1

2# -*- coding: utf-8 -*-

3

4 u'''Ellipsoidal and spherical earth models.

5

6 Classes L{a_f2Tuple}, L{Ellipsoid} and L{Ellipsoid2}, an L{Ellipsoids} registry and

7 2 dozen functions to convert I{equatorial} radius, I{polar} radius, I{eccentricities},

8 I{flattenings} and I{inverse flattening}.

9

10 See module L{datums} for L{Datum} and L{Transform} information and other details.

11

12 Following is the list of predefined L{Ellipsoid}s, all instantiated lazily.

13

14@var Ellipsoids.Airy1830: Ellipsoid(name='Airy1830', a=6377563.396, f=0.00334085, f_=299.3249646, b=6356256.90923729)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

63'''

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

65 from __future__ import division as _; del _ # PYCHOK semicolon

66

67# from pygeodesy.albers import AlbersEqualAreaCylindrical # _MODS

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

69 from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, INF, NINF, PI4, PI_2, PI_3, R_M, R_MA, R_FM, \

70 _EPSqrt, _EPStol as _TOL, _floatuple as _T, _isfinite, _over, \

71 _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \

72 _0_25, _3_0 # PYCHOK used!

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

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

75 hypot1, hypot2, sqrt3, Fsum

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

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

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

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

80 _height_, _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \

81 _Krassowsky1940_, _lat_, _meridional_, _negative_, _not_finite_, \

82 _prime_vertical_, _radius_, _Sphere_, _SPACE_, _vs_, _WGS72_, _WGS84_

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

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

85 _NamedEnumItem, _NamedTuple, _Pass, _ALL_LAZY, _MODS

86 from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple

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

88 deprecated_property_RO, property_RO, property_ROver

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

90# from pygeodesy.triaxials import _hartzell3 # _MODS

91 from pygeodesy.units import Azimuth, Bearing, Distance, Float, Float_, Height, Lamd, Lat, \

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

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

94

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

96

97__all__ = _ALL_LAZY.ellipsoids

98__version__ = '25.04.28'

99

100_f_0_0 = Float(f =_0_0) # zero flattening

101_f__0_0 = Float(f_=_0_0) # zero inverse flattening

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

103_f__WGS84 = Float(f_=_1_0 / (1000000000 / 298257223563)) # 298.257223562_999_97 vs 298.257223563

104

105

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

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

108 '''

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

110

111

112 def _s2_c2(phi):

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

114 '''

115 if phi:

116 s2 = sin(phi)**2

117 if s2 > EPS:

118 c2 = _1_0 - s2

119 if c2 > EPS:

120 if c2 < EPS1:

121 return s2, c2

122 else:

123 return _1_0, _0_0 # phi == PI_2

124 return _0_0, _1_0 # phi == 0

125

126

127 class a_f2Tuple(_NamedTuple):

128 '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial}

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

130

131 @see: Class L{Ellipsoid2}.

132 '''

133 _Names_ = (_a_, _f_) # name 'f' not 'f_'

134 _Units_ = (_Pass, _Pass)

135

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

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

138

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

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

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

142

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

144

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

146

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

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

149

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

151 '''

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

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

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

155 f = _f_0_0

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

157

158 @Property_RO

159 def b(self):

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

161 '''

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

163

164 def ellipsoid(self, **name):

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

166

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

168

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

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

171 '''

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

173

174 @Property_RO

175 def f_(self):

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

177 '''

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

179

180

181 class Circle4Tuple(_NamedTuple):

182 '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height},

183 both conventionally in C{meter} of a parallel I{circle of latitude} at

184 (geodetic) latitude C{lat} and the I{parametric (or reduced) auxiliary

185 latitude} C{beta}, both in C{degrees90}.

186

187 The C{height} is the (signed) distance along the z-axis between the

188 parallel and the equator. At near-polar C{lat}s, the C{radius} is C{0},

189 the C{height} is the ellipsoid's (signed) polar radius and C{beta}

190 equals C{lat}.

191 '''

192 _Names_ = (_radius_, _height_, _lat_, _beta_)

193 _Units_ = ( Radius, Height, Lat, Lat)

194

195

196 class Curvature2Tuple(_NamedTuple):

197 '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in

198 C{meter}, conventionally.

199 '''

200 _Names_ = (_meridional_, _prime_vertical_)

201 _Units_ = ( Meter, Meter)

202

203 @property_RO

204 def transverse(self):

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

206 '''

207 return self.prime_vertical

208

209

210 class Ellipsoid(_NamedEnumItem):

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

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

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

214 '''

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

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

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

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

219

220 _geodsolve = NN # means, use PYGEODESY_GEODSOLVE

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

222 _rhumbsolve = NN # means, use PYGEODESY_RHUMBSOLVE

223

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

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

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

227

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

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

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

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

232 spherical).

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

234

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

236

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

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

239

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

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

242 '''

243 ff_ = f, f_ # assertion below

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

245 try:

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

247 if not _isfinite(a):

248 raise ValueError(_SPACE_(_a_, _not_finite_))

249

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

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

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

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

254 elif f is not None:

255 f = Float(f=f)

256 b = a_f2b(a, f)

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

258 elif f_:

259 f_ = Float(f_=f_)

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

261 f = f_2f(f_)

262 else: # only a, spherical

263 f_ = f = 0

264 b = a # superfluous

265

266 if not f < _1_0: # sanity check, see .ecef.Ecef.__init__

267 raise ValueError(_SPACE_(_f_, _invalid_))

268 if not _isfinite(b):

269 raise ValueError(_SPACE_(_b_, _not_finite_))

270

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

272 b = a

273 f = _f_0_0

274 f_ = _f__0_0

275

276 except (TypeError, ValueError) as x:

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

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

279 raise _ValueError(t, cause=x)

280

281 self._a = a

282 self._b = b

283 self._f = f

284 self._f_ = f_

285

286 self._register(Ellipsoids, n)

287

288 if f and f_: # see test/testEllipsoidal

289 d = dict(eps=_TOL)

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

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

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

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

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

295

296 def __eq__(self, other):

297 '''Compare this and an other ellipsoid.

298

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

300

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

302 '''

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

304 self.a == other.a and

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

306

307 def __hash__(self):

308 return self._hash # memoized

309

310 @Property_RO

311 def a(self):

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

313 '''

314 return self._a

315

316 equatoradius = a # = Requatorial

317

318 @Property_RO

319 def a2(self):

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

321 '''

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

323

324 @Property_RO

325 def a2_(self):

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

327 '''

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

329

330 @Property_RO

331 def a_b(self):

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

333 '''

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

335

336 @Property_RO

337 def a2_b(self):

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

339

340 @see: U{Radii of Curvature

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

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

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

344

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

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

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

348 '''

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

350

351 @Property_RO

352 def a2_b2(self):

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

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

355 '''

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

357

358 @Property_RO

359 def a_f(self):

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

361 '''

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

363

364 a_f2 = a_f # synonym

365

366 @Property_RO

367 def A(self):

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

369

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

371

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

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

374 '''

375 A, n = self.a, self.n

376 if n:

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

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

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

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

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

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

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

384 return A

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

386# # q = 4 / self.rocPolar

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

388# # = (4 + e'2 * (-3 + e'2 * (45 / 16 + e'2 * (-175 / 64 + e'2 * 11025 / 4096)))) / q

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

390

391 @Property_RO

392 def _albersCyl(self):

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

394 '''

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

396

397 @Property_RO

398 def AlphaKs(self):

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

400 '''

401 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon

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

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

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

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

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

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

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

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

410 _T(1424729850961/743921418240)) # PYCHOK unaligned

411

412 @Property_RO

413 def area(self):

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

415

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

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

418 '''

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

420

421 @Property_RO

422 def areax(self):

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

424 accurate for very I{oblate} ellipsoids.

425

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

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

428 '''

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

430

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

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

433 '''

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

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

436 t = (v, _vs_, val)

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

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

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

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

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

442 eps=eps, f0=f0, **name_value))

443

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

445 '''Compute the I{authalic} auxiliary latitude or the I{inverse} thereof.

446

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

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

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

450

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

452

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

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

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

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

457 '''

458 if self.f:

459 f = self._albersCyl._tanf if inverse else \

460 self._albersCyl._txif # PYCHOK attr

461 lat = atan1d(f(tan(Phid(lat)))) # PYCHOK attr

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

463

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

465 '''Compute the I{conformal} auxiliary latitude or the I{inverse} thereof.

