# Coverage for pygeodesy/ellipsoids.py : 97%

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1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 | # -*- 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=10001867.69724907, 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=10001964.6315999, 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=9999999.11003639, 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=10002017.83655714, 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=10002012.0443814, 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 copysign, 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, _1_EPS, INF, NN, PI4, PI_2, R_M, _a_, \ _exceeds_eps_fmt_, _f_, _float, _floatuple as _T, \ _item_is, joined_, _lat_, _meridional_, _negative_, \ _prime_vertical_, _vs_, _0_0, _0_1, _0_5, \ _1_0, _2_0, _4_0, _90_0 from pygeodesy.interns import _0_125, _0_25, _3_0 # PYCHOK used! from pygeodesy.lazily import _ALL_LAZY from pygeodesy.named import _NamedEnum, _NamedEnumItem, _NamedTuple, _Pass from pygeodesy.namedTuples import Distance2Tuple from pygeodesy.streprs import _e, _g, instr, _Fmt, fstr, strs, unstr from pygeodesy.units import Bearing_, Distance, Float, Float_, Lam_, Lat, \ Meter, Phi, Phi_, Radius, Radius_, Scalar from pygeodesy.utily import atand, atan2b, degrees90, degrees2m, \ m2degrees, m2km, m2NM, m2SM from math import asinh, atan, atanh, 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 earth radius (C{meter}), WGS84, EPSG:3785 R_MB = Radius(R_MB=6356752.0) # polar 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__ = '20.10.29' _TOL = sqrt(_0_1 * EPS) # for Ellipsoid.estauf, in .ups, testEllipsoidal.py def _aux(lat, inverse, auxLat, clip=90): '''Return named auxiliary latitude degrees. ''' return Lat(lat, clip=clip, name=_lat_ if inverse else auxLat) def _s2_c2(phi): '''Return 2-tuple C{(sin(B{phi})**2, cos(B{phi})**2)}. ''' c2 = cos(phi)**2 return (_1_0 - c2), 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} in C{meter} and scalar I{flattening} C{f}. @see: Class L{Ellipsoid2}. ''' _Names_ = (_a_, _f_) _Units_ = (_Pass, _Pass) def __new__(cls, a, f): '''New L{a_f2Tuple} ellipsoid specification. @arg a: Equatorial radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: An L{a_f2Tuple}C{(a, f)} instance. @raise UnitError: Invalid B{C{a}} or B{C{f}}. @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. Negative C{B{f}} produces a I{prolate} ellipsoid. ''' 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 radius, semi-axis (C{meter}) _b = 0 # polar 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 .ecef, .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}) _a2_b2 = None # a**2 / b**2 = (a / b)**2 = _a_b**2 = 1 / (1 - e2) = 1 / _e12 _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 = 1 - _e2 = _e12 _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 radius, semi-axis (C{meter}). @arg b: Optional, polar 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} radius, semi-axis (C{meter}). ''' return self._a equatoradius = a # = Requatorial @property_RO def a2(self): '''Get the I{equatorial} 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} 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 the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b}. ''' 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 L{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 a2_b2(self): '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}), M{(a / b)**2} == M{1 / (1 - e**2)}. ''' if self._a2_b2 is None: self._a2_b2 = Float(a2_b2=self.a_b**2 if self.f else _1_0) return self._a2_b2 @property_RO def a_f(self): '''Get the 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>} (L{KsOrder}C{-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_0, **name_value): '''(INTERNAL) Assert a C{name=value} vs C{val}. ''' for n, v in name_value.items(): if abs(v - val) > eps: t = (v, _vs_, val) t = joined_(*strs(t, prec=12, fmt=_g)) t = _item_is(self._dot_(n), t) raise _AssertionError(t, txt=_exceeds_eps_fmt_ % (eps,)) return Float(v if self.f else f0, name=n) raise _AssertionError(unstr(self._dot_(self._assert.__name__), val, eps=eps, f0=f0, **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 = self.a2_b2 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 = self.a_b 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} radius, semi-axis (C{meter}). ''' return self._b polaradius = b # = Rpolar @property_RO def b_a(self): '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a} == 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_0) return self._b_a @property_RO def b2(self): '''Get the I{polar} 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 L{rocMeridional}C{(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 the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2} == M{(1 - f)**2} == M{1 - e**2}, see L{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>} (L{KsOrder}C{-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 L{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 L{a_b2e}. ''' if self._e is None: self._e = Float(e=sqrt(self.e2abs) 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 L{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 e2abs(self): '''Get C{abs} value of the I{(1st) eccentricity squared} (C{float}). ''' return abs(self._e2) @property_RO def e12(self): '''Get 1 less I{(1st) eccentricity squared} (C{float}), M{1 - e**2} == M{(1 - f)**2} == M{b**2 / a**2}, see L{b2_a2}. ''' if self._e12 is None: self._e12 = self._assert((_1_0 - self.f)**2, e12=_1_0 - self.e2, f0=_1_0) return self._e12 # 1 - e2 @property_RO def e22(self): '''Get the I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2) == (a / b)**2 - 1}, see L{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 e22abs(self): '''Get C{abs} value of the I{(2nd) eccentricity squared} (C{float}). ''' return abs(self._e22) @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 L{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 e32abs(self): '''Get C{abs} value of the I{(3rd) eccentricity squared} (C{float}). ''' return abs(self._e32) @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 helper 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: Sine 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_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>}. ''' # To lowest order in e^2, taup = (1 - e^2) * tau, so use starting # guess tau = taup / (1 - e^2) = taup * a^2 / b^2. This starting # guess is the geocentric latitude which, to first order in the # flattening, is equal to the conformal latitude, auxConformal. T_ = Scalar(taup=taup) tol = max(abs(T_), _1_0) * _TOL e = self.a2_b2 # == _1_0 / self.e12 t = T_ * e T = Fsum(t) for _ in range(9): a = self.es_taupf(t) d = (T_ - a) * (e + t**2) / (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, negative for prolate. ''' 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 L{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 L{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}), see L{AlphaKs}, L{BetaKs}.''') def KsOrder(self): '''Get the Krüger series order (C{int} 4, 6 or 8), see L{AlphaKs}, L{BetaKs}. ''' return self._KsOrder @KsOrder.setter # PYCHOK setter! def KsOrder(self, order): '''Set the I{Krüger} series' order, see L{AlphaKs}, L{BetaKs}. @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) * _0_25, fdot( (n, n2, n3), 