 @return: The prime-vertical radius of curvature (C{float}).

1607

1608 @note: The delta between both formulae with C{Ellipsoids.WGS84}

1609 is less than 2 nanometer over the entire latitude range.

1610

1611 @see: Method L{roc2_} and class L{EcefYou}.

1612 '''

1613 if sa and self.f: # .isEllipsoidal

1614 if ca is None:

1615 r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward

1616 n = sqrt(self.a2 / r) if r > EPS02 else _0_0

1617 elif ca: # derived from EcefYou.forward

1618 h = hypot(ca, self.b_a * sa)

1619 n = self.a / h

1620 else:

1621 n = self.a2_b / fabs(sa)

1622 else:

1623 n = self.a

1624 return n

1625

1626 def roc2(self, lat, scaled=False):

1627 '''Compute the I{meridional} and I{prime-vertical}, I{normal}

1628 radii of curvature of (geodetic) latitude.

1629

1630 @arg lat: Latitude (C{degrees90}).

1631 @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}).

1632

1633 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with

1634 the radii of curvature.

1635

1636 @raise ValueError: Invalid B{C{lat}}.

1637

1638 @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation

1639 <https://www.EdWilliams.org/avform.htm#flat>} and meridional and

1640 prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/

1641 Earth_radius#Radii_of_curvature>}.

1642 '''

1643 return self.roc2_(Phid(lat), scaled=scaled)

1644

1645 def roc2_(self, phi, scaled=False):

1646 '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of

1647 curvature of (geodetic) latitude.

1648

1649 @arg phi: Latitude (C{radians}).

1650 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}).

1651

1652 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the

1653 radii of curvature.

1654

1655 @raise ValueError: Invalid B{C{phi}}.

1656

1657 @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local,

1658 flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>}

1659 and the meridional and prime vertical U{Radii of Curvature

1660 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.

1661 '''

1662 p = fabs(Phi(phi))

1663 if self.f:

1664 r = self.e2s2(sin(p))

1665 if r > EPS02:

1666 n = sqrt(self.a2 / r)

1667 m = n * self.e21 / r

1668 else:

1669 m = n = _0_0

1670 else:

1671 m = n = self.a

1672 if scaled and p:

1673 n *= cos(p) if p < PI_2 else _0_0

1674 return Curvature2Tuple(m, n)

1675

1676 def rocAzimuth(self, lat, azimuth):

1677 '''Compute the I{directional} radius of curvature of (geodetic) latitude

1678 and C{azimuth} compass direction.

1679

1680 @see: Method L{rocBearing<Ellipsoid.rocBearing>} for details, using C{azimuth} for C{bearing}.

1681 '''

1682 return Radius(rocAzimuth=self._rocDirectional(lat, Azimuth(azimuth)))

1683

1684 def rocBearing(self, lat, bearing):

1685 '''Compute the I{directional} radius of curvature of (geodetic) latitude

1686 and C{bearing} compass direction.

1687

1688 @arg lat: Latitude (C{degrees90}).

1689 @arg bearing: Direction (compass C{degrees360}).

1690

1691 @return: Directional radius of curvature (C{meter}).

1692

1693 @raise RangeError: Latitude B{C{lat}} outside valid range and

1694 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1695

1696 @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}.

1697

1698 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}

1699 '''

1700 return Radius(rocBearing=self._rocDirectional(lat, Bearing(bearing)))

1701

1702 def _rocDirectional(self, lat, deg):

1703 '''(INTERNAL) Helper for C{rocAzimuth} and C{rocBearing}.

1704 '''

1705 if self.f:

1706 s2, c2 = _sin2cos2(radians(deg))

1707 m, n = self.roc2_(Phid(lat))

1708 if n < m: # == n / (c2 * n / m + s2)

1709 c2 *= n / m

1710 elif m < n: # == m / (c2 + s2 * m / n)

1711 s2 *= m / n

1712 n = m

1713 r = _over(n, c2 + s2) # == 1 / (c2 / m + s2 / n)

1714 else:

1715 r = self.b # == self.a

1716 return r

1717

1718 @Property_RO

1719 def rocEquatorial2(self):

1720 '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature

1721 at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}.

