# Coverage for pygeodesy/ellipsoids.py : 98%

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1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 | # -*- coding: utf-8 -*- u'''Classes L{a_f2Tuple}, L{Ellipsoid}, L{Ellipsoid2}, an L{Ellipsoids} registry and a dozen functions to convert I{equatorial} radius, I{polar} radius, I{eccentricities}, I{flattenings} and I{inverse flattening}. See module L{datums} for more information and other details. @newfield example: Example, Examples @var Ellipsoids.Airy1830: Ellipsoid(name='Airy1830', a=6377563.396, b=6356256.90923729, f_=299.3249646, f=0.00334085, f2=0.00335205, n=0.00167322, e=0.08167337, e2=0.00667054, e22=0.00671533, e32=0.00334643, L=10001126.0807165, R1=6370461.23374576, R2=6370459.65470808, R3=6370453.30994572) @var Ellipsoids.AiryModified: Ellipsoid(name='AiryModified', a=6377340.189, b=6356034.44793853, f_=299.3249646, f=0.00334085, f2=0.00335205, n=0.00167322, e=0.08167337, e2=0.00667054, e22=0.00671533, e32=0.00334643, L=10000776.05340819, R1=6370238.27531284, R2=6370236.69633043, R3=6370230.35179012) @var Ellipsoids.Australia1966: Ellipsoid(name='Australia1966', a=6378160, b=6356774.71919531, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, L=10002001.39064442, R1=6371031.5730651, R2=6371029.9824858, R3=6371023.59124343) @var Ellipsoids.Bessel1841: Ellipsoid(name='Bessel1841', a=6377397.155, b=6356078.962818, f_=299.1528128, f=0.00334277, f2=0.00335398, n=0.00167418, e=0.08169683, e2=0.00667437, e22=0.00671922, e32=0.00334836, L=10000855.76443237, R1=6370291.09093933, R2=6370289.51012659, R3=6370283.15821522) @var Ellipsoids.CPM1799: Ellipsoid(name='CPM1799', a=6375738.7, b=6356671.92557493, f_=334.39, f=0.00299052, f2=0.00299949, n=0.0014975, e=0.07727934, e2=0.0059721, e22=0.00600798, e32=0.00299499, L=10000017.52721564, R1=6369383.10852498, R2=6369381.8434158, R3=6369376.76247021) @var Ellipsoids.Clarke1866: Ellipsoid(name='Clarke1866', a=6378206.4, b=6356583.8, f_=294.97869821, f=0.00339008, f2=0.00340161, n=0.00169792, e=0.08227185, e2=0.00676866, e22=0.00681478, e32=0.00339582, L=10001888.04298286, R1=6370998.86666667, R2=6370997.240633, R3=6370990.70659881) @var Ellipsoids.Clarke1880: Ellipsoid(name='Clarke1880', a=6378249.145, b=6356514.86954978, f_=293.465, f=0.00340756, f2=0.00341921, n=0.00170669, e=0.0824834, e2=0.00680351, e22=0.00685012, e32=0.00341337, L=10001867.55164747, R1=6371004.38651659, R2=6371002.74366963, R3=6370996.1419165) @var Ellipsoids.Clarke1880IGN: Ellipsoid(name='Clarke1880IGN', a=6378249.2, b=6356515, f_=293.46602129, f=0.00340755, f2=0.0034192, n=0.00170668, e=0.08248326, e2=0.00680349, e22=0.00685009, e32=0.00341336, L=10001867.69724906, R1=6371004.46666667, R2=6371002.82383112, R3=6370996.22212394) @var Ellipsoids.Clarke1880Mod: Ellipsoid(name='Clarke1880Mod', a=6378249.145, b=6356514.96582849, f_=293.4663, f=0.00340755, f2=0.0034192, n=0.00170668, e=0.08248322, e2=0.00680348, e22=0.00685009, e32=0.00341335, L=10001867.62720001, R1=6371004.4186095, R2=6371002.77577708, R3=6370996.17408252) @var Ellipsoids.Delambre1810: Ellipsoid(name='Delambre1810', a=6376428, b=6355957.92616372, f_=311.5, f=0.00321027, f2=0.00322061, n=0.00160772, e=0.08006397, e2=0.00641024, e22=0.0064516, e32=0.00321543, L=9999998.98395793, R1=6369604.64205457, R2=6369603.18419749, R3=6369597.32739068) @var Ellipsoids.Engelis1985: Ellipsoid(name='Engelis1985', a=6378136.05, b=6356751.32272154, f_=298.2566, f=0.00335282, f2=0.0033641, n=0.00167922, e=0.08181928, e2=0.00669439, e22=0.00673951, e32=0.00335844, L=10001964.20447208, R1=6371007.80757385, R2=6371006.21707085, R3=6370999.82613572) @var Ellipsoids.Everest1969: Ellipsoid(name='Everest1969', a=6377295.664, b=6356094.667915, f_=300.8017, f=0.00332445, f2=0.00333554, n=0.00166499, e=0.08147298, e2=0.00663785, e22=0.0066822, e32=0.00332998, L=10000788.3115495, R1=6370228.665305, R2=6370227.10178537, R3=6370220.81951617) @var Ellipsoids.Fisher1968: Ellipsoid(name='Fisher1968', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995034) @var Ellipsoids.GEM10C: Ellipsoid(name='GEM10C', a=6378137, b=6356752.31424783, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, L=10001965.7293148, R1=6371008.77141594, R2=6371007.18091936, R3=6371000.79001004) @var Ellipsoids.GRS67: Ellipsoid(name='GRS67', a=6378160, b=6356774.51609071, f_=298.24716743, f=0.00335292, f2=0.0033642, n=0.00167928, e=0.08182057, e2=0.00669461, e22=0.00673973, e32=0.00335854, L=10002001.2312605, R1=6371031.50536357, R2=6371029.91475409, R3=6371023.52339014) @var Ellipsoids.GRS80: Ellipsoid(name='GRS80', a=6378137, b=6356752.31414035, f_=298.2572221, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, L=10001965.72923046, R1=6371008.77138012, R2=6371007.18088351, R3=6371000.78997413) @var Ellipsoids.Helmert1906: Ellipsoid(name='Helmert1906', a=6378200, b=6356818.16962789, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, L=10002066.93013953, R1=6371072.7232093, R2=6371071.13315272, R3=6371064.74401563) @var Ellipsoids.IERS1989: Ellipsoid(name='IERS1989', a=6378136, b=6356751.30156878, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669088, R3=6370999.78577296) @var Ellipsoids.IERS1992TOPEX: Ellipsoid(name='IERS1992TOPEX', a=6378136.3, b=6356751.61659215, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, L=10001964.63159783, R1=6371008.07219738, R2=6371006.48170097, R3=6371000.09079235) @var Ellipsoids.IERS2003: Ellipsoid(name='IERS2003', a=6378136.6, b=6356751.85797165, f_=298.25642, f=0.00335282, f2=0.0033641, n=0.00167922, e=0.0818193, e2=0.0066944, e22=0.00673951, e32=0.00335844, L=10001965.05683465, R1=6371008.35265722, R2=6371006.76215217, R3=6371000.37120876) @var Ellipsoids.Intl1924: Ellipsoid(name='Intl1924', a=6378388, b=6356911.94612795, f_=297, f=0.003367, f2=0.00337838, n=0.00168634, e=0.08199189, e2=0.00672267, e22=0.00676817, e32=0.00337267, L=10002288.29898944, R1=6371229.31537598, R2=6371227.71133444, R3=6371221.26587487) @var Ellipsoids.Intl1967: Ellipsoid(name='Intl1967', a=6378157.5, b=6356772.2, f_=298.24961539, f=0.0033529, f2=0.00336418, n=0.00167926, e=0.08182023, e2=0.00669455, e22=0.00673967, e32=0.00335852, L=10001997.44859308, R1=6371029.06666667, R2=6371027.47608389, R3=6371021.08482752) @var Ellipsoids.Krassovski1940: Ellipsoid(name='Krassovski1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439) @var Ellipsoids.Krassowsky1940: Ellipsoid(name='Krassowsky1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439) @var Ellipsoids.Maupertuis1738: Ellipsoid(name='Maupertuis1738', a=6397300, b=6363806.28272251, f_=191, f=0.0052356, f2=0.00526316, n=0.00262467, e=0.10219488, e2=0.01044379, e22=0.01055402, e32=0.00524931, L=10022566.69846922, R1=6386135.42757417, R2=6386131.54144847, R3=6386115.88628229) @var Ellipsoids.Mercury1960: Ellipsoid(name='Mercury1960', a=6378166, b=6356784.28360711, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, L=10002013.61254591, R1=6371038.76120237, R2=6371037.17115427, R3=6371030.78205124) @var Ellipsoids.Mercury1968Mod: Ellipsoid(name='Mercury1968Mod', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995034) @var Ellipsoids.NWL1965: Ellipsoid(name='NWL1965', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802666) @var Ellipsoids.OSU86F: Ellipsoid(name='OSU86F', a=6378136.2, b=6356751.51693008, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, L=10001964.47478349, R1=6371007.97231003, R2=6371006.38181364, R3=6370999.99090512) @var Ellipsoids.OSU91A: Ellipsoid(name='OSU91A', a=6378136.3, b=6356751.6165948, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, L=10001964.63159991, R1=6371008.07219827, R2=6371006.48170186, R3=6371000.09079324) @var