24, 24, 21) * _0_125, fdot( (n2, n3), 15, 15) * _0_125, 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 L{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 I{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: def _R2(a, b): r = (b / a)**2 if EPS < r < EPS1: e = sqrt( _1_0 - r) a *= sqrt((_1_0 + r * atanh(e) / e) * _0_5) return a self._R2 = Radius(R2=self._f_late(_R2)) 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>} ''' s2, c2 = _s2_c2(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=(sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2)) * self.a) 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 roc1_(self, sa, ca=None): '''Compute the I{prime-vertical}, I{normal} radius of curvature at the given latitude, I{unscaled}. @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]). @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0]) to use an alternate formula. @return: The prime-vertical radius of curvature (C{float}). @note: The delta between both formulae with C{Ellipsoids.WGS84} is less than 2 nanometer over the entire latitude range. @see: Method L{roc2_} and class L{EcefYou}. ''' if not self.f: # spherical n = self.a elif ca is None: r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward n = _0_0 if r < EPS else (self.a / sqrt(r)) elif ca: # derived from EcefYou.forward h = hypot(ca, self.b_a * sa) if sa else abs(ca) n = self.a / h elif sa: n = self.a2_b / abs(sa) else: n = self.a return n 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: Methods L{roc2_} and L{roc1_} and 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>}. ''' 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: Methods L{roc2} and L{roc1_} and 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 else: n = self.a / sqrt(r) m = n * self.e12 / r # PYCHOK attr 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>} ''' s2, c2 = _s2_c2(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 s2, c2 = _s2_c2(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: Methods L{roc2} and L{roc2_} and 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 L{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: Methods L{roc2}, L{roc2_} and L{roc1_} and 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_0) 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 radius, semi-axis (C{meter}). @arg f: Flattening: (C{float} < 1.0, negative for prolate). @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: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. Negative C{B{f}} produces a I{prolate} ellipsoid. ''' 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 radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The eccentricity (C{float} or C{0}), M{sqrt(1 - (b / a)**2)}. @note: The result is always non-negative and C{0} for I{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 radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The eccentricity I{squared} (C{float} or C{0}), M{1 - (b / a)**2}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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 radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The 2nd eccentricity I{squared} (C{float} or C{0}), M{(a / b)**2 - 1}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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 radius (C{scalar} > 0). @arg b: Polar 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 positive for I{oblate}, negative for I{prolate} or C{0} for I{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 radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The flattening (C{float} or C{0}), M{(a - b) / a}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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 radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The inverse flattening (C{float} or C{0}), M{a / (a - b)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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 radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The 2nd flattening (C{float} or C{0}), M{(a - b) / b}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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 radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The 3rd flattening (C{float} or C{0}), M{(a - b) / (a + b)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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 radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The polar 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 radius (C{scalar} > 0). @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The polar 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 radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The equatorial 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 radius (C{scalar} > 0). @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}. ''' t = f_ - _1_0 return Float(a=b if _spherical_(f_) or abs(t) < EPS or abs(t - f_) < 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, negative for I{prolate}). @return: The (1st) eccentricity I{squared} (C{float} < 1), M{f * (2 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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, negative for I{prolate}). @return: The 2nd eccentricity I{squared} (C{float} > -1 or C{INF}), M{f * (2 - f) / (1 - f)**2}. @note: The result is positive for I{oblate}, 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, negative for I{prolate}). @return: The 3rd eccentricity I{squared} (C{float}), M{f * (2 - f) / (1 + (1 - f)**2)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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 positive for I{oblate}, negative for I{prolate} or C{0} for I{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, negative for I{prolate}). @return: The inverse flattening (C{float} or C{0}), M{1 / f}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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, negative for I{prolate}). @return: The 2nd flattening (C{float} or C{INF}), M{f / (1 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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, negative for I{prolate}). @return: The 3rd flattening (-1 <= C{float} < 1), M{f / (2 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{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 positive for I{oblate}, negative for I{prolate} or C{0} for I{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(R_MA, 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 ) if __name__ == '__main__': for E in (Ellipsoids.WGS84, Ellipsoids.GRS80, # NAD83, Ellipsoids.Sphere, Ellipsoids.SpherePopular, Ellipsoid(Ellipsoids.WGS84.b, Ellipsoids.WGS84.a, name='_Prolate')): e = f2n(E.f) - E.n t = (E.toStr(prec=10), 'A=%r, e=%s, f_=%s, n=%s(%s)' % (E.A, fstr(E.e, prec=13, fmt=_e), 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_=298.257223563, n=0.0016792203864(0.0e+00), # 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=10001965.7292304579, R1=6371008.7713801153, R2=6371007.1808835138, R3=6371000.7899741307, # A=6367449.145771043, e=8.1819191042833e-02, f_=298.257222101, 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_=0.0, 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_=0.0, 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_=-297.2572235629972, 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)