466

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

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

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

470

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

472

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

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

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

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

477 '''

478 if self.f:

479 f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr

480 lat = atan1d(f(tan(Phid(lat)))) # PYCHOK attr

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

482

483 def auxGeocentric(self, lat, inverse=False):

484 '''Compute the I{geocentric} auxiliary latitude or the I{inverse} thereof.

485

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

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

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

489

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

491

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

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

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

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

496 '''

497 if self.f:

498 f = self.a2_b2 if inverse else self.b2_a2

499 lat = atan1d(tan(Phid(lat)) * f)

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

501

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

503 '''Compute the I{isometric} auxiliary latitude or the I{inverse} thereof.

504

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

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

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

508

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

510

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

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

513 thereof is the original geodetic latitude.

514

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

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

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

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

519 '''

520 if self.f:

521 r = Phid(lat, clip=0)

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

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

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

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

526

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

528 '''Compute the I{parametric} auxiliary latitude or the I{inverse} thereof.

529

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

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

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

533

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

535

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

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

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

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

540 '''

541 if self.f:

542 lat = self._beta(Lat(lat), inverse=inverse)

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

544

545 auxReduced = auxParametric # synonymous

546

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

548 '''Compute the I{rectifying} auxiliary latitude or the I{inverse} thereof.

549

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

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

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

553

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

555

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

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

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

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

560 '''

561 if self.f:

562 lat = Lat(lat)

563 if 0 < fabs(lat) < _90_0:

564 if inverse:

565 e = self._elliptic_e22

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

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

568 else:

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

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

571

572 @Property_RO

573 def b(self):

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

575 '''

576 return self._b

577

578 polaradius = b # = Rpolar

579

580 @Property_RO

581 def b_a(self):

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

583

584 @see: Property L{f1}.

585 '''

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

587

588 @Property_RO

589 def b2(self):

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

591 '''

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

593

594 @Property_RO

595 def b2_a(self):

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

597

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

599 '''

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

601

602 @Property_RO

603 def b2_a2(self):

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

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

606 '''

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

608

609 def _beta(self, lat, inverse=False):

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

611 '''

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

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

614 return atan1d(s, c)

615

616 @Property_RO

617 def BetaKs(self):

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

619 '''

620 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon

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

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

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

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

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

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

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

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

629 _T(191773887257/3719607091200)) # PYCHOK unaligned

630

631 @deprecated_Property_RO

632 def c(self): # PYCHOK no cover

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

634 return self.R2

635

636 @Property_RO

637 def c2(self):

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

639

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

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

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

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

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

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

646 '''

647 return self._c2f(False)

648

649 @Property_RO

650 def c2x(self):

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

652 ellipsoids.

653

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

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

656 '''

657 return self._c2f(True)

658

659 def _c2f(self, c2x):

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

661 '''

662 f, c2 = self.f, self.b2

663 if f:

664 e = self.e

665 if e > EPS0:

666 if f > 0: # .isOblate

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

668 elif f < 0: # .isProlate

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

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

671 return c2

672

673 def circle4(self, lat):

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

675

676 @arg lat: Geodetic latitude (C{degrees90}, C{str}).

677

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

679

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

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

682

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

684

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

686

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

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

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

690 and method C{Rlat}.

691 '''

692 lat = Lat(lat)

693 if lat:

694 b = lat

695 if fabs(lat) < _90_0:

696 if self.f:

697 b = self._beta(lat)

698 z, r = sincos2d(b)

699 r *= self.a

700 z *= self.b

701 else: # near-polar

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

703 else: # equator

704 r = self.a

705 z = lat = b = _0_0

706 return Circle4Tuple(r, z, lat, b)

707

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

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

710 along a parallel of (geodetic) latitude.

711

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

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

714

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

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

717

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

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

720

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

722 '''

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

724

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

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

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

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

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

730

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

732 and only between points not near-polar}.

733

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

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

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

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

738

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

740 in same units as this ellipsoid's axes.

741

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

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

744

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

746 '''

747 phi0 = Phid(lat0=lat0)

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

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

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

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

752

753 @Property_RO

754 def e(self):

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

756

757 @see: Property L{es}.

758 '''

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

760

761 @deprecated_Property_RO

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

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

764 return self.e21

765

766# @Property_RO

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

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

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

770

771 @Property_RO

772 def e2(self):

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

774 == 1 - (b / a)**2}, see C{a_b2e2}.

775 '''

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

777

778 @Property_RO

779 def e2abs(self):

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

781 '''

782 return fabs(self.e2)

783

784 @Property_RO

785 def e21(self):

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

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

788 '''

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

790

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

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

793

794 @Property_RO

795 def e22(self):

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

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

798 '''

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

800

801 @Property_RO

802 def e22abs(self):

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

804 '''

805 return fabs(self.e22)

806

807 @Property_RO

808 def e32(self):

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

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

811 '''

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

813

814 @Property_RO

815 def e32abs(self):

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

817 '''

818 return fabs(self.e32)

819

820 @Property_RO

821 def e4(self):

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

823 '''

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

825

826 eccentricity = e # eccentricity

827# eccentricity2 = e2 # eccentricity squared

828 eccentricity1st2 = e2 # first eccentricity squared, signed

829 eccentricity2nd2 = e22 # second eccentricity squared, signed

830 eccentricity3rd2 = e32 # third eccentricity squared, signed

831

832 def ecef(self, Ecef=None):

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

834

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

836

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

838

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

840

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

842 '''

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

844

845 @Property_RO

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

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

848 '''

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

850 return _MODS.elliptic.Elliptic(e12)

851

852 @Property_RO

853 def _elliptic_e22(self): # aka ._elliptic_ep2

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

855 '''

856 return _MODS.elliptic.Elliptic(-self.e22abs) # complex

857

858 equatoradius = a # Requatorial

859

860 def e2s(self, s):

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

862

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

864

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

866

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

868 '''

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

870

871 def e2s2(self, s):

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

873

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

875

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

877

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

879 '''

880 r = _1_0

881 if self.e2:

882 try:

883 r -= self.e2 * Scalar(s=s)**2

884 if r < 0:

885 raise ValueError(_negative_)

886 except (TypeError, ValueError) as x:

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

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

889 return r

890

891 @Property_RO

892 def es(self):

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

894

895 @see: Property L{e}.

896 '''

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

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

899

900 def es_atanh(self, x):

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

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

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

904

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

906

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

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

909 '''

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

911

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

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

914 '''

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

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

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

918 _0_0)) # .isSpherical

919

920 @Property_RO

921 def es_c(self):

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

923 '''

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

925

926 def es_tauf(self, taup):

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

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

929

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

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

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

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

934 '''

935 t = Scalar(taup=taup)

936 if self.f: # .isEllipsoidal

937 a = fabs(t)

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

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

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

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

942 if fabs(d) < s:

943 break

944 t = Scalar(tauf=T)

945 return t

946

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

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

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

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

951 '''

952 e = self._1_e21

953 _F2_ = Fsum(T).fsum2f_ # τ0

954 _tf2 = self._es_taupf2

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

956 a, h = _tf2(T)

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

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

959 T, d = _F2_(d) # τi, (τi - τi-1)

960 yield T, i, d

961

962 def es_taupf(self, tau):

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

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

965

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

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

968 '''

969 t = Scalar(tau=tau)

970 if self.f: # .isEllipsoidal

971 t, _ = self._es_taupf2(t)

972 t = Scalar(taupf=t)

973 return t

974

975 def _es_taupf2(self, tau):

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

977 '''

978 if _isfinite(tau):

979 h = hypot1(tau)

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

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

982 else:

983 a, h = tau, INF

984 return a, h

985

986 @Property_RO

987 def _exp_es_atanh_1(self):

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

989 '''

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

991

992 @Property_RO

993 def f(self):

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

995 '''

996 return self._f

997

998 @Property_RO

999 def f_(self):

1000 '''Get the I{inverse flattening} (C{scalar}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}.

1001 '''

1002 return self._f_

1003

1004 @Property_RO

1005 def f1(self):

1006 '''Get the I{1 - flattening} (C{float}), M{f1 == 1 - f == b / a}.

1007

1008 @see: Property L{b_a}.

1009 '''

1010 return Float(f1=_1_0 - self.f)

1011

1012 @Property_RO

1013 def f2(self):

1014 '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}.

1015 '''

1016 return self._assert(self.a_b - _1_0, f2=f2f2(self.f))

1017

1018 @deprecated_Property_RO

1019 def geodesic(self):

1020 '''DEPRECATED, use property C{geodesicw}.'''

1021 return self.geodesicw

1022

1023 def geodesic_(self, exact=True):

1024 '''Get the an I{exact} C{Geodesic...} instance for this ellipsoid.