1722

1723 @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a},

1724 L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature

1725 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.

1726 '''

1727 m = self.b2_a if self.f else self.a

1728 return Curvature2Tuple(m, self.a)

1729

1730 def rocGauss(self, lat):

1731 '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude.

1732

1733 @arg lat: Latitude (C{degrees90}).

1734

1735 @return: Gaussian radius of curvature (C{meter}).

1736

1737 @raise ValueError: Invalid B{C{lat}}.

1738

1739 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/

1740 Earth_radius#Radii_of_curvature>}

1741 '''

1742 # using ...

1743 # m, n = self.roc2_(Phid(lat))

1744 # return sqrt(m * n)

1745 # ... requires 1 or 2 sqrt

1746 g = self.b

1747 if self.f:

1748 s2, c2 = _sin2cos2(Phid(lat))

1749 g = _over(g, c2 + self.b2_a2 * s2)

1750 return Radius(rocGauss=g)

1751

1752 def rocMean(self, lat):

1753 '''Compute the I{mean} radius of curvature of (geodetic) latitude.

1754

1755 @arg lat: Latitude (C{degrees90}).

1756

1757 @return: Mean radius of curvature (C{meter}).

1758

1759 @raise ValueError: Invalid B{C{lat}}.

1760

1761 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/

1762 Earth_radius#Radii_of_curvature>}

1763 '''

1764 if self.f:

1765 m, n = self.roc2_(Phid(lat))

1766 m *= _over(n * _2_0, m + n) # == 2 / (1 / m + 1 / n)

1767 else:

1768 m = self.a

1769 return Radius(rocMean=m)

1770

1771 def rocMeridional(self, lat):

1772 '''Compute the I{meridional} radius of curvature of (geodetic) latitude.

1773

1774 @arg lat: Latitude (C{degrees90}).

1775

1776 @return: Meridional radius of curvature (C{meter}).

1777

1778 @raise ValueError: Invalid B{C{lat}}.

1779

1780 @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation

1781 <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of

1782 Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.

1783 '''

1784 r = self.roc2_(Phid(lat)) if lat else self.rocEquatorial2

1785 return Radius(rocMeridional=r.meridional)

1786

1787 rocPolar = a2_b # synonymous

1788

1789 def rocPrimeVertical(self, lat):

1790 '''Compute the I{prime-vertical}, I{normal} radius of curvature of

1791 (geodetic) latitude, aka the I{transverse} radius of curvature.

1792

1793 @arg lat: Latitude (C{degrees90}).

1794

1795 @return: Prime-vertical radius of curvature (C{meter}).

1796

1797 @raise ValueError: Invalid B{C{lat}}.

1798

1799 @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth

1800 approximation<https://www.EdWilliams.org/avform.htm#flat>} and

1801 U{Radii of Curvature<https://WikiPedia.org/wiki/

1802 Earth_radius#Radii_of_curvature>}.

1803 '''

1804 r = self.roc1_(sin(Phid(lat))) if lat else self.a

1805 return Radius(rocPrimeVertical=r)

1806

1807 rocTransverse = rocPrimeVertical # synonymous

1808

1809 @deprecated_Property_RO

1810 def Rquadratic(self): # PYCHOK no cover

1811 '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.'''

1812 return self.Rbiaxial

1813

1814 @deprecated_Property_RO

1815 def Rr(self): # PYCHOK no cover

1816 '''DEPRECATED, use property C{Rrectifying}.'''

1817 return self.Rrectifying

1818

1819 @Property_RO

1820 def Rrectifying(self):

1821 '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}.

1822

1823 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}.

1824 '''

1825 r = self.a

1826 if self.f:

1827 r *= cbrt2((sqrt3(self.b_a) + _1_0) * _0_5)

1828 return Radius(Rrectifying=r)

1829

1830 @deprecated_Property_RO

1831 def Rs(self): # PYCHOK no cover

1832 '''DEPRECATED, use property C{Rgeometric}.'''

1833 return self.Rgeometric

1834

1835 @Property_RO

1836 def Rtriaxial(self):

1837 '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}.

