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# -*- coding: utf-8 -*- 

 

u'''Classes L{Datum}, L{Ellipsoid} and L{Transform} and registries thereof 

L{Datums}, L{Ellipsoids} and L{Transforms}, respectively. 

 

Pure Python implementation of geodesy tools for ellipsoidal earth models, 

including datums and ellipsoid parameters for different geographic coordinate 

systems and methods for converting between them and to cartesian coordinates. 

Transcribed from JavaScript originals by I{(C) Chris Veness 2005-2016} and 

published under the same MIT Licence**, see U{latlon-ellipsoidal.js 

<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}. 

 

Historical geodetic datums: a latitude/longitude point defines a geographic 

location on or above/below the earth’s surface, measured in degrees from 

the equator and the International Reference Meridian and meters above the 

ellipsoid, and based on a given datum. The datum is based on a reference 

ellipsoid and tied to geodetic survey reference points. 

 

Modern geodesy is generally based on the WGS84 datum (as used for instance 

by GPS systems), but previously various reference ellipsoids and datum 

references were used. 

 

The UK Ordnance Survey National Grid References are still based on the otherwise 

historical OSGB36 datum, q.v. U{Ordnance Survey 'A guide to coordinate systems 

in Great Britain', Section 6<https://www.OrdnanceSurvey.co.UK/docs/support/ 

guide-coordinate-systems-great-britain.pdf>}. 

 

@newfield example: Example, Examples 

 

@var Datums.BD72: Datum(name='BD72', ellipsoid=Ellipsoids.Intl1924, transform=Transforms.BD72) 

@var Datums.DHDN: Datum(name='DHDN', ellipsoid=Ellipsoids.Bessel1841, transform=Transforms.DHDN) 

@var Datums.ED50: Datum(name='ED50', ellipsoid=Ellipsoids.Intl1924, transform=Transforms.ED50) 

@var Datums.GDA2020: Datum(name='GDA2020', ellipsoid=Ellipsoids.GRS80, transform=Transforms.WGS84) 

@var Datums.GRS80: Datum(name='GRS80', ellipsoid=Ellipsoids.GRS80, transform=Transforms.WGS84) 

@var Datums.Irl1975: Datum(name='Irl1975', ellipsoid=Ellipsoids.AiryModified, transform=Transforms.Irl1975) 

@var Datums.Krassovski1940: Datum(name='Krassovski1940', ellipsoid=Ellipsoids.Krassovski1940, transform=Transforms.Krassovski1940) 

@var Datums.Krassowsky1940: Datum(name='Krassowsky1940', ellipsoid=Ellipsoids.Krassowsky1940, transform=Transforms.Krassowsky1940) 

@var Datums.MGI: Datum(name='MGI', ellipsoid=Ellipsoids.Bessel1841, transform=Transforms.MGI) 

@var Datums.NAD27: Datum(name='NAD27', ellipsoid=Ellipsoids.Clarke1866, transform=Transforms.NAD27) 

@var Datums.NAD83: Datum(name='NAD83', ellipsoid=Ellipsoids.GRS80, transform=Transforms.NAD83) 

@var Datums.NTF: Datum(name='NTF', ellipsoid=Ellipsoids.Clarke1880IGN, transform=Transforms.NTF) 

@var Datums.OSGB36: Datum(name='OSGB36', ellipsoid=Ellipsoids.Airy1830, transform=Transforms.OSGB36) 

@var Datums.Potsdam: Datum(name='Potsdam', ellipsoid=Ellipsoids.Bessel1841, transform=Transforms.Bessel1841) 

@var Datums.Sphere: Datum(name='Sphere', ellipsoid=Ellipsoids.Sphere, transform=Transforms.WGS84) 

@var Datums.TokyoJapan: Datum(name='TokyoJapan', ellipsoid=Ellipsoids.Bessel1841, transform=Transforms.TokyoJapan) 

@var Datums.WGS72: Datum(name='WGS72', ellipsoid=Ellipsoids.WGS72, transform=Transforms.WGS72) 

@var Datums.WGS84: Datum(name='WGS84', ellipsoid=Ellipsoids.WGS84, transform=Transforms.WGS84) 

 

@var Ellipsoids.Airy1830: Ellipsoid(name='Airy1830', a=6377563.396, b=6356256.909, f_=299.3249646, f=0.003340851, e=0.081673374, e2=0.00667054, e12=0.99332946, e22=0.006715335, n=0.00167322, R1=6370461.233666666, R2=6370459.654589442, R3=6370453.309866445, Rr=6366914.608805893, Rs=6366901.239881964) 