1025

1026 @kwarg exact: If C{bool} return L{GeodesicExact}C{(exact=B{exact}, ...)},

1027 otherwise a L{Geodesic}, L{GeodesicExact} or L{GeodesicSolve}

1028 instance for I{this} ellipsoid.

1029

1030 @return: The C{exact} geodesic (C{Geodesic...}).

1031

1032 @raise TypeError: Invalid B{C{exact}}.

1033

1034 @raise ValueError: Incompatible B{C{exact}} ellipsoid.

1035 '''

1036 if isbool(exact): # for consistenccy with C{.rhumb_}

1037 g = _MODS.geodesicx.GeodesicExact(self, C4order=30 if exact else 24,

1038 name=self.name)

1039 else:

1040 g = exact

1041 E = _xattr(g, ellipsoid=None)

1042 if not (E is self and isinstance(g, self._Geodesics)):

1043 raise _ValueError(exact=g, ellipsoid=E, txt_not_=self.name)

1044 return g

1045

1046 @property_ROver

1047 def _Geodesics(self):

1048 '''(INTERNAL) Get all C{Geodesic...} classes, I{once}.

1049 '''

1050 t = (_MODS.geodesicx.GeodesicExact,

1051 _MODS.geodsolve.GeodesicSolve)

1052 try:

1053 t += (_MODS.geodesicw.Geodesic,

1054 _MODS.geodesicw._wrapped.Geodesic)

1055 except ImportError:

1056 pass

1057 return t # overwrite property_ROver

1058

1059 @property_RO

1060 def geodesicw(self):

1061 '''Get this ellipsoid's I{wrapped} U{geodesicw.Geodesic

1062 <https://GeographicLib.SourceForge.io/Python/doc/code.html>}, provided

1063 I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}

1064 package is installed.

1065 '''

1066 # if not self.isEllipsoidal:

1067 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1068 return _MODS.geodesicw.Geodesic(self)

1069

1070 @property_RO

1071 def geodesicx(self):

1072 '''Get this ellipsoid's I{exact} L{GeodesicExact}.

1073 '''

1074 # if not self.isEllipsoidal:

1075 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1076 return _MODS.geodesicx.GeodesicExact(self, name=self.name)

1077

1078 @property

1079 def geodsolve(self):

1080 '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility

1081 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},

1082 provided the path to the C{GeodSolve} executable is specified with env

1083 variable C{PYGEODESY_GEODSOLVE} or re-/set with this property..

1084 '''

1085 # if not self.isEllipsoidal:

1086 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1087 return _MODS.geodsolve.GeodesicSolve(self, path=self._geodsolve, name=self.name)

1088

1089 @geodsolve.setter # PYCHOK setter!

1090 def geodsolve(self, path):

1091 '''Re-/set the (fully qualified) path to the U{GeodSolve

1092 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,

1093 overriding env variable C{PYGEODESY_GEODSOLVE} (C{str}).

1094 '''

1095 self._geodsolve = path

1096

1097 def hartzell4(self, pov, los=None):

1098 '''Compute the intersection of this ellipsoid's surface and a Line-Of-Sight

1099 from a Point-Of-View in space.

1100

1101 @arg pov: Point-Of-View outside this ellipsoid (C{Cartesian}, L{Ecef9Tuple}

1102 or L{Vector3d}).

1103 @kwarg los: Line-Of-Sight, I{direction} to this ellipsoid (L{Los}, L{Vector3d})

1104 or C{True} for the I{normal, perpendicular, plumb} to the surface

1105 of this ellipsoid or C{False} or C{None} to point to its center.

1106

1107 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},

1108 C{y} and C{z} of the projection on or the intersection with this

1109 ellipsoid and the I{distance} C{h} from B{C{pov}} to C{(x, y, z)}

1110 along B{C{los}}, all in C{meter}, conventionally.

1111

1112 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, or B{C{pov}}

1113 is inside this ellipsoid or B{C{los}} points

1114 outside this ellipsoid or in opposite direction.

1115

1116 @raise TypeError: Invalid B{C{pov}} or B{C{los}}.

1117

1118 @see: U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.

1119 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and

1120 methods L{Ellipsoid.height4} and L{Triaxial.hartzell4}.

1121 '''

1122 try:

1123 v, d, i = _MODS.triaxials._hartzell3(pov, los, self._triaxial)

1124 except Exception as x:

1125 raise IntersectionError(pov=pov, los=los, cause=x)

1126 return Vector4Tuple(v.x, v.y, v.z, d, iteration=i, name__=self.hartzell4)

1127

1128 @Property_RO

1129 def _hash(self):

1130 return hash((self.a, self.f))

1131

1132 def height4(self, xyz, normal=True):

1133 '''Compute the projection on and the height of a cartesian above or below

1134 this ellipsoid's surface.

1135

1136 @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d},

1137 L{Vector3Tuple} or L{Vector4Tuple}).

1138 @kwarg normal: If C{True}, the projection is perpendicular to (the nearest

1139 point on) this ellipsoid's surface, otherwise the C{radial}

1140 line to this ellipsoid's center (C{bool}).

1141

1142 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},

1143 C{y} and C{z} of the projection on and the height C{h} above or

1144 below this ellipsoid's surface, all in C{meter}, conventionally.

1145

1146 @raise ValueError: Null B{C{xyz}}.

1147

1148 @raise TypeError: Non-cartesian B{C{xyz}}.

1149

1150 @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>}

1151 and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse and

1152 methods L{Ellipsoid.hartzell4} and L{Triaxial.height4}.

1153 '''

1154 v = _MODS.vector3d._otherV3d(xyz=xyz)

1155 r = v.length

1156

1157 a, b, i = self.a, self.b, None

1158 if r < EPS0: # EPS

1159 v = v.times(_0_0)

1160 h = -a

1161

1162 elif self.isSpherical:

1163 v = v.times(a / r)

1164 h = r - a

1165

1166 elif normal: # perpendicular to ellipsoid

1167 x, y = hypot(v.x, v.y), fabs(v.z)

1168 if x < EPS0: # PYCHOK no cover

1169 z = copysign0(b, v.z)

1170 v = Vector3Tuple(v.x, v.y, z)

1171 h = y - b # polar

1172 elif y < EPS0: # PYCHOK no cover

1173 t = a / r

1174 v = v.times_(t, t, 0) # force z=0.0

1175 h = x - a # equatorial

1176 else: # normal in 1st quadrant

1177 x, y, i = _MODS.triaxials._plumbTo3(x, y, self)

1178 t, v = v, v.times_(x, x, y)

1179 h = t.minus(v).length

1180

1181 else: # radial to ellipsoid's center

1182 h = hypot_(a * v.z, b * v.x, b * v.y)

1183 t = (a * b / h) if h > EPS0 else _0_0 # EPS

1184 v = v.times(t)

1185 h = r * (_1_0 - t)

1186

1187 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name__=self.height4)

1188

1189 def _hubeny_2(self, phi2, phi1, lam21, scaled=True, squared=True):

1190 '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_},

1191 returning the I{angular} distance in C{radians squared} or C{radians}

1192 '''

1193 m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=scaled)

1194 h, r = (hypot2, self.a2_) if squared else (hypot, _1_0 / self.a)

1195 return h(m * (phi2 - phi1), n * lam21) * r

1196

1197 @Property_RO

1198 def isEllipsoidal(self):

1199 '''Is this model I{ellipsoidal} (C{bool})?

1200 '''

1201 return self.f != 0

1202

1203 @Property_RO

1204 def isOblate(self):

1205 '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or

1206 spherical otherwise.

1207 '''

1208 return self.f > 0

1209

1210 @Property_RO

1211 def isProlate(self):

1212 '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or

1213 spherical otherwise.

1214 '''

1215 return self.f < 0

1216

1217 @Property_RO

1218 def isSpherical(self):

1219 '''Is this ellipsoid I{spherical} (C{bool})?

1220 '''

1221 return self.f == 0

1222

1223 def _Kseries(self, *AB8Ks):

1224 '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha

1225 or Beta series coefficients per I{Karney}'s U{equations (35)

1226 and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}.

1227

1228 @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series

1229 coefficient tuples.

1230

1231 @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}).

1232

1233 @see: I{Karney}'s 30-th order U{TMseries30

1234 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}.

1235 '''

1236 k = self.KsOrder

1237 if self.n:

1238 ns = fpowers(self.n, k)

1239 ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k))

1240 else:

1241 ks = _0_0s(k)

1242 return ks

1243

1244 @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''')

1245 def KsOrder(self):

1246 '''Get the I{Krüger} series' order (C{int} 4, 6 or 8).