1838

1839 @see: C{Rbiaxial}

1840 '''

1841 q, b = self.a, self.b

1842 if b < q:

1843 q *= sqrt((self.b2_a2 + _3_0) * _0_25)

1844 elif b > q:

1845 q = sqrt((self.a2_b2 * _3_0 + _1_0) * _0_25) * b

1846 return Radius(Rtriaxial=q)

1847

1848 def toEllipse(self, **name):

1849 '''Get this ellipsoid's meridional ellipse L{Ellipse<pygeodesy.Ellipse>}.

1850

1851 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).

1852 '''

1853 return _MODS.ellipses.Ellipse(self.a, self.b, **name)

1854

1855 def toEllipsoid2(self, **name):

1856 '''Get a copy of this ellipsoid as an L{Ellipsoid2}.

1857

1858 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).

1859

1860 @see: Property C{a_f}.

1861 '''

1862 return Ellipsoid2(self, None, **name)

1863

1864 def toStr(self, prec=8, terse=5, **sep_name): # PYCHOK expected

1865 '''Return this ellipsoid as a text string.

1866

1867 @kwarg prec: Number of decimal digits, unstripped (C{int}).

1868 @kwarg terse: Limit the number of items (C{int}, 0...18),

1869 use C{B{terse}=0} or C{=None} for all.

1870 @kwarg sep_name: Optional C{B{name}=NN} (C{str}) or C{None}

1871 to exclude this ellipsoid's name and separator

1872 C{B{sep}=", "} to join the items (C{str}).

1873

1874 @return: This C{Ellipsoid}'s attributes (C{str}).

1875 '''

1876 E = Ellipsoid

1877 t = (E.a, E.f, E.e, E.b, E.f_, E.e2,

1878 E.e21, E.e22, E.e32, E.n,

1879 E.A, E.L, E.R1, E.R2, E.R3,

1880 E.Rbiaxial, E.Rtriaxial)

1881 if terse:

1882 t = t[:terse]

1883 return self._instr(prec=prec, props=t, **sep_name)

1884

1885 def toTriaxial(self, **name):

1886 '''Convert this ellipsoid to a L{Triaxial_}.

1887

1888 @kwarg name: Optional C{B{name}=NN} (C{str}).

1889

1890 @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis

1891 pointing east and C{Z} pointing north.

1892

1893 @see: Method L{Triaxial_.toEllipsoid}.

1894 '''

1895 T = self._triaxial

1896 return T.copy(**name) if name else T

1897

1898 @property_RO

1899 def _triaxial(self):

1900 '''(INTERNAL) Get this ellipsoid's un-/ordered C{Triaxial/_}.

1901 '''

1902 a, b, t = self.a, self.b, _MODS.triaxials

1903 T = t.Triaxial if a > b else t.Triaxial_

1904 return T(a, a, b, name=self.name)

1905

1906 @Property_RO

1907 def volume(self):

1908 '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}.

1909

1910 @see: C{R3}.

1911 '''

1912 return Meter3(volume=self.a2 * self.b * PI_3 * _4_0)

1913

1914

1915class Ellipsoid2(Ellipsoid):

1916 '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}.

1917 '''

1918 def __init__(self, a, f=None, **name):

1919 '''New L{Ellipsoid2}.

1920

1921 @arg a: Equatorial radius, semi-axis (C{meter}) or a previous

1922 L{Ellipsoid} instance.

1923 @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}),

1924 if B{C{a}} is in C{meter}.

1925 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).

1926

1927 @raise NameError: Ellipsoid with that B{C{name}} already exists.

1928

1929 @raise ValueError: Invalid B{C{a}} or B{C{f}}.

1930

1931 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.

1932 Negative C{B{f}} produces a I{prolate} ellipsoid.

1933 '''

1934 if f is None and isinstance(a, Ellipsoid):

1935 Ellipsoid.__init__(self, a.a, f =a.f,

1936 b=a.b, f_=a.f_, **name)

1937 else:

1938 Ellipsoid.__init__(self, a, f=f, **name)

1939

1940

1941def _ispherical_a_b(a, b):

1942 '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}.