Ellipsoids.Plessis1817: Ellipsoid(name='Plessis1817', a=6376523, b=6355862.93325557, f_=308.64, f=0.00324002, f2=0.00325055, n=0.00162264, e=0.08043347, e2=0.00646954, e22=0.00651167, e32=0.00324527, L=9999999.1100364, R1=6369636.31108519, R2=6369634.82608583, R3=6369628.85999667) @var Ellipsoids.SGS85: Ellipsoid(name='SGS85', a=6378136, b=6356751.30156878, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669087, R3=6370999.78577296) @var Ellipsoids.SoAmerican1969: Ellipsoid(name='SoAmerican1969', a=6378160, b=6356774.71919531, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, L=10002001.39064442, R1=6371031.5730651, R2=6371029.98248581, R3=6371023.59124343) @var Ellipsoids.Sphere: Ellipsoid(name='Sphere', a=6371008.771415, b=6371008.771415, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, L=10007557.17611675, R1=6371008.771415, R2=6371008.771415, R3=6371008.771415) @var Ellipsoids.SphereAuthalic: Ellipsoid(name='SphereAuthalic', a=6371000, b=6371000, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, L=10007543.39801029, R1=6371000, R2=6371000, R3=6371000) @var Ellipsoids.SpherePopular: Ellipsoid(name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, L=10018754.17139462, R1=6378137, R2=6378137, R3=6378137) @var Ellipsoids.Struve1860: Ellipsoid(name='Struve1860', a=6378298.3, b=6356657.14266956, f_=294.73, f=0.00339294, f2=0.00340449, n=0.00169935, e=0.0823065, e2=0.00677436, e22=0.00682056, e32=0.00339869, L=10002017.83655713, R1=6371084.58088985, R2=6371082.95208988, R3=6371076.40691418) @var Ellipsoids.WGS60: Ellipsoid(name='WGS60', a=6378165, b=6356783.28695944, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, L=10002012.04438139, R1=6371037.76231981, R2=6371036.17227197, R3=6371029.78316993) @var Ellipsoids.WGS66: Ellipsoid(name='WGS66', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802666) @var Ellipsoids.WGS72: Ellipsoid(name='WGS72', a=6378135, b=6356750.52001609, f_=298.26, f=0.00335278, f2=0.00336406, n=0.0016792, e=0.08181881, e2=0.00669432, e22=0.00673943, e32=0.0033584, L=10001962.74919858, R1=6371006.84000536, R2=6371005.24953886, R3=6370998.85875069) @var Ellipsoids.WGS84: Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000915) ''' # make sure int/int division yields float quotient from __future__ import division division = 1 / 2 # double check int division, see .datums, .fmath, .utily if not division: raise ImportError('%s 1/2 == %d' % ('division', division)) del division from pygeodesy.basics import property_doc_, property_RO, _xinstanceof from pygeodesy.errors import _AssertionError, _ValueError from pygeodesy.fmath import cbrt, cbrt2, fdot, fpowers, Fsum, fsum_, \ hypot1, hypot2, sqrt3 from pygeodesy.interns import EPS, EPS1, INF, NN, PI4 as _PI4, PI_2, R_M, \ _float, _floatuple as _T, _lat_, _n_a_, _negative_, \ _0_0, _0_1, _0_5, _1_0, _2_0, _3_0, _4_0, _90_0 from pygeodesy.interns import _8_0 # PYCHOK used! from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import _NamedEnum, _NamedEnumItem, _NamedTuple, _Pass from pygeodesy.namedTuples import Distance2Tuple from pygeodesy.streprs import _e, instr, _Fmt, fstr from pygeodesy.units import Bearing_, Distance, Float, Float_, Lam_, Lat, \ Meter, Phi, Phi_, Radius, Radius_, Scalar from pygeodesy.utily import atan2b, atan2d, degrees90, degrees2m, \ m2degrees, m2km, m2NM, m2SM from math import asinh, atan, atanh, copysign, cos, degrees, \ exp, hypot, sin, sinh, sqrt, tan R_M = Radius(R_M =R_M) # mean (spherical) earth radius (C{meter}) R_MA = Radius(R_MA=6378137.0) # equatorial (major) earth radius (C{meter}), WGS84, EPSG:3785 R_MB = Radius(R_MB=6356752.0) # polar (minor) earth radius (C{meter}), WGS84, EPSG:3785 R_KM = Radius(R_KM=m2km(R_M)) # mean (spherical) earth radius (C{KM}, kilo meter) R_NM = Radius(R_NM=m2NM(R_M)) # mean (spherical) earth radius (C{NM}, nautical miles) R_SM = Radius(R_SM=m2SM(R_M)) # mean (spherical) earth radius (C{SM}, statute miles) # See <https://www.EdWilliams.org/avform.htm>, # <https://www.DTIC.mil/dtic/tr/fulltext/u2/a216843.pdf> and # <https://GitHub.com/NASA/MultiDop/blob/master/src/share/man/man3/geog_lib.3> # based on International Standard Nautical Mile of 1,852 meter (1' latitude) R_FM = Radius(R_FM=6371000.0) # former FAI Sphere earth radius (C{meter}) R_VM = Radius(R_VM=6366707.0194937) # Aviation/Navigation earth radius (C{meter}) # R_ = Radius(R_ =6372797.560856) # XXX some other earth radius??? __all__ = _ALL_LAZY.ellipsoids __version__ = '20.10.05' _1_EPS = _1_0 / EPS # largest abs(f_) _PI4_3 = _PI4 / _3_0 # PYCHOK used! _TOL = sqrt(EPS * _0_1) # for Ellipsoid.estauf, in .ups, testEllipsoidal.py def _atand(y_x): '''Return C{atan} angle in C{degrees}. ''' return atan2d(y_x, 1) def _aux(lat, inverse, auxLat, clip=90): '''Return named auxiliary latitude degrees. ''' return Lat(lat, clip=clip, name=_lat_ if inverse else auxLat) def _c2_s2(phi): '''Return 2-tuple C{(cos(phi)**2, sin(phi)**2)}. ''' c2 = cos(phi)**2 return c2, (_1_0 - c2) def _4Ecef(this, Ecef): '''Return an ECEF converter. ''' from pygeodesy.ecef import EcefKarney, EcefVeness, EcefYou if Ecef is None: Ecef = EcefKarney else: _xinstanceof(EcefKarney, EcefVeness, EcefYou, Ecef=Ecef) return Ecef(this, name=this.name) # datum or ellipsoid class a_f2Tuple(_NamedTuple): '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial} radius C{a} (C{meter}) and scalar I{flattening} C{f}. @note: M{abs(f) < EPS} is forced to M{f = 0}, spherical. @see: Class L{Ellipsoid2}. ''' _Names_ = ('a', 'f') _Units_ = (_Pass, _Pass) def __new__(cls, a, f): '''New L{a_f2Tuple} ellipsoid specification. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1). @return: An L{a_f2Tuple}C{(a, f)} instance. @raise UnitError: Invalid B{C{a}} or B{C{f}}. @note: M{abs(f) < EPS} is forced to M{f = 0}, spherical. ''' a = Radius_(a=a) f = Float_( f=f, low=None, high=EPS1) if abs(f) < EPS: # force spherical f = Float(f=_0_0) return _NamedTuple.__new__(cls, a, f) @property_RO def b(self): '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}. ''' return a_f2b(self.a, self.f) # PYCHOK .a and .f @property_RO def f_(self): '''Get the I{inverse} flattening (C{float}), M{1 / f} == M{a / (a - b)}. ''' return f2f_(self.f) # PYCHOK .f class Curvature2Tuple(_NamedTuple): '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in C{meter}, conventionally. ''' _Names_ = ('meridional', 'prime_vertical') _Units_ = ( Meter, Meter) class Ellipsoid(_NamedEnumItem): '''Ellipsoid with I{equatorial} and I{polar} radius, I{flattening}, I{inverse flattening} and other, often used, cached attributes, supporting I{spherical} and I{oblate} and I{prolate} ellipsoidal models. ''' _a = 0 # equatorial (major) radius, semi-axis (C{meter}) _b = 0 # polar (minor) radius, semi-axis (C{meter}): a * (f - 1) / f _f = 0 # (1st) flattening: (a - b) / a _f_ = 0 # inverse flattening: 1 / f = a / (a - b) _f2 = None # 2nd flattening: (a - b) / b # rarely used _n = None # 3rd flattening: f / (2 - f) = (a - b) / (a + b) # for .A and .utm _a2 = None # a**2 _a2_ = None # (1 / a**2) for .ellipsiodalNvector.Cartesian.toNvector _a_b = None # (a / b) = 1 / (1 - f) for .ellipsoidalNvector.Nvector.toCartesian _b2 = None # b**2 _b_a = None # (b / a) = 1 - f for .formy, .R2, .Rgeocentric below # curvatures <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature> _a2_b = None # meridional radius of curvature at poles: a**2 / b (C{meter}) _b2_a = None # meridional radius of curvature at equator: b**2 / a (C{meter}) # eccentricities _e = None # (1st) eccentricity: sqrt(1 - (b / a)**2)) # for utm _e2 = None # (1st) eccentricity squared: f * (2 - f) = 1 - (b / a)**2 _e22 = None # 2nd eccentricity squared: e2 / (1 - e2) = e2 / (1 - f)**2 = (a / b)**2 - 1 _e32 = None # 3rd eccentricity squared: e2 / (2 - e2) = e2 / (1 + (1 - f)**2) = (a**2 - b**2) / (a**2 + b**2) _e12 = None # 1 - e2 = (1 - f)**2 for .ellipsoidalNvector.Cartesian.toNvector, .ecef .utm _e4 = None # e**4 = e2**2 for .ellipsoidalNvector.Cartesian.toNvector, .ecef _es_c = None # M{(1 - f) * exp(es_atanh(1))} # fixed earth radii from <https://WikiPedia.org/wiki/Earth_radius> _A = None # UTM meridional radius _L = None # quarter meridian: b * Elliptic(-e2 / (1 - e2)).E (C{meter}) _R1 = None # mean earth radius: (2 * a + b) / 3 per IUGG definition (C{meter}) _R2 = None # authalic radius: sqrt((a**2 + b**2 * atanh(e) / e) / 2) (C{meter}) # _c = None # authalic radius: equ (60) in Karney's "Algorithms for Geodesics" _R3 = None # volumetric radius: (a * a * b)**(1/3) (C{meter}) _Rb = None # biaxial mean earth radius: sqrt((a**2 * b**2) / 2) (C{meter}) _Rg = None # geometric mean earth radius: sqrt(a * b) (C{meter}) _Rr = None # rectifying radius: ((a**(3/2) + b**(3/2)) / 2)**(2/3) (C{meter}) _ab_90 = None # (a - b) / 90 # for .Rlat below _area = None # surface area: 4 * PI * R2**2 _b2_a2 = None # b**2 / a**2 = (b / a)**2 = _b_a**2 = (1 - f)**2 _volume = None # volume: 4 / 3 * PI * a**2 * b _AlphaKs = None # up to 8th-order Krüger Alpha series _BetaKs = None # up to 8th-order Krüger Beta series _KsOrder = 8 # Krüger series order (4, 6 or 8) _Mabcd = None # OSGR meridional coefficients _albersC = None # cached L{AlbersEqualAreaCylindrical} instance _elliptic = None # cached elliptic function L{Elliptic} instance _geodesic = None # cached C{karney._wrapped_.Geodesic} instance _Math = None # cached C{geographiclib.geomath.Math} module def __init__(self, a, b=None, f_=None, name=NN): '''New L{Ellipsoid} from I{equatorial} and I{polar} radius or I{equatorial} radius and I{inverse flattening}. @arg a: Equatorial (major) radius, semi-axis (C{meter}). @arg b: Optional, polar (minor) radius, semi-axis (C{meter}). @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0). @kwarg name: Optional, unique name (C{str}). @raise NameError: Ellipsoid with that B{C{name}} already exists. @raise ValueError: Invalid B{C{a}}, B{C{b}} or B{C{f_}}. @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced to M{1 / f_ = 0}, spherical. ''' try: a = Radius_(a=_float(a)) # low=EPS if b: b = Radius_(b=_float(b)) # low=EPS f = a_b2f(a, b) if f_ is None: f_ = f2f_(f) elif f_: b = a_f_2b(a, f_) # a * (f_ - 1) / f_ b = Radius_(b=_float(b)) # low=EPS f = a_b2f(a, b) else: # only a, spherical f = f_ = 0 b = a # superfluous if abs(f) < EPS or a == b or not f_: # spherical b = a f = f_ = 0 elif f > EPS1: # sanity check, see .ecef.Ecef.__init__ raise ValueError except (TypeError, ValueError) as x: raise _ValueError(instr(self, a=a, b=b, f_=f_), txt=str(x)) self._a = a self._b = b self._f = Float(f =_float(f)) self._f_ = Float(f_=_float(f_)) self._register(Ellipsoids, name) if f and f_: # see .test/testEllipsoidal.py self._assert(_1_0 / f, f_=f_, eps=_TOL) self._assert(_1_0 / f_, f=f, eps=_TOL) self._assert(self.b2_a2, e12=self.e12, eps=EPS) def __eq__(self, other): '''Compare this and an other ellipsoid. @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). @return: C{True} if equal, C{False} otherwise. ''' return self is other or (isinstance(other, Ellipsoid) and self.a == other.a and (self.b == other.b or self.f == other.f)) def _Kseries(self, *AB8Ks): '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha or Beta series coefficients per I{Karney} 2011, 'Transverse Mercator with an accuracy of a few nanometers', U{page 7, equations 35 and 36<https://Arxiv.org/pdf/1002.1417v3.pdf>}. @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series coefficient tuples. @return: I{Krüger} series coefficients (C{KsOrder}-tuple). @see: I{Karney}s 30-th order U{TMseries30 <https://GeographicLib.SourceForge.io/html/tmseries30.html>}. ''' k = self.KsOrder ns = fpowers(self.n, k) return tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k)) @property_RO def a(self): '''Get the I{equatorial} (major) radius, semi-axis (C{meter}). ''' return self._a equatoradius = a # = Requatorial @property_RO def a2(self): '''Get the I{equatorial} (major) radius I{squared} (C{float}), M{a**2}. ''' if self._a2 is None: self._a2 = Float(a2=self.a**2) return self._a2 @property_RO def a2_(self): '''Get the inverse of the I{equatorial} (major) radius I{squared} (C{float}), M{1 / a**2}. ''' if self._a2_ is None: self._a2_ = Float(a2_=_1_0 / self.a2) return self._a2_ # (1 / a**2) @property_RO def a_b(self): '''Get ratio M{a / b} (C{float}). ''' if self._a_b is None: self._a_b = Float(a_b=self.a / self.b if self.f else _1_0) return self._a_b @property_RO def a2_b(self): '''Get the I{polar} meridional radius of curvature (C{meter}), M{a**2 / b}, see C{rocPolar}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} and U{Moritz, H. (1980), Geodetic Reference System 1980 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}. @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' if self._a2_b is None: self._a2_b = Radius(a2_b=self.a2 / self.b if self.f else self.a) return self._a2_b @property_RO def a_f(self): '''Get I{equatorial} radius and I{flattening} (L{a_f2Tuple}). ''' return self._xnamed(a_f2Tuple(self.a, self.f)) @property_RO def A(self): '''Get the UTM I{meridional} radius (C{meter}). ''' if self._A is None: # <https://GeographicLib.SourceForge.io/html/transversemercator.html> A, n = self.a, self.n if n and self.f: A = A / (_1_0 + n) * (fsum_(65536, 16384 * n**2, 1024 * n**4, 256 * n**6, 100 * n**8, 49 * n**10) / 65536) # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> # A = self.a / (1 + n) * (fhorner(n**2, 16384, 4096, 256, 64, 25) / 16384) self._A = Radius(A=A) return self._A @property_RO def area(self): '''Get the ellipsoid's surface area (C{meter**2}), M{4 * PI * R2**2} or M{4 * PI * a**2}. ''' if self._area is None: r = self.R2 if self.f else self.a # authalic radius self._area = Float(area=_PI4 * r**2) return self._area @property_RO def AlphaKs(self): '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/html/tmseries30.html>} (C{KsOrder}-tuple). ''' if self._AlphaKs is None: self._AlphaKs = self._Kseries( # XXX int/int quotients may require from __future__ import division # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200), _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned _T(1424729850961/743921418240)) # PYCHOK unaligned return self._AlphaKs def _assert(self, val, eps=_TOL, f0=0, **name_value): '''(INTERNAL) Assert a C{name=value} vs C{val}. ''' for n, v in name_value.items(): if abs(v - val) > eps: raise _AssertionError('%s=%.12g vs %.12g' % (self._dot_(n), v, val)) return Float(v if self.f else f0, name=n) raise _AssertionError(self._dot_(self._assert.__name__), name_value) def auxAuthalic(self, lat, inverse=False): '''Compute the I{authalic} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and return the geodetic latitude (C{bool}). @return: The I{authalic} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/AuthalicLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and U{Snyder<https://pubs.er.USGS.gov/djvu/PP/PP_1395.pdf>}, p 16. ''' if self.isEllipsoidal: if self._albersC is None: from pygeodesy.albers import AlbersEqualAreaCylindrical as _AC self._albersC = _AC(datum=self, name=self.name) f = self._albersC._tanf if inverse else self._albersC._txif # PYCHOK attr lat = _atand(f(tan(Phi_(lat)))) # PYCHOK attr return _aux(lat, inverse, Ellipsoid.auxAuthalic.