1247 '''

1248 return self._KsOrder

1249

1250 @KsOrder.setter # PYCHOK setter!

1251 def KsOrder(self, order):

1252 '''Set the I{Krüger} series' order (C{int} 4, 6 or 8).

1253

1254 @raise ValueError: Invalid B{C{order}}.

1255 '''

1256 if not (isint(order) and _isin(order, 4, 6, 8)):

1257 raise _ValueError(order=order)

1258 if self._KsOrder != order:

1259 Ellipsoid.AlphaKs._update(self)

1260 Ellipsoid.BetaKs._update(self)

1261 self._KsOrder = order

1262

1263 @Property_RO

1264 def L(self):

1265 '''Get the I{quarter meridian} C{L}, aka the C{polar distance}

1266 along a meridian between the equator and a pole (C{meter}),

1267 M{b * Elliptic(-e2 / (1 - e2)).cE} or M{b * PI / 2}.

1268 '''

1269 r = self._elliptic_e22.cE if self.f else PI_2

1270 return Distance(L=self.b * r)

1271

1272 def Llat(self, lat):

1273 '''Return the I{meridional length}, the distance along a meridian

1274 between the equator and a (geodetic) latitude, see C{L}.

1275

1276 @arg lat: Geodetic latitude (C{degrees90}).

1277

1278 @return: The meridional length at B{C{lat}}, negative on southern

1279 hemisphere (C{meter}).

1280 '''

1281 r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phid(lat)

1282 return Distance(Llat=self.b * r)

1283

1284 Lmeridian = Llat # meridional distance

1285

1286 @property_RO

1287 def _Lpd(self):

1288 '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}.

1289 '''

1290 return Meter(_Lpd=self.L / _90_0)

1291

1292 @property_RO

1293 def _Lpr(self):

1294 '''Get the I{quarter meridian} per radian (C{meter}), M{self.L / PI_2}.

1295 '''

1296 return Meter(_Lpr=self.L / PI_2)

1297

1298 @deprecated_Property_RO

1299 def majoradius(self): # PYCHOK no cover

1300 '''DEPRECATED, use property C{a} or C{Requatorial}.'''

1301 return self.a

1302

1303 def m2degrees(self, distance, lat=0):

1304 '''Convert a distance to an angle along the equator or along

1305 a parallel of (geodetic) latitude.

1306

1307 @arg distance: Distance (C{meter}).

1308 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).

1309

1310 @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}.

1311

1312 @raise RangeError: Latitude B{C{lat}} outside valid range and

1313 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1314

1315 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.

1316 '''

1317 return degrees(self.m2radians(distance, lat=lat))

1318

1319 def m2radians(self, distance, lat=0):

1320 '''Convert a distance to an angle along the equator or along

1321 a parallel of (geodetic) latitude.

1322

1323 @arg distance: Distance (C{meter}).

1324 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).

1325

1326 @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}.

1327

1328 @raise RangeError: Latitude B{C{lat}} outside valid range and

1329 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1330

1331 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.

1332 '''

1333 r = self.circle4(lat).radius if lat else self.a

1334 return m2radians(distance, radius=r, lat=0)

1335

1336 @deprecated_Property_RO

1337 def minoradius(self): # PYCHOK no cover

1338 '''DEPRECATED, use property C{b}, C{polaradius} or C{Rpolar}.'''

1339 return self.b

1340

1341 @Property_RO

1342 def n(self):

1343 '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}.

1344 '''

1345 return self._assert(a_b2n(self.a, self.b), n=f2n(self.f))

1346

1347 flattening = f

1348 flattening1st = f

1349 flattening2nd = f2

1350 flattening3rd = n

1351

1352 polaradius = b # Rpolar

1353

1354# Q = A # I{meridian arc unit} C{Q}, the mean, meridional length I{per radian}

1355

1356 @deprecated_Property_RO

1357 def quarteradius(self): # PYCHOK no cover

1358 '''DEPRECATED, use property C{L} or method C{Llat}.'''

1359 return self.L

1360

1361 @Property_RO

1362 def R1(self):

1363 '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}.

1364

1365 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}

1366 and method C{Rgeometric}.

1367 '''

1368 r = Fsum(self.a, self.a, self.b).fover(_3_0) if self.f else self.a

1369 return Radius(R1=r)

1370

1371 Rmean = R1

1372

1373 @Property_RO

1374 def R2(self):

1375 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}.

1376

1377 @see: C{R2x}, C{c2}, C{area} and U{Earth radius

1378 <https://WikiPedia.org/wiki/Earth_radius>}.

1379 '''

1380 return Radius(R2=sqrt(self.c2) if self.f else self.a)

1381# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>

1382# # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar

1383# # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3

1384# return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar)

1385

1386 Rauthalic = R2

1387

1388 @Property_RO

1389 def R2x(self):

1390 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}.

1391

1392 @see: C{R2}, C{c2x} and C{areax}.

1393 '''

1394 return Radius(R2x=sqrt(self.c2x) if self.f else self.a)

1395

1396 Rauthalicx = R2x

1397

1398 @Property_RO

1399 def R3(self):

1400 '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}.

1401

1402 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}.

1403 '''

1404 r = (cbrt(self.b_a) * self.a) if self.f else self.a

1405 return Radius(R3=r)

1406

1407 Rvolumetric = R3

1408

1409 def radians2m(self, rad, lat=0):

1410 '''Convert an angle to the distance along the equator or along

1411 a parallel of (geodetic) latitude.

1412

1413 @arg rad: The angle (C{radians}).

1414 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).

1415

1416 @return: Distance (C{meter}, same units as the equatorial

1417 and polar radii) or C{0} for near-polar B{C{lat}}.

1418

1419 @raise RangeError: Latitude B{C{lat}} outside valid range and

1420 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1421

1422 @raise ValueError: Invalid B{C{rad}} or B{C{lat}}.

1423 '''

1424 r = self.circle4(lat).radius if lat else self.a

1425 return radians2m(rad, radius=r, lat=0)

1426

1427 @Property_RO

1428 def Rbiaxial(self):

1429 '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}.

1430

1431 @see: C{Rtriaxial}

1432 '''

1433 a, b = self.a, self.b

1434 if b < a:

1435 b = sqrt(_0_5 + self.b2_a2 * _0_5) * a

1436 elif b > a:

1437 b *= sqrt(_0_5 + self.a2_b2 * _0_5)

1438 return Radius(Rbiaxial=b)

1439

1440 Requatorial = a # for consistent naming

1441

1442 def Rgeocentric(self, lat):

1443 '''Compute the I{geocentric} earth radius of (geodetic) latitude.

1444

1445 @arg lat: Latitude (C{degrees90}).

1446

1447 @return: Geocentric earth radius (C{meter}).

1448

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

1450

1451 @see: U{Geocentric Radius

1452 <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>}

1453 '''

1454 r, a = self.a, Phid(lat)

1455 if a and self.f:

1456 if fabs(a) < PI_2:

1457 s2, c2 = _s2_c2(a)

1458 b2_a2_s2 = self.b2_a2 * s2

1459 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2))

1460 # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2)))

1461 # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2))

1462 # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2))

1463 # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2))

1464 r *= sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2))

1465 else:

1466 r = self.b

1467 return Radius(Rgeocentric=r)

1468

1469 @Property_RO

1470 def Rgeometric(self):

1471 '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}.

1472

1473 @see: C{R1}.

1474 '''

1475 g = sqrt(self.a * self.b) if self.f else self.a

1476 return Radius(Rgeometric=g)

1477

1478 def rhumb_(self, exact=True):

1479 '''Get the an I{exact} C{Rhumb...} instance for this ellipsoid.

1480

1481 @kwarg exact: If C{bool} or C{None} return L{Rhumb}C{(exact=B{exact}, ...)},

1482 otherwise a L{Rhumb}, L{RhumbAux} or L{RhumbSolve} instance

1483 for I{this} ellipsoid.

1484

1485 @return: The C{exact} rhumb (C{Rhumb...}).

1486

1487 @raise TypeError: Invalid B{C{exact}}.

1488

1489 @raise ValueError: Incompatible B{C{exact}} ellipsoid.

1490 '''

1491 if isbool(exact): # use Rhumb for backward compatibility

1492 r = _MODS.rhumb.ekx.Rhumb(self, exact=exact, name=self.name)

1493 else:

1494 r = exact

1495 E = _xattr(r, ellipsoid=None)

1496 if not (E is self and isinstance(r, self._Rhumbs)):

1497 raise _ValueError(exact=r, ellipsosid=E, txt_not_=self.name)

1498 return r

1499

1500 @property_RO

1501 def rhumbaux(self):

1502 '''Get this ellipsoid's I{Auxiliary} C{rhumb.RhumbAux}.