1943 '''

1944 return a < EPS0 or b < EPS0 or fabs(a - b) < EPS0

1945

1946

1947def _ispherical_f(f):

1948 '''(INTERNAL) C{True} for spherical or invalid C{f}.

1949 '''

1950 return f > EPS1 or fabs(f) < EPS

1951

1952

1953def _ispherical_f_(f_):

1954 '''(INTERNAL) C{True} for spherical or invalid C{f_}.

1955 '''

1956 f_ = fabs(f_)

1957 return f_ < EPS or f_ > _1_EPS

1958

1959

1960def a_b2e(a, b):

1961 '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius.

1962

1963 @arg a: Equatorial radius (C{scalar} > 0).

1964 @arg b: Polar radius (C{scalar} > 0).

1965

1966 @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}), M{sqrt(1 - (b / a)**2)}.

1967

1968 @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids.

1969 '''

1970 e2 = _a2b2e2(a, b, b2=False)

1971 return Float(e=sqrt(fabs(e2)) if e2 else _0_0) # == sqrt(fabs((a - b) * (a + b))) / a

1972

1973

1974def a_b2e2(a, b):

1975 '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius.

1976

1977 @arg a: Equatorial radius (C{scalar} > 0).

1978 @arg b: Polar radius (C{scalar} > 0).

1979

1980 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}), M{1 - (b / a)**2}.

1981

1982 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

1983 for I{near-spherical} ellipsoids.

1984 '''

1985 return Float(e2=_a2b2e2(a, b, b2=False))

1986

1987

1988def a_b2e22(a, b):

1989 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius.

1990

1991 @arg a: Equatorial radius (C{scalar} > 0).

1992 @arg b: Polar radius (C{scalar} > 0).

1993

1994 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}), M{(a / b)**2 - 1}.

1995

1996 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

1997 for I{near-spherical} ellipsoids.

1998 '''

1999 return Float(e22=_a2b2e2(a, b, a2=False))

2000

2001

2002def a_b2e32(a, b):

2003 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius.

2004

2005 @arg a: Equatorial radius (C{scalar} > 0).

2006 @arg b: Polar radius (C{scalar} > 0).

2007

2008 @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}),

2009 M{(a**2 - b**2) / (a**2 + b**2)}.

2010

2011 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2012 for I{near-spherical} ellipsoids.

2013 '''

2014 return Float(e32=_a2b2e2(a, b))

2015

2016

2017def _a2b2e2(a, b, a2=True, b2=True):

2018 '''(INTERNAL) Helper for C{a_b2e}, C{a_b2e2}, C{a_b2e22} and C{a_b2e32}.

2019 '''

2020 if _ispherical_a_b(a, b):

2021 e2 = _0_0

2022 else: # a > 0, b > 0

2023 a, b = (_1_0, b / a) if a > b else (a / b, _1_0)

2024 a2b2 = float(a - b) * (a + b)

2025 e2 = _over(a2b2, (a**2 if a2 else _0_0) +

2026 (b**2 if b2 else _0_0)) if a2b2 else _0_0

2027 return e2

2028

2029

2030def a_b2f(a, b):

2031 '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius.

2032

2033 @arg a: Equatorial radius (C{scalar} > 0).

2034 @arg b: Polar radius (C{scalar} > 0).

2035

2036 @return: The flattening (C{scalar} or C{0}), M{(a - b) / a}.

2037

2038 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2039 for I{near-spherical} ellipsoids.

2040 '''

2041 f = 0 if _ispherical_a_b(a, b) else _over(float(a - b), a)

2042 return _f_0_0 if _ispherical_f(f) else Float(f=f)

2043

2044

2045def a_b2f_(a, b):

2046 '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius.

2047

2048 @arg a: Equatorial radius (C{scalar} > 0).

2049 @arg b: Polar radius (C{scalar} > 0).

2050

2051 @return: The inverse flattening (C{scalar} or C{0}), M{a / (a - b)}.

2052

2053 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2054 for I{near-spherical} ellipsoids.