__name__) def auxConformal(self, lat, inverse=False): '''Compute the I{conformal} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and return the geodetic latitude (C{bool}). @return: The I{conformal} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/ConformalLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and U{Snyder<https://pubs.er.USGS.gov/djvu/PP/PP_1395.pdf>}, pp 15-16. ''' if self.isEllipsoidal: f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr lat = _atand(f(tan(Phi_(lat)))) # PYCHOK attr return _aux(lat, inverse, Ellipsoid.auxConformal.__name__) def auxGeocentric(self, lat, inverse=False): '''Compute the I{geocentric} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and return the I{geocentric} latitude (C{bool}). @return: The I{geocentric} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/GeocentricLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and U{Snyder<<https://pubs.er.USGS.gov/djvu/PP/PP_1395.pdf>}, pp 17-18. ''' if self.isEllipsoidal: f = (_1_0 / self.b2_a2) if inverse else self.b2_a2 lat = _atand(f * tan(Phi_(lat))) return _aux(lat, inverse, Ellipsoid.auxGeocentric.__name__) def auxIsometric(self, lat, inverse=False): '''Compute the I{isometric} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}). @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and return the geodetic latitude (C{bool}). @return: The I{isometric} (or geodetic) latitude in C{degrees}. @note: The I{isometric} latitude for geodetic C{+/-90} is far outside the C{[-90..+90]} range but the inverse thereof is the original geodetic latitude. @see: U{Inverse-/IsometricLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and U{Snyder<https://pubs.er.USGS.gov/djvu/PP/PP_1395.pdf>}, pp 15-16. ''' if self.isEllipsoidal: r = Phi_(lat, clip=0) lat = degrees( atan(self.es_tauf(sinh(r))) if inverse else asinh(self.es_taupf(tan(r)))) # clip=0, since auxIsometric(+/-90) is far outside [-90..+90] return _aux(lat, inverse, Ellipsoid.auxIsometric.__name__, clip=0) def auxParametric(self, lat, inverse=False): '''Compute the I{parametric} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and return the geodetic latitude (C{bool}). @return: The I{parametric} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/ParametricLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>}, and U{Snyder<https://pubs.er.USGS.gov/djvu/PP/PP_1395.pdf>}, p 18. ''' if self.isEllipsoidal: f = (_1_0 / self.b_a) if inverse else self.b_a lat = _atand(f * tan(Phi_(lat))) return _aux(lat, inverse, Ellipsoid.auxParametric.__name__) auxReduced = auxParametric # synonyms def auxRectifying(self, lat, inverse=False): '''Compute the I{rectifying} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and return the geodetic latitude (C{bool}). @return: The I{rectifying} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/RectifyingLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and U{Snyder<https://pubs.er.USGS.gov/djvu/PP/PP_1395.pdf>}, pp 16-17. ''' lat = Lat(lat) if 0 < abs(lat) < _90_0 and self.isEllipsoidal: if inverse: e = self._elliptic_e22 lat = degrees90(e.fEinv(e.cE * lat / _90_0)) lat = self.auxParametric(lat, inverse=True) else: lat = _90_0 * self.Llat(lat) / self.L return _aux(lat, inverse, Ellipsoid.auxRectifying.__name__) @property_RO def b(self): '''Get the I{polar} (minor) radius, semi-axis (C{meter}). ''' return self._b polaradius = b # = Rpolar @property_RO def b_a(self): '''Get ratio M{b / a} (C{float}), M{1 - f}. ''' if self._b_a is None: self._b_a = self._assert(self.b / self.a, b_a=_1_0 - self.f, f0=1) return self._b_a @property_RO def b2(self): '''Get the I{polar} (minor) radius I{squared} (C{float}), M{b**2}. ''' if self._b2 is None: self._b2 = Float(b2=self.b**2) return self._b2 @property_RO def b2_a(self): '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see C{rocMeridional(0)}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' if self._b2_a is None: self._b2_a = Radius(b2_a=self.b2 / self.a if self.f else self.b) return self._b2_a @property_RO def b2_a2(self): '''Get ratio M{(b / a)**2} (C{float}), M{(1 - f)**2}, see C{e12}. ''' if self._b2_a2 is None: self._b2_a2 = Float(b2_a2=self.b_a**2 if self.f else _1_0) return self._b2_a2 @property_RO def BetaKs(self): '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/html/tmseries30.html>} (C{KsOrder}-tuple). ''' if self._BetaKs is None: self._BetaKs = self._Kseries( # XXX int/int quotients may require from __future__ import division # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600), _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned _T(191773887257/3719607091200)) # PYCHOK unaligned return self._BetaKs @property_RO def c(self): '''Get the I{authalic} earth radius (C{meter}), see C{R2}. @note: Symbol C{c} in U{equation 60 <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>}. @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see C{rocPolar}. ''' return self.R2 # if self._R2 is None else self._R2 def degrees2m(self, deg, lat=0): '''Convert an angle to the distance along the equator or along a parallel at the given latitude. @arg deg: Angle (C{degrees}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Angle (C{degrees}). @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise ValueError: Invalid B{C{deg}} or B{C{lat}}. ''' return degrees2m(deg, radius=self.a, lat=lat) def distance2(self, lat0, lon0, lat1, lon1): '''I{Approximate} the distance and (initial) bearing between two points based on the U{local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. I{Suitable only for distances of several hundred Km or Miles and only between points not near-polar}. @arg lat0: From latitude (C{degrees}). @arg lon0: From longitude (C{degrees}). @arg lat1: To latitude (C{degrees}). @arg lon1: To longitude (C{degrees}). @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance} in same units as this ellipsoid's axes. @note: The meridional and prime_vertical radii of curvature are taken and scaled I{at the initial latitude}, see C{roc2}. @see: Function L{flatLocal}/L{hubeny}. ''' phi0 = Phi_(lat0=lat0) m, n = self.roc2_(phi0, scaled=True) m *= Phi_(lat1=lat1) - phi0 n *= Lam_(lon1=lon1) - Lam_(lon0=lon0) return Distance2Tuple(hypot(m, n), atan2b(n, m)) @property_RO def e(self): '''Get the I{(1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}. ''' if self._e is None: self._e = Float(e=sqrt(abs(self.e2)) if self.f else _0_0) return self._e eccentricity = e # eccentricity @property_RO def e2(self): '''Get the I{(1st) eccentricity squared} (C{float}), M{f * (2 - f) == 1 - (b / a)**2}, see C{a_b2e2}. ''' if self._e2 is None: self._e2 = self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f)) return self._e2 eccentricity2 = e2 # eccentricity squared @property_RO def e22(self): '''Get the I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2) == (a / b)**2 - 1}, see C{a_b2e22}. ''' if self._e22 is None: self._e22 = self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f)) return self._e22 eccentricity2nd2 = e2 # second eccentricity squared @property_RO def e32(self): '''Get the I{3rd eccentricity squared} (C{float}), M{e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2)}, see C{a_b2e32}. ''' if self._e32 is None: self._e32 = self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f)) return self._e32 eccentricity3rd2 = e32 # third eccentricity squared @property_RO def e12(self): '''Get M{1 - e**2} (C{float}), M{(1 - f)**2}, M{b**2 / a**2}, see C{b2_a2}. ''' if self._e12 is None: self._e12 = self._assert((_1_0 - self.f)**2, e12=_1_0 - self.e2, f0=1) return self._e12 # 1 - e2 @property_RO def e4(self): '''Get the I{(1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}. ''' if self._e4 is None: self._e4 = Float(e4=self.e2**2 if self.f else _0_0) return self._e4 def ecef(self, Ecef=None): '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter. @kwarg Ecef: ECEF class to use (L{EcefKarney}, L{EcefVeness} or L{EcefYou}). @return: An ECEF converter for this C{ellipsoid} (L{EcefKarney}, L{EcefVeness} or L{EcefYou}). @raise TypeError: Invalid B{C{Ecef}}. ''' return _4Ecef(self, Ecef) @property_RO def _elliptic_e22(self): '''(INTERNAL) Elliptic function for C{auxRectifying}, C{L}, C{Llat}. ''' if self._elliptic is None: from pygeodesy.elliptic import Elliptic self._elliptic = Elliptic(-abs(self.e22)) return self._elliptic def e2s(self, s): '''Compute norm M{sqrt(1 - e2 * s**2)}. @arg s: S value (C{scalar}). @return: Norm (C{float}). @raise ValueError: Invalid B{C{s}}. ''' return sqrt(self.e2s2(s)) def e2s2(self, s): '''Compute M{1 - e2 * s**2}. @arg s: S value (C{scalar}). @return: Result (C{float}). @raise ValueError: Invalid B{C{s}}. ''' try: r = _1_0 - self.e2 * Scalar(s=s)**2 if r < 0: raise ValueError(_negative_) except (TypeError, ValueError) as x: raise _ValueError(self._dot_(Ellipsoid.e2s2.__name__), s, txt=str(x)) return r @property_RO def es(self): '''Get the I{(1st) eccentricity signed} (C{float}). ''' # note, self.e is always non-negative return Float(es=copysign(self.e, self.f)) # see .ups def es_atanh(self, x): '''Compute M{es * atanh(es * x)} where I{es} is the I{signed} (1st) eccentricity. @raise ValueError: Invalid B{C{x}}. @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Math.html>}. ''' # note, self.e is always non-negative if self.f > 0: # oblate r = self.e * atanh(self.e * Scalar(x=x)) elif self.f < 0: # prolate r = -self.e * atan(self.e * Scalar(x=x)) else: r = 0 return r @property_RO def es_c(self): '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}. ''' if self._es_c is None: self._es_c = Float(es_c=(self.b_a * exp(self.es_atanh(_1_0))) if self.f else _1_0) return self._es_c def es_tauf(self, taup): '''Compute I{Karney}'s U{equations (19), (20) and (21) <https://ArXiv.org/abs/1002.1417>}. @see: U{Math::tauf<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Math.html>}. ''' T_ = Scalar(taup=taup) tol = max(abs(T_), _1_0) * _TOL e12 = self.e12 t = T_ / e12 T = Fsum(t) for _ in range(9): a = self.es_taupf(t) d = (T_ - a) * (_1_0 + e12 * t**2) / (e12 * hypot1(t) * hypot1(a)) t, d = T.fsum2_(d) if abs(d) < tol: break return t def es_taupf(self, tau): '''Compute I{Karney}'s U{equations (7), (8) and (9) <https://ArXiv.org/abs/1002.1417>}. @see: U{Math::taupf<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Math.html>}. ''' T = Scalar(tau=tau) t = hypot1(T) s = sinh(self.es_atanh(T / t)) return hypot1(s) * T - s * t @property_RO def f(self): '''Get the I{flattening} (C{float}), M{(a - b) / a}, C{0} for spherical. ''' return self._f flattening = f @property_RO def f_(self): '''Get the I{inverse flattening} (C{float}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}. ''' return self._f_ @property_RO def f2(self): '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}. ''' if self._f2 is None: self._f2 = self._assert(self.a_b - _1_0, f2=f2f2(self.f)) return self._f2 flattening2nd = f2 def _f_late(self, _f_a_b): '''(INTERNAL) Handle oblate, prolate, spherical case. ''' if self.f > 0: # oblate, a > b return _f_a_b(self.a, self.b) elif self.f < 0: # prolate, a < b return _f_a_b(self.b, self.a) else: return self.a @property_RO def geodesic(self): '''Get this ellipsoid's I{wrapped Karney} U{Geodesic <https://GeographicLib.SourceForge.io/html/python/code.html>}, provided the U{geographiclib<https://PyPI.org/project/geographiclib>} package is installed. ''' if self._geodesic is None: # if not self.isEllipsoidal: # raise _IsnotError('ellipsoidal', ellipsoid=self) from pygeodesy.karney import _wrapped self._geodesic = _wrapped.Geodesic(self.a, self.f) return self._geodesic @property_RO def _geodesic_Math2(self): '''(INTERNAL) Get this ellipsoid's I{wrapped Karney} C{Geodesic} and I{Karney}'s C{Math} class. ''' if Ellipsoid._Math is None: from pygeodesy.karney import _wrapped Ellipsoid._Math = _wrapped.Math return self.geodesic, Ellipsoid._Math def _hubeny2_(self, phi2, phi1, lam21): '''(INTERNAL) like function L{flatLocal_}/L{hubeny_} but returning the I{angular} distance in C{radians squared}. ''' m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=True) return hypot2(m * (phi2 - phi1), n * lam21) * self.a2_ @property_RO def isEllipsoidal(self): '''Is this model I{ellipsoidal} (C{bool})? ''' return self.f != 0 @property_RO def isOblate(self): '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or spherical otherwise. ''' return self.f > 0 @property_RO def isProlate(self): '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or spherical otherwise. ''' return self.f < 0 @property_RO def isSpherical(self): '''Is this model I{spherical} (C{bool})? ''' return self.f == 0 @property_doc_(''' the I{Krüger} series' order (C{int}).''') def KsOrder(self): '''Get the Krüger series order (C{int} 4, 6 or 8). ''' return self._KsOrder @KsOrder.setter # PYCHOK setter! def KsOrder(self, order): '''Set the I{Krüger} series' order. @arg order: New I{Krüger} series' order (C{int} 4, 6 or 8). @raise ValueError: Invalid B{C{order}}. ''' if order not in (4, 6, 8): raise _ValueError(order=order) if order != self._KsOrder: if self._AlphaKs: self._AlphaKs = None if self._BetaKs: self._BetaKs = None self._KsOrder = order @property_RO def L(self): '''Get the I{quarter meridian} C{L} (aka polar distance) the distance along a meridian between the equator and a pole (C{meter}), M{b * Elliptic(-e2 / (1 - e2)).E} or M{a * PI / 2}. ''' if self._L is None: if self.f: # complete integral 2nd ... # kind: Elliptic(-e2 / (1 - e2)).E L = self.b * self._elliptic_e22.cE else: # spherical L = self.a * PI_2 self._L = Distance(L=L) return self._L quarteradius = L # -meridian '''DEPRECATED, use C{L}.''' def Llat(self, lat): '''Return the I{meridional length}, the distance along a meridian between the equator and the given latitude (C{Meter}). @arg lat: Geodetic latitude (C{degrees90}). @return: The meridional length at B{C{lat}}, negative on southern hemisphere (C{meter}). ''' L = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phi_(lat) return Distance(Llat=self.b * L) Lmeridian = Llat # meridional distance def m2degrees(self, meter, lat=0): '''Convert distance to angle along equator or along a parallel at an other latitude. @arg meter: Distance (C{meter}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Angle (C{degrees}). @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise ValueError: Invalid B{C{meter}} or B{C{lat}}. ''' return m2degrees(meter, radius=self.a, lat=lat) @property_RO def Mabcd(self): '''Get the OSGR meridional coefficients (C{4-Tuple}), C{Airy130} only. ''' if self._Mabcd is None: n, n2, n3 = fpowers(self.n, 3) # PYCHOK false! self._Mabcd = (fdot((1, n, n2, n3), 4, 4, 5, 5) / _4_0, fdot( (n, n2, n3), 24, 24, 21) / _8_0, fdot( (n2, n3), 15, 15) / _8_0, 35 * n3 / 24.0) return self._Mabcd majoradius = a '''DEPRECATED, use C{a} or C{Requatorial}.''' minoradius = b '''DEPRECATED, use C{b} or C{Rpolar}.''' @property_RO def n(self): '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}. ''' if self._n is None: self._n = self._assert(a_b2n(self.a, self.b), n=f2n(self.f)) return self._n flattening3rd = n @property_RO def R1(self): '''Get the I{mean} earth radius per IUGG (C{meter}), M{(2 * a + b) / 3}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and method C{Rgeometric}. ''' if self._R1 is None: def _R1(a, b): return fsum_(a, a, b) / _3_0 self._R1 = Radius(R1=self._f_late(_R1)) return self._R1 Rmean = R1 @property_RO def R2(self): '''Get the I{authalic} earth radius (C{meter}), M{sqrt((a**2 + b**2 * atanh(e) / e) / 2)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}, C{area} and U{c<https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>}. ''' if self._R2 is None: r = self.b2_a2 if r < _1_0 and self.f > 0: # oblate e = self.e a = self.a elif r > _1_0 and self.f < 0: # prolate r = _1_0 / r # swap a and b e = sqrt(abs(_1_0 - r)) a = self.b else: # spherical e = 0 a = self.a r = a if e < EPS else \ a * sqrt((_1_0 + r * atanh(e) / e) * _0_5) self._R2 = Radius(R2=r) return self._R2 Rauthalic = R2 @property_RO def R3(self): '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}. ''' if self._R3 is None: def _R3(a, b): return cbrt(a**2 * b) self._R3 = Radius(R3=self._f_late(_R3)) return self._R3 Rvolumetric = R3 @property_RO def Rbiaxial(self): '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}. ''' if self._Rb is None: self._Rb = Radius(Rbiaxial=sqrt((self.a2 + self.b2) * _0_5) if self.f else self.a) return self._Rb Requatorial = a # for consistent naming def Rgeocentric(self, lat): '''Compute the I{geocentric} earth radius at the given latitude. @arg lat: Latitude (C{degrees90}). @return: Geocentric earth radius (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Geocentric Radius <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>} ''' c2, s2 = _c2_s2(Phi_(lat)) b2_a2_s2 = self.b2_a2 * s2 # R_ == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2)) # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2))) # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2)) # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2)) # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2)) return Radius(Rgeocentric=(self.a * sqrt((c2 + self.b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2))) if self.f else self.a) @property_RO def Rgeometric(self): '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}. @see: Method C{R1}. ''' if self._Rg is None: def _Rg(a, b): return sqrt(a * b) self._Rg = Radius(Rgeometric=self._f_late(_Rg)) return self._Rg Rs = Rgeometric # for backward compatibility '''DEPRECATED, use C{Rgeometric}.''' Rpolar = b # for consistent naming def Rlat(self, lat): '''Approximate the earth radius at the given latitude. @arg lat: Latitude (C{degrees90}). @return: Approximate earth radius (C{meter}). @raise ValueError: Invalid B{C{lat}}. ''' if self._ab_90 is None: self._ab_90 = (self.a - self.b) / _90_0 # r = a - (a - b) * |lat| / 90 r = self.a if lat: r -= self._ab_90 * min(abs(Lat(lat, clip=0)), _90_0) return Radius(Rlat=r) Rquadratic = Rbiaxial # synonyms @property_RO def Rrectifying(self): '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}. ''' if self._Rr is None: def _Rr(a, b): return cbrt2((sqrt3(a) + sqrt3(b)) * _0_5) self._Rr = Radius(Rrectifying=self._f_late(_Rr)) return self._Rr Rr = Rrectifying '''DEPRECATED, use C{Rrectifying}.''' def roc2(self, lat, scaled=False): '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of curvature at the given latitude. @arg lat: Latitude (C{degrees90}). @kwarg scaled: Scale prime_vertical by B{C{cos(radians(lat))}} (C{bool}). @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the radii of curvature. @raise ValueError: Invalid B{C{lat}}. @see: Method C{roc2_}, U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat), scaled=scaled) def roc2_(self, phi, scaled=False): '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of curvature at the given latitude. @arg phi: Latitude (C{radians}). @kwarg scaled: Scale prime_vertical by B{C{cos(phi)}} (C{bool}). @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the radii of curvature. @raise ValueError: Invalid B{C{phi}}. @see: Method C{roc2}, U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} and meridional and prime vertical U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' a = abs(Phi(phi)) r = self.e2s2(sin(a) if a < PI_2 else _1_0) if r < EPS: m = n = 0 # PYCHOK attr elif r < EPS1: n = self.a / sqrt(r) m = n * self.e12 / r # PYCHOK attr else: n = self.a m = n * self.e12 if scaled: n *= cos(a) if a < PI_2 else _0_0 return Curvature2Tuple(Radius(rocMeridional=m), Radius(rocPrimeVertical=n)) def rocBearing(self, lat, bearing): '''Compute the I{directional} radius of curvature at a given latitude and compass direction. @arg lat: Latitude (C{degrees90}). @arg bearing: Direction (compass C{degrees360}). @return: Directional radius of curvature (C{meter}). @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} ''' c2, s2 = _c2_s2(Bearing_(bearing)) m, n = self.roc2_(Phi_(lat)) if n < m: # == n / (c2 * n / m + s2) c2 *= n / m elif m < n: # == m / (c2 + s2 * m / n) s2 *= m / n n = m return Radius(rocBearing=n / (c2 + s2)) # == 1 / (c2 / m + s2 / n) def rocGauss(self, lat): '''Compute the I{Gaussian} radius of curvature at the given latitude. @arg lat: Latitude (C{degrees90}). @return: Gaussian radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} ''' # using ... # m, n = self.roc2_(Phi_(lat)) # return sqrt(m * n) # ... requires 1 or 2 sqrt c2, s2 = _c2_s2(Phi_(lat)) return Radius(rocGauss=self.b / (c2 + self.b2_a2 * s2)) def rocMean(self, lat): '''Compute the I{mean} radius of curvature at the given latitude. @arg lat: Latitude (C{degrees90}). @return: Mean radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} ''' m, n = self.roc2_(Phi_(lat)) return Radius(rocMean=2 * m * n / (m + n)) # == 2 / (1 / m + 1 / n) def rocMeridional(self, lat): '''Compute the I{meridional} radius of curvature at the given latitude. @arg lat: Latitude (C{degrees90}). @return: Meridional radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat)).meridional @property_RO def rocPolar(self): '''Get the I{polar} radius of curvature (C{meter}), see C{a2_b}. ''' return self.a2_b # if self._a2_b is None else self._a2_b def rocPrimeVertical(self, lat): '''Compute the I{prime-vertical}, I{normal} radius of curvature at the given latitude, aka the transverse radius of curvature. @arg lat: Latitude (C{degrees90}). @return: Prime-vertical radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat)).prime_vertical rocTransverse = rocPrimeVertical # synonyms def toStr(self, prec=8): # PYCHOK expected '''Return this ellipsoid as a text string. @kwarg prec: Optional number of decimals, unstripped (C{int}). @return: Ellipsoid attributes (C{str}). ''' return self._instr(prec, 'a', 'b', 'f_', 'f', 'f2', 'n', 'e', 'e2', 'e22', 'e32', 'L', 'R1', 'R2', 'R3') @property_RO def volume(self): '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * a**2 * b}. ''' if self._volume is None: self._volume = Float(volume=((self.a2 * self.b) if self.f > 0 else (self.b2 * self.a)) * _PI4_3) return self._volume class Ellipsoid2(Ellipsoid): '''Like L{Ellipsoid}, but specified by I{equatorial} radius and I{flattening}. ''' def __init__(self, a, f, name=NN): '''New L{Ellipsoid2}. @arg a: Equatorial (major) radius, semi-axis (C{meter}). @arg f: Flattening: (C{float} < 1.0). @kwarg name: Optional, unique name (C{str}). @raise NameError: Ellipsoid with that B{C{name}} already exists. @raise ValueError: Invalid B{C{a}} or B{C{f}}. @note: M{abs(f) < EPS} is forced to M{f = 0}, spherical. ''' try: a = Float_(a=a, low=EPS, high=None) # like Radius_ f = Float_(f=f, low=None, high=EPS1) Ellipsoid.