1503 '''

1504 # if not self.isEllipsoidal:

1505 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1506 return _MODS.rhumb.aux_.RhumbAux(self, name=self.name)

1507

1508 @property_RO

1509 def rhumbekx(self):

1510 '''Get this ellipsoid's I{Elliptic, Krüger} C{rhumb.Rhumb}.

1511 '''

1512 # if not self.isEllipsoidal:

1513 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1514 return _MODS.rhumb.ekx.Rhumb(self, name=self.name)

1515

1516 @property_ROver

1517 def _Rhumbs(self):

1518 '''(INTERNAL) Get all C{Rhumb...} classes, I{once}.

1519 '''

1520 r = _MODS.rhumb

1521 return (r.aux_.RhumbAux, # overwrite property_ROver

1522 r.ekx.Rhumb, r.solve.RhumbSolve)

1523

1524 @property

1525 def rhumbsolve(self):

1526 '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility

1527 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},

1528 provided the path to the C{RhumbSolve} executable is specified with env

1529 variable C{PYGEODESY_RHUMBSOLVE} or re-/set with this property.

1530 '''

1531 # if not self.isEllipsoidal:

1532 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)

1533 return _MODS.rhumb.solve.RhumbSolve(self, path=self._rhumbsolve, name=self.name)

1534

1535 @rhumbsolve.setter # PYCHOK setter!

1536 def rhumbsolve(self, path):

1537 '''Re-/set the (fully qualified) path to the U{RhumbSolve

1538 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,

1539 overriding env variable C{PYGEODESY_RHUMBSOLVE} (C{str}).

1540 '''

1541 self._rhumbsolve = path

1542

1543 @deprecated_property_RO

1544 def rhumbx(self):

1545 '''DEPRECATED on 2023.11.28, use property C{rhumbekx}. '''

1546 return self.rhumbekx

1547

1548 def Rlat(self, lat):

1549 '''I{Approximate} the earth radius of (geodetic) latitude.

1550

1551 @arg lat: Latitude (C{degrees90}).

1552

1553 @return: Approximate earth radius (C{meter}).

1554

1555 @raise RangeError: Latitude B{C{lat}} outside valid range and

1556 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1557

1558 @raise TypeError: Invalid B{C{lat}}.

1559

1560 @raise ValueError: Invalid B{C{lat}}.

1561

1562 @note: C{Rlat(B{90})} equals C{Rpolar}.

1563

1564 @see: Method C{circle4}.

1565 '''

1566 # r = a - (a - b) * |lat| / 90

1567 r = self.a

1568 if self.f and lat: # .isEllipsoidal

1569 r -= (r - self.b) * fabs(Lat(lat)) / _90_0

1570 r = Radius(Rlat=r)

1571 return r

1572

1573 Rpolar = b # for consistent naming

1574

1575 def roc1_(self, sa, ca=None):

1576 '''Compute the I{prime-vertical}, I{normal} radius of curvature

1577 of (geodetic) latitude, I{unscaled}.

1578

1579 @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]).

1580 @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0])

1581 to use an alternate formula.

1582

1583 @return: The prime-vertical radius of curvature (C{float}).

1584

1585 @note: The delta between both formulae with C{Ellipsoids.WGS84}

1586 is less than 2 nanometer over the entire latitude range.

1587

1588 @see: Method L{roc2_} and class L{EcefYou}.

1589 '''

1590 if not self.f: # .isSpherical

1591 n = self.a

1592 elif ca is None:

1593 r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward

1594 n = sqrt(self.a2 / r) if r > EPS02 else _0_0

1595 elif ca: # derived from EcefYou.forward

1596 h = hypot(ca, self.b_a * sa) if sa else fabs(ca)

1597 n = self.a / h

1598 elif sa:

1599 n = self.a2_b / fabs(sa)

1600 else:

1601 n = self.a

1602 return n

1603

1604 def roc2(self, lat, scaled=False):

1605 '''Compute the I{meridional} and I{prime-vertical}, I{normal}

1606 radii of curvature of (geodetic) latitude.

1607

1608 @arg lat: Latitude (C{degrees90}).

1609 @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}).

1610

1611 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with

1612 the radii of curvature.

1613

1614 @raise ValueError: Invalid B{C{lat}}.

1615

1616 @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation

1617 <https://www.EdWilliams.org/avform.htm#flat>} and meridional and

1618 prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/

1619 Earth_radius#Radii_of_curvature>}.

1620 '''

1621 return self.roc2_(Phid(lat), scaled=scaled)

1622

1623 def roc2_(self, phi, scaled=False):

1624 '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of

1625 curvature of (geodetic) latitude.

1626

1627 @arg phi: Latitude (C{radians}).

1628 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}).

1629

1630 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the

1631 radii of curvature.

1632

1633 @raise ValueError: Invalid B{C{phi}}.

1634

1635 @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local,

1636 flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>}

1637 and the meridional and prime vertical U{Radii of Curvature

1638 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.

1639 '''

1640 a = fabs(Phi(phi))

1641 if self.f:

1642 r = self.e2s2(sin(a))

1643 if r > EPS02:

1644 n = self.a / sqrt(r)

1645 m = n * self.e21 / r

1646 else:

1647 m = n = _0_0

1648 else:

1649 m = n = self.a

1650 if scaled and a:

1651 n *= cos(a) if a < PI_2 else _0_0

1652 return Curvature2Tuple(m, n)

1653

1654 def rocAzimuth(self, lat, azimuth):

1655 '''Compute the I{directional} radius of curvature of (geodetic) latitude

1656 and C{azimuth} compass direction.

1657

1658 @see: Method L{rocBearing<Ellipsoid.rocBearing>} for details, using C{azimuth} for C{bearing}.

1659 '''

1660 return Radius(rocAzimuth=self._rocDirectional(lat, Azimuth(azimuth)))

1661

1662 def rocBearing(self, lat, bearing):

1663 '''Compute the I{directional} radius of curvature of (geodetic) latitude

1664 and C{bearing} compass direction.

1665

1666 @arg lat: Latitude (C{degrees90}).

1667 @arg bearing: Direction (compass C{degrees360}).

1668

1669 @return: Directional radius of curvature (C{meter}).

1670

1671 @raise RangeError: Latitude B{C{lat}} outside valid range and

1672 L{rangerrors<pygeodesy.rangerrors>} is C{True}.

1673

1674 @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}.

1675

1676 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}

1677 '''

1678 return Radius(rocBearing=self._rocDirectional(lat, Bearing(bearing)))

1679

1680 def _rocDirectional(self, lat, deg):

1681 '''(INTERNAL) Helper for C{rocAzimuth} and C{rocBearing}.

1682 '''

1683 if self.f:

1684 s2, c2 = _s2_c2(radians(deg))

1685 m, n = self.roc2_(Phid(lat))

1686 if n < m: # == n / (c2 * n / m + s2)

1687 c2 *= n / m

1688 elif m < n: # == m / (c2 + s2 * m / n)

1689 s2 *= m / n

1690 n = m

1691 r = _over(n, c2 + s2) # == 1 / (c2 / m + s2 / n)

1692 else:

1693 r = self.b # == self.a

1694 return r

1695

1696 @Property_RO

1697 def rocEquatorial2(self):

1698 '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature

1699 at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}.

1700

1701 @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a},

1702 L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature

1703 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.

1704 '''

1705 m = self.b2_a if self.f else self.a

1706 return Curvature2Tuple(m, self.a)

1707

1708 def rocGauss(self, lat):

1709 '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude.

1710

1711 @arg lat: Latitude (C{degrees90}).

1712

1713 @return: Gaussian radius of curvature (C{meter}).

1714

1715 @raise ValueError: Invalid B{C{lat}}.

1716

1717 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/

1718 Earth_radius#Radii_of_curvature>}

1719 '''

1720 # using ...

1721 # m, n = self.roc2_(Phid(lat))

1722 # return sqrt(m * n)

1723 # ... requires 1 or 2 sqrt

1724 g = self.b

1725 if self.f:

1726 s2, c2 = _s2_c2(Phid(lat))

1727 g = _over(g, c2 + self.b2_a2 * s2)

1728 return Radius(rocGauss=g)

1729

1730 def rocMean(self, lat):

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

1732

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

1734

1735 @return: Mean 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 if self.f:

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

1744 m *= _over(n * _2_0, m + n) # == 2 / (1 / m + 1 / n)

1745 else:

1746 m = self.a

1747 return Radius(rocMean=m)

1748

1749 def rocMeridional(self, lat):

1750 '''Compute the I{meridional} radius of curvature of (geodetic) latitude.