2055 '''

2056 f_ = 0 if _ispherical_a_b(a, b) else _over(a, float(a - b))

2057 return _f__0_0 if _ispherical_f_(f_) else Float(f_=f_)

2058

2059

2060def a_b2f2(a, b):

2061 '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius.

2062

2063 @arg a: Equatorial radius (C{scalar} > 0).

2064 @arg b: Polar radius (C{scalar} > 0).

2065

2066 @return: The I{signed}, 2nd flattening (C{scalar} or C{0}), M{(a - b) / b}.

2067

2068 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2069 for I{near-spherical} ellipsoids.

2070 '''

2071 t = 0 if _ispherical_a_b(a, b) else float(a - b)

2072 return Float(f2=_0_0 if fabs(t) < EPS0 else _over(t, b))

2073

2074

2075def a_b2n(a, b):

2076 '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius.

2077

2078 @arg a: Equatorial radius (C{scalar} > 0).

2079 @arg b: Polar radius (C{scalar} > 0).

2080

2081 @return: The I{signed}, 3rd flattening (C{scalar} or C{0}), M{(a - b) / (a + b)}.

2082

2083 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2084 for I{near-spherical} ellipsoids.

2085 '''

2086 t = 0 if _ispherical_a_b(a, b) else float(a - b)

2087 return Float(n=_0_0 if fabs(t) < EPS0 else _over(t, a + b))

2088

2089

2090def a_f2b(a, f):

2091 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}.

2092

2093 @arg a: Equatorial radius (C{scalar} > 0).

2094 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2095

2096 @return: The polar radius (C{float}), M{a * (1 - f)}.

2097 '''

2098 b = a if _ispherical_f(f) else (a * (_1_0 - f))

2099 return Radius_(b=a if _ispherical_a_b(a, b) else b)

2100

2101

2102def a_f_2b(a, f_):

2103 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}.

2104

2105 @arg a: Equatorial radius (C{scalar} > 0).

2106 @arg f_: Inverse flattening (C{scalar} >>> 1).

2107

2108 @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}.

2109 '''

2110 b = a if _ispherical_f_(f_) else _over(a * (f_ - _1_0), f_)

2111 return Radius_(b=a if _ispherical_a_b(a, b) else b)

2112

2113

2114def b_f2a(b, f):

2115 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}.

2116

2117 @arg b: Polar radius (C{scalar} > 0).

2118 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2119

2120 @return: The equatorial radius (C{float}), M{b / (1 - f)}.

2121 '''

2122 t = _1_0 - f

2123 a = b if fabs(t) < EPS0 else _over(b, t)

2124 return Radius_(a=b if _ispherical_a_b(a, b) else a)

2125

2126

2127def b_f_2a(b, f_):

2128 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}.

2129

2130 @arg b: Polar radius (C{scalar} > 0).

2131 @arg f_: Inverse flattening (C{scalar} >>> 1).

2132

2133 @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}.

2134 '''

2135 t = f_ - _1_0

2136 a = b if _ispherical_f_(f_) or fabs(t) < EPS0 \

2137 or fabs(t - f_) < EPS0 else _over(b * f_, t)

2138 return Radius_(a=b if _ispherical_a_b(a, b) else a)

2139

2140

2141def e2f(e):

2142 '''Return C{f}, the I{flattening} for a given I{1st eccentricity}.

2143

2144 @arg e: The (1st) eccentricity (0 <= C{float} < 1)

2145

2146 @return: The flattening (C{scalar} or C{0}).

2147

2148 @see: Function L{e22f}.

2149 '''

2150 return e22f(e**2)

2151

2152

2153def e22f(e2):

2154 '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}.

2155

2156 @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1)

2157

2158 @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(1 - e2) + 1)}.

2159 '''

2160 return Float(f=_over(e2, sqrt(_1_0 - e2) + _1_0)) if e2 and e2 < _1_0 else _f_0_0

2161

2162

2163def f2e2(f):

2164 '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}.

2165

2166 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2167

2168 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1), M{f * (2 - f)}.

2169

2170 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2171 for I{near-spherical} ellipsoids.

2172

2173 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2174 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening

2175 <https://WikiPedia.org/wiki/Flattening>}.