__init__(self, a, a_f2b(a, f), name=name) except (TypeError, ValueError) as x: raise _ValueError(instr(self, a=a, f=f), txt=str(x)) def _spherical(f): '''(INTERNAL) C{True} for spherical or invalid C{f}. ''' return abs(f) < EPS or f > EPS1 def _spherical_(f_): '''(INTERNAL) C{True} for spherical or invalid C{f_}. ''' return abs(f_) < EPS or abs(f_) > _1_EPS def _spherical_a_b(a, b): '''(INTERNAL) C{True} for spherical or invalid C{a} and C{b}. ''' return a < EPS or b < EPS or abs(a - b) < EPS def a_b2e(a, b): '''Return C{e}, the I{eccentricity} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg b: Polar (minor) radius (C{scalar} > 0). @return: The eccentricity (C{float} or C{0}), M{sqrt(1 - (b / a)**2)}. @note: The result is C{0} for I{prolate} and near-spherical ellipsoids. ''' return Float(e=sqrt(abs(a_b2e2(a, b)))) def a_b2e2(a, b): '''Return C{e2}, the I{eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg b: Polar (minor) radius (C{scalar} > 0). @return: The eccentricity I{squared} (C{float} or C{0}), M{1 - (b / a)**2}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' return Float(e2=_0_0 if _spherical_a_b(a, b) else (_1_0 - (b / a)**2)) def a_b2e22(a, b): '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg b: Polar (minor) radius (C{scalar} > 0). @return: The 2nd eccentricity I{squared} (C{float} or C{0}), M{(a / b)**2 - 1}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' return Float(e22=_0_0 if _spherical_a_b(a, b) else ((a / b)**2 - _1_0)) def a_b2e32(a, b): '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg b: Polar (minor) radius (C{scalar} > 0). @return: The 3rd eccentricity I{squared} (C{float} or C{0}), M{(a**2 - b**2) / (a**2 + b**2)}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' a2, b2 = a**2, b**2 return Float(e32=_0_0 if _spherical_a_b(a2, b2) else ((a2 - b2) / (a2 + b2))) def a_b2f(a, b): '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg b: Polar (minor) radius (C{scalar} > 0). @return: The flattening (C{float} or C{0}), M{(a - b) / a}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' f = 0 if _spherical_a_b(a, b) else (float(a - b) / a) return Float(f=_0_0 if _spherical(f) else f) def a_b2f_(a, b): '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg b: Polar (minor) radius (C{scalar} > 0). @return: The inverse flattening (C{float} or C{0}), M{a / (a - b)}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' f_ = 0 if _spherical_a_b(a, b) else (a / float(a - b)) return Float(f_=_0_0 if _spherical_(f_) else f_) def a_b2f2(a, b): '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg b: Polar (minor) radius (C{scalar} > 0). @return: The 2nd flattening (C{float} or C{0}), M{(a - b) / b}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' t = 0 if _spherical_a_b(a, b) else float(a - b) return Float(f2=_0_0 if abs(t) < EPS else (t / b)) def a_b2n(a, b): '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg b: Polar (minor) radius (C{scalar} > 0). @return: The 3rd flattening (C{float} or C{0}), M{(a - b) / (a + b)}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' t = 0 if _spherical_a_b(a, b) else float(a - b) return Float(n=_0_0 if abs(t) < EPS else (t / (a + b))) def a_f2b(a, f): '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1). @return: The polar (minor) radius (C{float}), M{a * (1 - f)}. ''' return Float(b=a if _spherical(f) else (a * (_1_0 - f))) def a_f_2b(a, f_): '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}. @arg a: Equatorial (major) radius (C{scalar} > 0). @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The polar (minor) radius (C{float}), M{a * (f_ - 1) / f_}. ''' return Float(b=a if _spherical_(f_) else (a * (f_ - _1_0) / f_)) def b_f2a(b, f): '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}. @arg b: Polar (minor) radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1). @return: The equatorial (major) radius (C{float}), M{b / (1 - f)}. ''' t = _1_0 - f return Float(a=b if _spherical(f) or abs(t) < EPS else (b / t)) def b_f_2a(b, f_): '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}. @arg b: Polar (minor) radius (C{scalar} > 0). @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The equatorial (major) radius (C{float}), M{b * f_ / (f_ - 1)}. ''' t = f_ - _1_0 return Float(a=b if _spherical_(f_) or abs(t) < EPS else (b * f_ / t)) def f2e2(f): '''Return C{e2}, the I{eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1). @return: The (1st) eccentricity I{squared} (C{float} < 1), M{f * (2 - f)}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' return Float(e2=_0_0 if _spherical(f) else (f * (_2_0 - f))) def f2e22(f): '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1). @return: The 2nd eccentricity I{squared} (C{float} > -1 or C{INF}), M{f * (2 - f) / (1 - f)**2}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}. ''' # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2 t = (_1_0 - f)**2 return Float(e22=INF if t < EPS else (f2e2(f) / t)) # PYCHOK type def f2e32(f): '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1). @return: The 3rd eccentricity I{squared} (C{float}), M{f * (2 - f) / (1 + (1 - f)**2)}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}. ''' # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2) e2 = f2e2(f) return Float(e32=e2 / (_2_0 - e2)) def f_2f(f_): '''Return C{f}, the I{flattening} for a given I{inverse flattening}. @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The flattening (C{float} or C{0}), M{1 / f_}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' f = 0 if _spherical_(f_) else _1_0 / f_ return Float(f=_0_0 if _spherical(f) else f) # PYCHOK type def f2f_(f): '''Return C{f_}, the I{inverse flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1). @return: The inverse flattening (C{float} or C{0}), M{1 / f}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. ''' f_ = 0 if _spherical(f) else _1_0 / f return Float(f_=_0_0 if _spherical_(f_) else f_) # PYCHOK type def f2f2(f): '''Return C{f2}, the I{2nd flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1). @return: The 2nd flattening (C{float} or C{INF}), M{f / (1 - f)}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' t = _1_0 - f return Float(f2=_0_0 if _spherical(f) else (INF if abs(t) < EPS else (f / t))) # PYCHOK type def f2n(f): '''Return C{n}, the I{3rd flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1). @return: The 3rd flattening (-1 < C{float} < 1), M{f / (2 - f)}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' return Float(n=_0_0 if _spherical(f) else (f / float(_2_0 - f))) def n2e2(n): '''Return C{e2}, the I{eccentricity squared} for a given I{3rd flattening}. @arg n: The 3rd flattening (-1 < C{scalar} < 1). @return: The (1st) eccentricity I{squared} (C{float} or -INF), M{4 * n / (1 + n)**2}. @note: The result is negative for I{prolate} or C{0} for near-spherical ellipsoids. @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}. ''' t = (_1_0 + n)**2 return Float(e2=_0_0 if abs(n) < EPS else (-INF if t < EPS else (_4_0 * n / t))) def n2f(n): '''Return C{f}, the I{flattening} for a given I{3rd flattening}. @arg n: The 3rd flattening (-1 <= C{scalar} <= 1). @return: The flattening (C{float} or -INF), M{2 * n / (1 + n)}. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' t = n + _1_0 f = 0 if abs(n) < EPS else (-INF if t < EPS else _2_0 * n / t) return Float(f=_0_0 if _spherical(f) else f) Ellipsoids = _NamedEnum('Ellipsoids', Ellipsoid) # registered ellipsods # <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html> # <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/ # LogicalDictionary/StandardValues/ellipsoid.html> # <https://kb.OSU.edu/dspace/handle/1811/77986> Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid> Airy1830 = Ellipsoid(6377563.396, None, 299.3249646, 'Airy1830'), # b=6356256.909 AiryModified = Ellipsoid(6377340.189, None, 299.3249646, 'AiryModified'), # b=6356034.448 # ANS = Ellipsoid(6378160.0, None, 298.25, 'ANS'), # b=6356774.719 Australia1966 = Ellipsoid(6378160.0, None, 298.25, 'Australia1966'), # b=6356774.719 # Bessel1841 = Ellipsoid(6377397.155, 6356078.963, 299.152815351, 'Bessel1841'), Bessel1841 = Ellipsoid(6377397.155, 6356078.962818, 299.1528128, 'Bessel1841'), Clarke1866 = Ellipsoid(6378206.4, 6356583.8, 294.978698214, 'Clarke1866'), Clarke1880 = Ellipsoid(6378249.145, 6356514.86954978, 293.465, 'Clarke1880'), Clarke1880IGN = Ellipsoid(6378249.2, 6356515.0, 293.466021294, 'Clarke1880IGN'), Clarke1880Mod = Ellipsoid(6378249.145, 6356514.96582849, 293.4663, 'Clarke1880Mod'), CPM1799 = Ellipsoid(6375738.7, 6356671.92557493, 334.39, 'CPM1799'), # Comm. des Poids et Mesures Delambre1810 = Ellipsoid(6376428.0, 6355957.92616372, 311.5, 'Delambre1810'), # Belgium Engelis1985 = Ellipsoid(6378136.05, 6356751.32272154, 298.2566, 'Engelis1985'), Everest1969 = Ellipsoid(6377295.664, 6356094.667915, 300.8017, 'Everest1969'), Fisher1968 = Ellipsoid(6378150.0, 6356768.33724438, 298.3, 'Fisher1968'), GEM10C = Ellipsoid(6378137.0, 6356752.31424783, 298.2572236, 'GEM10C'), GRS67 = Ellipsoid(6378160.0, None, 298.247167427, 'GRS67'), # Lucerne b=6356774.516 GRS80 = Ellipsoid(6378137.0, 6356752.314140347, 298.257222101, 'GRS80'), # ITRS, ETRS89 Helmert1906 = Ellipsoid(6378200.0, 6356818.16962789, 298.3, 'Helmert1906'), IERS1989 = Ellipsoid(6378136.0, None, 298.257, 'IERS1989'), # b=6356751.302 IERS1992TOPEX = Ellipsoid(6378136.3, 6356751.61659215, 298.257223563, 'IERS1992TOPEX'), # IERS/TOPEX/Poseidon/McCarthy IERS2003 = Ellipsoid(6378136.6, 6356751.85797165, 298.25642, 'IERS2003'), Intl1924 = Ellipsoid(6378388.0, None, 297.0, 'Intl1924'), # aka Hayford b=6356911.946 Intl1967 = Ellipsoid(6378157.5, 6356772.2, 298.24961539, 'Intl1967'), # New Int'l Krassovski1940 = Ellipsoid(6378245.0, 6356863.01877305, 298.3, 'Krassovski1940'), # spelling Krassowsky1940 = Ellipsoid(6378245.0, 6356863.01877305, 298.3, 'Krassowsky1940'), # spelling Maupertuis1738 = Ellipsoid(6397300.0, 6363806.28272251, 191.0, 'Maupertuis1738'), # France Mercury1960 = Ellipsoid(6378166.0, 6356784.28360711, 298.3, 'Mercury1960'), Mercury1968Mod = Ellipsoid(6378150.0, 6356768.33724438, 298.3, 'Mercury1968Mod'), NWL1965 = Ellipsoid(6378145.0, 6356759.76948868, 298.25, 'NWL1965'), # Naval Weapons Lab. OSU86F = Ellipsoid(6378136.2, 6356751.51693008, 298.2572236, 'OSU86F'), OSU91A = Ellipsoid(6378136.3, 6356751.6165948, 298.2572236, 'OSU91A'), # Plessis1817 = Ellipsoid(6397523.0, 6355863.0, 153.56512242, 'Plessis1817'), # XXX incorrect? Plessis1817 = Ellipsoid(6376523.0, 6355862.93325557, 308.64, 'Plessis1817'), # XXX IGN France 1972 SGS85 = Ellipsoid(6378136.0, 6356751.30156878, 298.257, 'SGS85'), # Soviet Geodetic System SoAmerican1969 = Ellipsoid(6378160.0, 6356774.71919531, 298.25, 'SoAmerican1969'), # South American Struve1860 = Ellipsoid(6378298.3, 6356657.14266956, 294.73, 'Struve1860'), WGS60 = Ellipsoid(6378165.0, 6356783.28695944, 298.3, 'WGS60'), WGS66 = Ellipsoid(6378145.0, 6356759.76948868, 298.25, 'WGS66'), WGS72 = Ellipsoid(6378135.0, None, 298.26, 'WGS72'), # b=6356750.52 WGS84 = Ellipsoid(6378137.0, None, 298.257223563, 'WGS84'), # GPS b=6356752.31425 # Prolate = Ellipsoid(6356752.3, 6378137.0, None, 'Prolate'), Sphere = Ellipsoid(R_M, R_M, _0_0, 'Sphere'), # pseudo SphereAuthalic = Ellipsoid(R_FM, R_FM, _0_0, 'SphereAuthalic'), # pseudo SpherePopular = Ellipsoid(R_MA, R_MA, _0_0, 'SpherePopular'), # EPSG:3857 Spheroid ) __all__ += _ALL_DOCS(Curvature2Tuple) if __name__ == '__main__': for E in (Ellipsoids.WGS84, Ellipsoids.GRS80, # NAD83, Ellipsoids.Sphere, Ellipsoids.SpherePopular, Ellipsoid(Ellipsoids.WGS84.b, Ellipsoids.WGS84.a, name='_Prolate')): if E.f: f_ = 'f_=1/%.10F' % (_1_0 / E.f,) else: f_ = 'f_=' + _n_a_ e = (E.a - E.b) / (E.a + E.b) - E.n t = (E.toStr(prec=10), 'A=%r, e=%s, %s, n=%s(%s)' % (E.A, fstr(E.e, prec=13, fmt=_e), f_, fstr(E.n, prec=13, fmt=_Fmt), fstr(e, prec=3, fmt=_e),), '%s=(%s)' % (Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20),), '%s= (%s)' % (Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20),), '%s= %s' % ('KsOrder', E.KsOrder), '%s= (%s)' % (Ellipsoid.Mabcd.name, fstr(E.Mabcd, prec=20),)) print('\nEllipsoid.%s: %s' % (E.name, ',\n '.join(t))) # __doc__ of this file for e in (Ellipsoids,): t = [NN] + repr(e).split('\n') print('\n@var '.join(i.strip(',') for i in t)) # **) MIT License # # Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE. # % python -m pygeodesy.ellipsoids # Ellipsoid.WGS84: name='WGS84', a=6378137, b=6356752.3142451793, f_=298.257223563, f=0.0033528107, f2=0.0033640898, n=0.0016792204, e=0.0818191908, e2=0.00669438, e22=0.0067394967, e32=0.0033584313, L=10001965.7293127235, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.790009154, # A=6367449.145823414, e=8.1819190842622e-02, f_=1/298.2572235630, n=0.0016792203864(-2.168e-19), # AlphaKs=(0.00083773182062447786, 0.00000076085277735726, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0), # BetaKs= (0.00083773216405795667, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0), # KsOrder= 8, # Mabcd= (1.00168275103155868244, 0.00504613293193333871, 0.00000529596776243457, 0.00000000690525779769) # Ellipsoid.GRS80: name='GRS80', a=6378137, b=6356752.3141403468, f_=298.257222101, f=0.0033528107, f2=0.0033640898, n=0.0016792204, e=0.081819191, e2=0.00669438, e22=0.0067394968, e32=0.0033584313, L=10001965.7292304561, R1=6371008.7713801153, R2=6371007.1808835138, R3=6371000.7899741307, # A=6367449.145771043, e=8.1819191042833e-02, f_=1/298.2572221009, n=0.0016792203946(0.0e+00), # AlphaKs=(0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0), # BetaKs= (0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0), # KsOrder= 8, # Mabcd= (1.00168275103983916985, 0.0050461329567537995, 0.00000529596781448937, 0.00000000690525789941) # Ellipsoid.Sphere: name='Sphere', a=6371008.7714149999, b=6371008.7714149999, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, L=10007557.1761167478, R1=6371008.7714149999, R2=6371008.7714149999, R3=6371008.7714149999, # A=6371008.771415, e=0.0e+00, f_=n/a, n=0.0(0.0e+00), # AlphaKs=(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # BetaKs= (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # KsOrder= 8, # Mabcd= (1.0, 0.0, 0.0, 0.0) # Ellipsoid.SpherePopular: name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, L=10018754.171394622, R1=6378137, R2=6378137, R3=6378137, # A=6378137.0, e=0.0e+00, f_=n/a, n=0.0(0.0e+00), # AlphaKs=(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # BetaKs= (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # KsOrder= 8, # Mabcd= (1.0, 0.0, 0.0, 0.0) # Ellipsoid._Prolate: name='_Prolate', a=6356752.3142451793, b=6378137, f_=-297.257223563, f=-0.0033640898, f2=-0.0033528107, n=-0.0016792204, e=0.0820944379, e2=-0.0067394967, e22=-0.00669438, e32=-0.0033584313, L=10035500.5204500332, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.790009154, # A=6367449.145823415, e=8.2094437949696e-02, f_=1/-297.2572235630, n=-0.0016792203864(0.0e+00), # AlphaKs=(-0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0), # BetaKs= (-0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0), # KsOrder= 8, # Mabcd= (0.99832429842120640195, -0.00502921424529705757, 0.00000527821138524052, -0.00000000690525779769)