1751

1752 @arg lat: Latitude (C{degrees90}).

1753

1754 @return: Meridional radius of curvature (C{meter}).

1755

1756 @raise ValueError: Invalid B{C{lat}}.

1757

1758 @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation

1759 <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of

1760 Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.

1761 '''

1762 r = self.roc2_(Phid(lat)) if lat else self.rocEquatorial2

1763 return Radius(rocMeridional=r.meridional)

1764

1765 rocPolar = a2_b # synonymous

1766

1767 def rocPrimeVertical(self, lat):

1768 '''Compute the I{prime-vertical}, I{normal} radius of curvature of

1769 (geodetic) latitude, aka the I{transverse} radius of curvature.

1770

1771 @arg lat: Latitude (C{degrees90}).

1772

1773 @return: Prime-vertical radius of curvature (C{meter}).

1774

1775 @raise ValueError: Invalid B{C{lat}}.

1776

1777 @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth

1778 approximation<https://www.EdWilliams.org/avform.htm#flat>} and

1779 U{Radii of Curvature<https://WikiPedia.org/wiki/

1780 Earth_radius#Radii_of_curvature>}.

1781 '''

1782 r = self.roc2_(Phid(lat)) if lat else self.rocEquatorial2

1783 return Radius(rocPrimeVertical=r.prime_vertical)

1784

1785 rocTransverse = rocPrimeVertical # synonymous

1786

1787 @deprecated_Property_RO

1788 def Rquadratic(self): # PYCHOK no cover

1789 '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.'''

1790 return self.Rbiaxial

1791

1792 @deprecated_Property_RO

1793 def Rr(self): # PYCHOK no cover

1794 '''DEPRECATED, use property C{Rrectifying}.'''

1795 return self.Rrectifying

1796

1797 @Property_RO

1798 def Rrectifying(self):

1799 '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}.

1800

1801 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}.

1802 '''

1803 r = self.a

1804 if self.f:

1805 r *= cbrt2((sqrt3(self.b_a) + _1_0) * _0_5)

1806 return Radius(Rrectifying=r)

1807

1808 @deprecated_Property_RO

1809 def Rs(self): # PYCHOK no cover

1810 '''DEPRECATED, use property C{Rgeometric}.'''

1811 return self.Rgeometric

1812

1813 @Property_RO

1814 def Rtriaxial(self):

1815 '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}.

1816

1817 @see: C{Rbiaxial}

1818 '''

1819 q, b = self.a, self.b

1820 if b < q:

1821 q *= sqrt((self.b2_a2 + _3_0) * _0_25)

1822 elif b > q:

1823 q = sqrt((self.a2_b2 * _3_0 + _1_0) * _0_25) * b

1824 return Radius(Rtriaxial=q)

1825

1826 def toEllipsoid2(self, **name):

1827 '''Get a copy of this ellipsoid as an L{Ellipsoid2}.

1828

1829 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).

1830

1831 @see: Property C{a_f}.

1832 '''

1833 return Ellipsoid2(self, None, **name)

1834

1835 def toStr(self, prec=8, terse=4, **sep_name): # PYCHOK expected

1836 '''Return this ellipsoid as a text string.

1837

1838 @kwarg prec: Number of decimal digits, unstripped (C{int}).

1839 @kwarg terse: Limit the number of items (C{int}, 0...18),

1840 use C{B{terse}=0} or C{=None} for all.

1841 @kwarg sep_name: Optional C{B{name}=NN} (C{str}) or C{None}

1842 to exclude this ellipsoid's name and separator

1843 C{B{sep}=", "} to join the items (C{str}).

1844

1845 @return: This C{Ellipsoid}'s attributes (C{str}).

1846 '''

1847 E = Ellipsoid

1848 t = (E.a, E.f, E.f_, E.b, E.f2, E.n, E.e,

1849 E.e2, E.e21, E.e22, E.e32,

1850 E.A, E.L, E.R1, E.R2, E.R3,

1851 E.Rbiaxial, E.Rtriaxial)

1852 if terse:

1853 t = t[:terse]

1854 return self._instr(prec=prec, props=t, **sep_name)

1855

1856 def toTriaxial(self, **name):

1857 '''Convert this ellipsoid to a L{Triaxial_}.

1858

1859 @kwarg name: Optional C{B{name}=NN} (C{str}).

1860

1861 @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis

1862 pointing east and C{Z} pointing north.

1863

1864 @see: Method L{Triaxial_.toEllipsoid}.

1865 '''

1866 T = self._triaxial

1867 return T.copy(**name) if name else T

1868

1869 @property_RO

1870 def _triaxial(self):

1871 '''(INTERNAL) Get this ellipsoid's un-/ordered C{Triaxial/_}.

1872 '''

1873 a, b, m = self.a, self.b, _MODS.triaxials

1874 T = m.Triaxial if a > b else m.Triaxial_

1875 return T(a, a, b, name=self.name)

1876

1877 @Property_RO

1878 def volume(self):

1879 '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}.

1880

1881 @see: C{R3}.

1882 '''

1883 return Meter3(volume=self.a2 * self.b * PI_3 * _4_0)

1884

1885

1886 class Ellipsoid2(Ellipsoid):

1887 '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}.

1888 '''

1889 def __init__(self, a, f=None, **name):

1890 '''New L{Ellipsoid2}.

1891

1892 @arg a: Equatorial radius, semi-axis (C{meter}) or a previous

1893 L{Ellipsoid} instance.

1894 @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}),

1895 if B{C{a}} is in C{meter}.

1896 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).

1897

1898 @raise NameError: Ellipsoid with that B{C{name}} already exists.

1899

1900 @raise ValueError: Invalid B{C{a}} or B{C{f}}.

1901

1902 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.

1903 Negative C{B{f}} produces a I{prolate} ellipsoid.

1904 '''

1905 if f is None and isinstance(a, Ellipsoid):

1906 Ellipsoid.__init__(self, a.a, f =a.f,

1907 b=a.b, f_=a.f_, **name)

1908 else:

1909 Ellipsoid.__init__(self, a, f=f, **name)

1910

1911

1912 def _ispherical_a_b(a, b):

1913 '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}.

1914 '''

1915 return a < EPS0 or b < EPS0 or fabs(a - b) < EPS0

1916

1917

1918 def _ispherical_f(f):

1919 '''(INTERNAL) C{True} for spherical or invalid C{f}.

1920 '''

1921 return f > EPS1 or fabs(f) < EPS

1922

1923

1924 def _ispherical_f_(f_):

1925 '''(INTERNAL) C{True} for spherical or invalid C{f_}.

1926 '''

1927 f_ = fabs(f_)

1928 return f_ < EPS or f_ > _1_EPS

1929

1930

1931 def a_b2e(a, b):

1932 '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius.

1933

1934 @arg a: Equatorial radius (C{scalar} > 0).

1935 @arg b: Polar radius (C{scalar} > 0).

1936

1937 @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}), M{sqrt(1 - (b / a)**2)}.

1938

1939 @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids.

1940 '''

1941 e2 = _a2b2e2(a, b, b2=False)

1942 return Float(e=sqrt(fabs(e2)) if e2 else _0_0) # == sqrt(fabs((a - b) * (a + b))) / a

1943

1944

1945 def a_b2e2(a, b):

1946 '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius.

1947

1948 @arg a: Equatorial radius (C{scalar} > 0).

1949 @arg b: Polar radius (C{scalar} > 0).

1950

1951 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}), M{1 - (b / a)**2}.

1952

1953 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

1954 for I{near-spherical} ellipsoids.

1955 '''

1956 return Float(e2=_a2b2e2(a, b, b2=False))

1957

1958

1959 def a_b2e22(a, b):

1960 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius.

1961

1962 @arg a: Equatorial radius (C{scalar} > 0).

1963 @arg b: Polar radius (C{scalar} > 0).

1964

1965 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}), M{(a / b)**2 - 1}.

1966

1967 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

1968 for I{near-spherical} ellipsoids.

1969 '''

1970 return Float(e22=_a2b2e2(a, b, a2=False))

1971

1972

1973 def a_b2e32(a, b):

1974 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius.

1975

1976 @arg a: Equatorial radius (C{scalar} > 0).

1977 @arg b: Polar radius (C{scalar} > 0).