2176 '''

2177 return Float(e2=_0_0 if _ispherical_f(f) else (f * (_2_0 - f)))

2178

2179

2180def f2e22(f):

2181 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}.

2182

2183 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2184

2185 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or C{INF}),

2186 M{f * (2 - f) / (1 - f)**2}.

2187

2188 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2189 for near-spherical ellipsoids.

2190

2191 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2192 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.

2193 '''

2194 # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2

2195 t = (_1_0 - f)**2

2196 return Float(e22=INF if t < EPS0 else _over(f2e2(f), t)) # PYCHOK type

2197

2198

2199def f2e32(f):

2200 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}.

2201

2202 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2203

2204 @return: The I{signed}, 3rd eccentricity I{squared} (C{float}),

2205 M{f * (2 - f) / (1 + (1 - f)**2)}.

2206

2207 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2208 for I{near-spherical} ellipsoids.

2209

2210 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2211 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.

2212 '''

2213 # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2)

2214 e2 = f2e2(f)

2215 return Float(e32=_over(e2, _2_0 - e2))

2216

2217

2218def f_2f(f_):

2219 '''Return C{f}, the I{flattening} for a given I{inverse flattening}.

2220

2221 @arg f_: Inverse flattening (C{scalar} >>> 1).

2222

2223 @return: The flattening (C{scalar} or C{0}), M{1 / f_}.

2224

2225 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2226 for I{near-spherical} ellipsoids.

2227 '''

2228 f = 0 if _ispherical_f_(f_) else _over(_1_0, f_)

2229 return _f_0_0 if _ispherical_f(f) else Float(f=f) # PYCHOK type

2230

2231

2232def f2f_(f):

2233 '''Return C{f_}, the I{inverse flattening} for a given I{flattening}.

2234

2235 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2236

2237 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.

2238

2239 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2240 for I{near-spherical} ellipsoids.

2241 '''

2242 f_ = 0 if _ispherical_f(f) else _over(_1_0, f)

2243 return _f__0_0 if _ispherical_f_(f_) else Float(f_=f_) # PYCHOK type

2244

2245

2246def f2f2(f):

2247 '''Return C{f2}, the I{2nd flattening} for a given I{flattening}.

2248

2249 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2250

2251 @return: The I{signed}, 2nd flattening (C{scalar} or C{INF}), M{f / (1 - f)}.

2252

2253 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2254 for I{near-spherical} ellipsoids.

2255

2256 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2257 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening

2258 <https://WikiPedia.org/wiki/Flattening>}.

2259 '''

2260 t = _1_0 - f

2261 return Float(f2=_0_0 if _ispherical_f(f) else

2262 (INF if fabs(t) < EPS else _over(f, t))) # PYCHOK type

2263

2264

2265def f2n(f):

2266 '''Return C{n}, the I{3rd flattening} for a given I{flattening}.

2267

2268 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2269

2270 @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1),

2271 M{f / (2 - f)}.

2272

2273 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2274 for I{near-spherical} ellipsoids.

2275

2276 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2277 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening

2278 <https://WikiPedia.org/wiki/Flattening>}.

2279 '''

2280 return Float(n=_0_0 if _ispherical_f(f) else _over(f, float(_2_0 - f)))

2281

2282

2283def n2e2(n):

2284 '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}.

2285

2286 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).

2287

2288 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF),

2289 M{4 * n / (1 + n)**2}.

2290

2291 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2292 for I{near-spherical} ellipsoids.

2293

2294 @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}.

2295 '''

2296 t = (n + _1_0)**2

2297 return Float(e2=_0_0 if fabs(n) < EPS0 else

2298 (NINF if t < EPS0 else _over(_4_0 * n, t)))

2299

2300

2301def n2f(n):

2302 '''Return C{f}, the I{flattening} for a given I{3rd flattening}.

2303

2304 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).

2305

2306 @return: The flattening (C{scalar} or NINF), M{2 * n / (1 + n)}.

2307

2308 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2309 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening

2310 <https://WikiPedia.org/wiki/Flattening>}.