1978

1979 @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}),

1980 M{(a**2 - b**2) / (a**2 + b**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(e32=_a2b2e2(a, b))

1986

1987

1988 def _a2b2e2(a, b, a2=True, b2=True):

1989 '''(INTERNAL) Helper for C{a_b2e}, C{a_b2e2}, C{a_b2e22} and C{a_b2e32}.

1990 '''

1991 if _ispherical_a_b(a, b):

1992 e2 = _0_0

1993 else: # a > 0, b > 0

1994 a, b = (_1_0, b / a) if a > b else (a / b, _1_0)

1995 a2b2 = float(a - b) * (a + b)

1996 e2 = _over(a2b2, (a**2 if a2 else _0_0) +

1997 (b**2 if b2 else _0_0)) if a2b2 else _0_0

1998 return e2

1999

2000

2001 def a_b2f(a, b):

2002 '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius.

2003

2004 @arg a: Equatorial radius (C{scalar} > 0).

2005 @arg b: Polar radius (C{scalar} > 0).

2006

2007 @return: The flattening (C{scalar} or C{0}), M{(a - b) / a}.

2008

2009 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2010 for I{near-spherical} ellipsoids.

2011 '''

2012 f = 0 if _ispherical_a_b(a, b) else _over(float(a - b), a)

2013 return _f_0_0 if _ispherical_f(f) else Float(f=f)

2014

2015

2016 def a_b2f_(a, b):

2017 '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius.

2018

2019 @arg a: Equatorial radius (C{scalar} > 0).

2020 @arg b: Polar radius (C{scalar} > 0).

2021

2022 @return: The inverse flattening (C{scalar} or C{0}), M{a / (a - b)}.

2023

2024 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2025 for I{near-spherical} ellipsoids.

2026 '''

2027 f_ = 0 if _ispherical_a_b(a, b) else _over(a, float(a - b))

2028 return _f__0_0 if _ispherical_f_(f_) else Float(f_=f_)

2029

2030

2031 def a_b2f2(a, b):

2032 '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius.

2033

2034 @arg a: Equatorial radius (C{scalar} > 0).

2035 @arg b: Polar radius (C{scalar} > 0).

2036

2037 @return: The I{signed}, 2nd flattening (C{scalar} or C{0}), M{(a - b) / b}.

2038

2039 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2040 for I{near-spherical} ellipsoids.

2041 '''

2042 t = 0 if _ispherical_a_b(a, b) else float(a - b)

2043 return Float(f2=_0_0 if fabs(t) < EPS0 else _over(t, b))

2044

2045

2046 def a_b2n(a, b):

2047 '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius.

2048

2049 @arg a: Equatorial radius (C{scalar} > 0).

2050 @arg b: Polar radius (C{scalar} > 0).

2051

2052 @return: The I{signed}, 3rd flattening (C{scalar} or C{0}), M{(a - b) / (a + b)}.

2053

2054 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2055 for I{near-spherical} ellipsoids.

2056 '''

2057 t = 0 if _ispherical_a_b(a, b) else float(a - b)

2058 return Float(n=_0_0 if fabs(t) < EPS0 else _over(t, a + b))

2059

2060

2061 def a_f2b(a, f):

2062 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}.

2063

2064 @arg a: Equatorial radius (C{scalar} > 0).

2065 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2066

2067 @return: The polar radius (C{float}), M{a * (1 - f)}.

2068 '''

2069 b = a if _ispherical_f(f) else (a * (_1_0 - f))

2070 return Radius_(b=a if _ispherical_a_b(a, b) else b)

2071

2072

2073 def a_f_2b(a, f_):

2074 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}.

2075

2076 @arg a: Equatorial radius (C{scalar} > 0).

2077 @arg f_: Inverse flattening (C{scalar} >>> 1).

2078

2079 @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}.

2080 '''

2081 b = a if _ispherical_f_(f_) else _over(a * (f_ - _1_0), f_)

2082 return Radius_(b=a if _ispherical_a_b(a, b) else b)

2083

2084

2085 def b_f2a(b, f):

2086 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}.

2087

2088 @arg b: Polar radius (C{scalar} > 0).

2089 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2090

2091 @return: The equatorial radius (C{float}), M{b / (1 - f)}.

2092 '''

2093 t = _1_0 - f

2094 a = b if fabs(t) < EPS0 else _over(b, t)

2095 return Radius_(a=b if _ispherical_a_b(a, b) else a)

2096

2097

2098 def b_f_2a(b, f_):

2099 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}.

2100

2101 @arg b: Polar radius (C{scalar} > 0).

2102 @arg f_: Inverse flattening (C{scalar} >>> 1).

2103

2104 @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}.

2105 '''

2106 t = f_ - _1_0

2107 a = b if _ispherical_f_(f_) or fabs(t) < EPS0 \

2108 or fabs(t - f_) < EPS0 else _over(b * f_, t)

2109 return Radius_(a=b if _ispherical_a_b(a, b) else a)

2110

2111

2112 def e2f(e):

2113 '''Return C{f}, the I{flattening} for a given I{1st eccentricity}.

2114

2115 @arg e: The (1st) eccentricity (0 <= C{float} < 1)

2116

2117 @return: The flattening (C{scalar} or C{0}).

2118

2119 @see: Function L{e22f}.

2120 '''

2121 return e22f(e**2)

2122

2123

2124 def e22f(e2):

2125 '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}.

2126

2127 @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1)

2128

2129 @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(1 - e2) + 1)}.

2130 '''

2131 return Float(f=_over(e2, sqrt(_1_0 - e2) + _1_0)) if e2 and e2 < _1_0 else _f_0_0

2132

2133

2134 def f2e2(f):

2135 '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}.

2136

2137 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2138

2139 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1), M{f * (2 - f)}.

2140

2141 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2142 for I{near-spherical} ellipsoids.

2143

2144 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2145 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening

2146 <https://WikiPedia.org/wiki/Flattening>}.

2147 '''

2148 return Float(e2=_0_0 if _ispherical_f(f) else (f * (_2_0 - f)))

2149

2150

2151 def f2e22(f):

2152 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}.

2153

2154 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2155

2156 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or C{INF}),

2157 M{f * (2 - f) / (1 - f)**2}.

2158

2159 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2160 for near-spherical ellipsoids.

2161

2162 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2163 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.

2164 '''

2165 # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2

2166 t = (_1_0 - f)**2

2167 return Float(e22=INF if t < EPS0 else _over(f2e2(f), t)) # PYCHOK type

2168

2169

2170 def f2e32(f):

2171 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}.

2172

2173 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2174

2175 @return: The I{signed}, 3rd eccentricity I{squared} (C{float}),

2176 M{f * (2 - f) / (1 + (1 - f)**2)}.

2177

2178 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2179 for I{near-spherical} ellipsoids.

2180

2181 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2182 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.

2183 '''

2184 # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2)

2185 e2 = f2e2(f)

2186 return Float(e32=_over(e2, _2_0 - e2))

2187

2188

2189 def f_2f(f_):

2190 '''Return C{f}, the I{flattening} for a given I{inverse flattening}.

2191

2192 @arg f_: Inverse flattening (C{scalar} >>> 1).

2193

2194 @return: The flattening (C{scalar} or C{0}), M{1 / f_}.

2195

2196 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2197 for I{near-spherical} ellipsoids.

2198 '''

2199 f = 0 if _ispherical_f_(f_) else _over(_1_0, f_)

2200 return _f_0_0 if _ispherical_f(f) else Float(f=f) # PYCHOK type

2201

2202

2203 def f2f_(f):

2204 '''Return C{f_}, the I{inverse flattening} for a given I{flattening}.

2205

2206 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2207

2208 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.

2209

2210 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2211 for I{near-spherical} ellipsoids.

2212 '''

2213 f_ = 0 if _ispherical_f(f) else _over(_1_0, f)

2214 return _f__0_0 if _ispherical_f_(f_) else Float(f_=f_) # PYCHOK type

2215

2216

2217 def f2f2(f):

2218 '''Return C{f2}, the I{2nd flattening} for a given I{flattening}.

2219

2220 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2221

2222 @return: The I{signed}, 2nd flattening (C{scalar} or C{INF}), M{f / (1 - f)}.

2223

2224 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2225 for I{near-spherical} ellipsoids.

2226

2227 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2228 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening

2229 <https://WikiPedia.org/wiki/Flattening>}.

2230 '''

2231 t = _1_0 - f

2232 return Float(f2=_0_0 if _ispherical_f(f) else

2233 (INF if fabs(t) < EPS else _over(f, t))) # PYCHOK type

2234

2235

2236 def f2n(f):

2237 '''Return C{n}, the I{3rd flattening} for a given I{flattening}.

2238

2239 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).

2240

2241 @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1),

2242 M{f / (2 - f)}.

2243

2244 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2245 for I{near-spherical} ellipsoids.

2246

2247 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2248 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening

2249 <https://WikiPedia.org/wiki/Flattening>}.

2250 '''

2251 return Float(n=_0_0 if _ispherical_f(f) else _over(f, float(_2_0 - f)))

2252

2253

2254 def n2e2(n):

2255 '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}.

2256

2257 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).

2258

2259 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF),

2260 M{4 * n / (1 + n)**2}.

2261

2262 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}

2263 for I{near-spherical} ellipsoids.

2264

2265 @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}.

2266 '''

2267 t = (n + _1_0)**2

2268 return Float(e2=_0_0 if fabs(n) < EPS0 else

2269 (NINF if t < EPS0 else _over(_4_0 * n, t)))

2270

2271

2272 def n2f(n):

2273 '''Return C{f}, the I{flattening} for a given I{3rd flattening}.

2274

2275 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).

2276

2277 @return: The flattening (C{scalar} or NINF), M{2 * n / (1 + n)}.

2278

2279 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/

2280 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening

2281 <https://WikiPedia.org/wiki/Flattening>}.

2282 '''

2283 t = n + _1_0

2284 f = 0 if fabs(n) < EPS0 else (NINF if t < EPS0 else _over(_2_0 * n, t))

2285 return _f_0_0 if _ispherical_f(f) else Float(f=f)

2286

2287

2288 def n2f_(n):

2289 '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}.

2290

2291 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).

2292

2293 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.

2294

2295 @see: L{n2f} and L{f2f_}.

2296 '''

2297 return f2f_(n2f(n))

2298

2299

2300 class Ellipsoids(_NamedEnum):

2301 '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class

2302 to accommodate the L{_LazyNamedEnumItem} properties.

2303 '''

2304 def _Lazy(self, a, b, f_, **kwds):

2305 '''(INTERNAL) Instantiate the L{Ellipsoid}.

2306 '''

2307 return Ellipsoid(a, b=b, f_=f_, **kwds)

2308

2309 Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton

2310'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.'''

2311# <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html>

2312# <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS>

2313# <https://kb.OSU.edu/dspace/handle/1811/77986>

2314# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids>

2315# <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html>

2316# <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java>

2317 Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid>

2318 Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909

2319 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448

2320# APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965

2321# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid

2322# AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America

2323 Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192

2324 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System"

2325 Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)),

2326 BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)),

2327# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)),

2328 CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC)

2329# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)),

2330 Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)),

2331 Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)),

2332 Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)),

2333 Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc

2334 CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures

2335 Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium

2336 Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)),

2337# Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)),

2338# Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)),

2339# Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)),

2340 Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)),

2341 Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)),

2342 Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)),

2343# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)),

2344 GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)),

2345 GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid"

2346 GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516

2347# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)),

2348 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.25722210088)), # IUGG, ITRS, ETRS89

2349# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297

2350 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)),

2351# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)),

2352 IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union

2353 IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302

2354 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy

2355 IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)),

2356 Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795

2357 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l

2358 Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling

2359 Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling

2360# Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961

2361# Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979

2362 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France

2363 Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)),

2364 Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)),

2365# MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983

2366# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab.

2367 NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab.

2368# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965

2369 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)),

2370 OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)),

2371# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect?

2372 Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972

2373# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)),

2374 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11

2375# SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia

2376 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System

2377 SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American

2378 Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo

2379 SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo

2380 SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)), # EPSG:3857 Spheroid

2381 Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)),

2382# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)),

2383# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)),

2384 WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)),

2385 WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)),

2386 WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52

2387 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # GPS b=6356752.3142451793

2388# U{NOAA/NOS/NGS/inverse<https://GitHub.com/noaa-ngs/inverse/blob/main/invers3d.f>}

2389 WGS84_NGS = _lazy('WGS84_NGS', *_T(R_MA, _0_0, 298.257222100882711243162836600094))

2390)

2391

2392_EWGS84 = Ellipsoids.WGS84 # (INTERNAL) shared

2393

2394 if __name__ == _DMAIN_:

2395

2396 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_

2397 from pygeodesy import nameof, printf

2398

2399 for E in (_EWGS84, Ellipsoids.GRS80, # NAD83,

2400 Ellipsoids.Sphere, Ellipsoids.SpherePopular,

2401 Ellipsoid(_EWGS84.b, _EWGS84.a, name='_Prolate')):

2402 e = f2n(E.f) - E.n

2403 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10, terse=0), nl=1)

2404 printf('# e=%s, f_=%s, f=%s, n=%s (%s)', fstr(E.e, prec=13, fmt=Fmt.e),

2405 fstr(E.f_, prec=13, fmt=Fmt.e),

2406 fstr(E.f, prec=13, fmt=Fmt.e),

2407 fstr(E.n, prec=13, fmt=Fmt.e),

2408 fstr(e, prec=9, fmt=Fmt.e))

2409 printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20))

2410 printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20))

2411 printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property

2412

2413 # __doc__ of this file, force all into registry

2414 t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_)

2415 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))

2416

2417# % python3.13 -m pygeodesy.ellipsoids

2418

2419# Ellipsoids.WGS84: name='WGS84', a=6378137, f=0.0033528107, f_=298.257223563, b=6356752.3142451793, f2=0.0033640898, n=0.0016792204, e=0.0818191908, e2=0.00669438, e21=0.99330562, e22=0.0067394967, e32=0.0033584313, A=6367449.1458234144, L=10001965.7293127235, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.7900091587, Rbiaxial=6367453.6345163295, Rtriaxial=6372797.5559594007

2420# e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00)

2421# AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0

2422# BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0

2423# KsOrder 8

2424

2425# Ellipsoids.GRS80: name='GRS80', a=6378137, f=0.0033528107, f_=298.2572221009, b=6356752.3141403468, f2=0.0033640898, n=0.0016792204, e=0.081819191, e2=0.00669438, e21=0.99330562, e22=0.0067394968, e32=0.0033584313, A=6367449.1457710434, L=10001965.7292304561, R1=6371008.7713801153, R2=6371007.1808835147, R3=6371000.7899741363, Rbiaxial=6367453.6344640013, Rtriaxial=6372797.5559332585

2426# e=8.1819191042833e-02, f_=2.9825722210088e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00)

2427# AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0

2428# BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0

2429# KsOrder 8

2430

2431# Ellipsoids.Sphere: name='Sphere', a=6371008.7714149999, f=0, f_=0, b=6371008.7714149999, f2=0, n=0, e=0, e2=0, e21=1, e22=0, e32=0, A=6371008.7714149999, L=10007557.1761167478, R1=6371008.7714149999, R2=6371008.7714149999, R3=6371008.7714149999, Rbiaxial=6371008.7714149999, Rtriaxial=6371008.7714149999

2432# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)

2433# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0

2434# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0

2435# KsOrder 8

2436

2437# Ellipsoids.SpherePopular: name='SpherePopular', a=6378137, f=0, f_=0, b=6378137, f2=0, n=0, e=0, e2=0, e21=1, e22=0, e32=0, A=6378137, L=10018754.171394622, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137

2438# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)

2439# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0

2440# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0

2441# KsOrder 8

2442

2443# Ellipsoids._Prolate: name='_Prolate', a=6356752.3142451793, f=-0.0033640898, f_=-297.257223563, b=6378137, f2=-0.0033528107, n=-0.0016792204, e=0.0820944379, e2=-0.0067394967, e21=1.0067394967, e22=-0.00669438, e32=-0.0033584313, A=6367449.1458234144, L=10035500.5204500332, R1=6363880.5428301189, R2=6363878.9413582645, R3=6363872.5644020075, Rbiaxial=6367453.6345163295, Rtriaxial=6362105.2243882557

2444# e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00)

2445# AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0

2446# BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0

2447# KsOrder 8

2448

2449# **) MIT License

2450#

2451# Copyright (C) 2016-2025 -- mrJean1 at Gmail -- All Rights Reserved.

2452#

2453# Permission is hereby granted, free of charge, to any person obtaining a

2454# copy of this software and associated documentation files (the "Software"),

2455# to deal in the Software without restriction, including without limitation

2456# the rights to use, copy, modify, merge, publish, distribute, sublicense,

2457# and/or sell copies of the Software, and to permit persons to whom the

2458# Software is furnished to do so, subject to the following conditions:

2459#

2460# The above copyright notice and this permission notice shall be included

2461# in all copies or substantial portions of the Software.

2462#

2463# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

2464# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

2465# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

2466# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

2467# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

2468# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

2469# OTHER DEALINGS IN THE SOFTWARE.