2311 '''

2312 t = n + _1_0

2313 f = 0 if fabs(n) < EPS0 else (NINF if t < EPS0 else _over(_2_0 * n, t))

2314 return _f_0_0 if _ispherical_f(f) else Float(f=f)

2315

2316

2317def n2f_(n):

2318 '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}.

2319

2320 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).

2321

2322 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.

2323

2324 @see: L{n2f} and L{f2f_}.

2325 '''

2326 return f2f_(n2f(n))

2327

2328

2329class Ellipsoids(_NamedEnum):

2330 '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class

2331 to accommodate the L{_LazyNamedEnumItem} properties.

2332 '''

2333 def _Lazy(self, a, b, f_, **kwds):

2334 '''(INTERNAL) Instantiate the L{Ellipsoid}.

2335 '''

2336 return Ellipsoid(a, b=b, f_=f_, **kwds)

2337

2338Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton

2339'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.'''

2340# <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html>

2341# <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS>

2342# <https://kb.OSU.edu/dspace/handle/1811/77986>

2343# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids>

2344# <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html>

2345# <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java>

2346Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid>

2347 Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909

2348 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448

2349# APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965

2350# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid

2351# AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America

2352 Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192

2353 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System"

2354 Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)),

2355 BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)),

2356# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)),

2357 CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC)

2358# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)),

2359 Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)),

2360 Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)),

2361 Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)),

2362 Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc

2363 CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures

2364 Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium

2365 Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)),

2366# Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)),

2367# Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)),

2368# Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)),

2369 Everest1967 = _lazy('Everest1967', *_T(6377298.556, _0_0, 300.801699997)),

2370 Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)),

2371 Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)),

2372 Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)),

2373# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)),

2374 GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)),

2375 GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid"

2376 GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516

2377# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)),

2378 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.25722210088)), # IUGG, ITRS, ETRS89

2379# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297

2380 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)),

2381# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)),

2382 IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union

2383 IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302

2384 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy

2385 IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)),

2386 Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795

2387 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l

2388 Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling

2389 Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling

2390# Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961

2391# Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979

2392 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France

2393 Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)),

2394 Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)),

2395# MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983

2396# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab.

2397 NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab.

2398# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965

2399 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)),

2400 OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)),

2401# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect?

2402 Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972

2403# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)),

2404 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11

2405# SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia

2406 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System

2407 SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American

2408 Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo

2409 SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo

2410 SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)), # EPSG:3857 Spheroid

2411 Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)),

2412# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)),

2413# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)),

2414 WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)),

2415 WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)),

2416 WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52

2417 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # b=6356752.3142451793

2418# U{NOAA/NOS/NGS/inverse<https://GitHub.com/noaa-ngs/inverse/blob/main/invers3d.f>}

2419 WGS84_NGS = _lazy('WGS84_NGS', *_T(R_MA, _0_0, 298.257222100882711243162836600094))

2420)

2421

2422_EWGS84 = Ellipsoids.WGS84 # (INTERNAL) shared

2423

2424if __name__ == _DMAIN_:

2425

2426 from pygeodesy import nameof, printf

2427

2428 def _e13(f):

2429 return fstr(f, prec=13, fmt=Fmt.e)

2430

2431 for E in (_EWGS84, Ellipsoids.GRS80, # NAD83,

2432 Ellipsoids.Sphere, Ellipsoids.SpherePopular,

2433 Ellipsoid(_EWGS84.b, _EWGS84.a, name='_Prolate')):

2434 e = f2n(E.f) - E.n

2435 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10, terse=0), nl=1)

2436 printf('# (e-fmt) f_=%s, f=%s, e=%s, e2=%s, n=%s (%s)', _e13(E.f_), _e13(E.f),

2437 _e13(E.e), _e13(E.e2),

2438 _e13(E.n), fstr(e, prec=9, fmt=Fmt.e))

2439 printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20))

2440 printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20))

2441 printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property

2442

2443 from pygeodesy.internals import _pregistry

2444 # __doc__ of this file, force all into registry

2445 _pregistry(Ellipsoids)

2446

2447# % python3.13 -m pygeodesy.ellipsoids

2448

2449# Ellipsoids.WGS84: name='WGS84', a=6378137, f=0.0033528107, e=0.0818191908, b=6356752.3142451793, f_=298.257223563, e2=0.00669438, e21=0.99330562, e22=0.0067394967, e32=0.0033584313, n=0.0016792204, A=6367449.1458234144, L=10001965.7293127235, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.7900091587, Rbiaxial=6367453.6345163295, Rtriaxial=6372797.5559594007

2450# (e-fmt) f_=2.98257223563e+02, f=3.3528106647475e-03, e=8.1819190842622e-02, e2=6.6943799901413e-03, n=1.6792203863837e-03 (0.0e+00)

2451# AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0

2452# BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0

2453# KsOrder 8

2454

2455# Ellipsoids.GRS80: name='GRS80', a=6378137, f=0.0033528107, e=0.081819191, b=6356752.3141403468, f_=298.2572221009, e2=0.00669438, e21=0.99330562, e22=0.0067394968, e32=0.0033584313, n=0.0016792204, A=6367449.1457710434, L=10001965.7292304561, R1=6371008.7713801153, R2=6371007.1808835147, R3=6371000.7899741363, Rbiaxial=6367453.6344640013, Rtriaxial=6372797.5559332585

2456# (e-fmt) f_=2.9825722210088e+02, f=3.3528106811837e-03, e=8.1819191042833e-02, e2=6.6943800229036e-03, n=1.6792203946295e-03 (0.0e+00)

2457# AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0

2458# BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0

2459# KsOrder 8

2460

2461# Ellipsoids.Sphere: name='Sphere', a=6371008.7714149999, f=0, e=0, b=6371008.7714149999, f_=0, e2=0, e21=1, e22=0, e32=0, n=0, A=6371008.7714149999, L=10007557.1761167478, R1=6371008.7714149999, R2=6371008.7714149999, R3=6371008.7714149999, Rbiaxial=6371008.7714149999, Rtriaxial=6371008.7714149999

2462# (e-fmt) f_=0.0e+00, f=0.0e+00, e=0.0e+00, e2=0.0e+00, n=0.0e+00 (0.0e+00)

2463# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0

2464# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0

2465# KsOrder 8

2466

2467# Ellipsoids.SpherePopular: name='SpherePopular', a=6378137, f=0, e=0, b=6378137, f_=0, e2=0, e21=1, e22=0, e32=0, n=0, A=6378137, L=10018754.171394622, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137

2468# (e-fmt) f_=0.0e+00, f=0.0e+00, e=0.0e+00, e2=0.0e+00, n=0.0e+00 (0.0e+00)

2469# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0

2470# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0

2471# KsOrder 8

2472

2473# Ellipsoids._Prolate: name='_Prolate', a=6356752.3142451793, f=-0.0033640898, e=0.0820944379, b=6378137, f_=-297.257223563, e2=-0.0067394967, e21=1.0067394967, e22=-0.00669438, e32=-0.0033584313, n=-0.0016792204, A=6367449.1458234144, L=10001965.7293127235, R1=6363880.5428301189, R2=6363878.9413582645, R3=6363872.5644020075, Rbiaxial=6367453.6345163295, Rtriaxial=6362105.2243882557

2474# (e-fmt) f_=-2.97257223563e+02, f=-3.3640898209765e-03, e=8.2094437949696e-02, e2=-6.7394967422765e-03, n=-1.6792203863837e-03 (0.0e+00)

2475# AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0

2476# BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0

2477# KsOrder 8

2478

2479# **) MIT License

2480#

2482#

2483# Permission is hereby granted, free of charge, to any person obtaining a

2484# copy of this software and associated documentation files (the "Software"),

2485# to deal in the Software without restriction, including without limitation

2486# the rights to use, copy, modify, merge, publish, distribute, sublicense,

2487# and/or sell copies of the Software, and to permit persons to whom the

2488# Software is furnished to do so, subject to the following conditions:

2489#

2490# The above copyright notice and this permission notice shall be included

2491# in all copies or substantial portions of the Software.

2492#

2493# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

2494# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

2495# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

2496# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

2497# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

2498# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

2499# OTHER DEALINGS IN THE SOFTWARE.

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