@var Ellipsoids.AiryModified: Ellipsoid(name='AiryModified', a=6377340.189, b=6356034.448, f_=299.3249646, f=0.003340851, e=0.081673374, e2=0.00667054, e12=0.99332946, e22=0.006715335, n=0.00167322, R1=6370238.275333334, R2=6370236.696361165, R3=6370230.351810658, Rr=6366691.774649803, Rs=6366678.406194146) 

@var Ellipsoids.Australia1966: Ellipsoid(name='Australia1966', a=6378160, b=6356774.719, f_=298.25, f=0.003352892, e=0.08182018, e2=0.006694542, e12=0.993305458, e22=0.006739661, n=0.001679261, R1=6371031.573, R2=6371029.982388151, R3=6371023.591178183, Rr=6367471.848433915, Rs=6367458.38162583) 

@var Ellipsoids.Bessel1841: Ellipsoid(name='Bessel1841', a=6377397.155, b=6356078.962818, f_=299.1528128, f=0.003342773, e=0.081696831, e2=0.006674372, e12=0.993325628, e22=0.006719219, n=0.001674185, R1=6370291.090939333, R2=6370289.510126558, R3=6370283.158215224, Rr=6366742.520233163, Rs=6366729.136254413) 

@var Ellipsoids.CPM1799: Ellipsoid(name='CPM1799', a=6375738.7, b=6356671.92557493, f_=334.39, f=0.00299052, e=0.077279343, e2=0.005972097, e12=0.994027903, e22=0.006007977, n=0.001497499, R1=6369383.108524977, R2=6369381.843415799, R3=6369376.762470212, Rr=6366208.881847335, Rs=6366198.174663714) 

@var Ellipsoids.Clarke1866: Ellipsoid(name='Clarke1866', a=6378206.4, b=6356583.8, f_=294.978698214, f=0.003390075, e=0.082271854, e2=0.006768658, e12=0.993231342, e22=0.006814785, n=0.001697916, R1=6370998.866666667, R2=6370997.240632997, R3=6370990.706598808, Rr=6367399.689168951, Rs=6367385.921655473) 

@var Ellipsoids.Clarke1880: Ellipsoid(name='Clarke1880', a=6378249.145, b=6356514.86954978, f_=293.465, f=0.003407561, e=0.0824834, e2=0.006803511, e12=0.993196489, e22=0.006850116, n=0.001706689, R1=6371004.386516593, R2=6371002.743669633, R3=6370996.141916499, Rr=6367386.643979664, Rs=6367372.733858579) 

@var Ellipsoids.Clarke1880IGN: Ellipsoid(name='Clarke1880IGN', a=6378249.2, b=6356515, f_=293.466021294, f=0.00340755, e=0.082483257, e2=0.006803488, e12=0.993196512, e22=0.006850092, n=0.001706683, R1=6371004.466666666, R2=6371002.823831111, R3=6370996.22212394, Rr=6367386.736672513, Rs=6367372.826648208) 

@var Ellipsoids.Clarke1880Mod: Ellipsoid(name='Clarke1880Mod', a=6378249.145, b=6356514.96582849, f_=293.4663, f=0.003407546, e=0.082483218, e2=0.006803481, e12=0.993196519, e22=0.006850086, n=0.001706681, R1=6371004.418609496, R2=6371002.775777077, R3=6370996.174082516, Rr=6367386.692077904, Rs=6367372.782080163) 

@var Ellipsoids.Delambre1810: Ellipsoid(name='Delambre1810', a=6376428, b=6355957.92616372, f_=311.5, f=0.003210273, e=0.080063974, e2=0.00641024, e12=0.99358976, e22=0.006451596, n=0.001607717, R1=6369604.642054573, R2=6369603.184197493, R3=6369597.327390675, Rr=6366197.076842674, Rs=6366184.735554905) 

@var Ellipsoids.Engelis1985: Ellipsoid(name='Engelis1985', a=6378136.05, b=6356751.32272154, f_=298.2566, f=0.003352818, e=0.081819276, e2=0.006694394, e12=0.993305606, e22=0.006739511, n=0.001679224, R1=6371007.807573847, R2=6371006.217070852, R3=6370999.826135725, Rr=6367448.175078915, Rs=6367434.70891814) 

@var Ellipsoids.Everest1969: Ellipsoid(name='Everest1969', a=6377295.664, b=6356094.667915, f_=300.8017, f=0.003324449, e=0.081472981, e2=0.006637847, e12=0.993362153, e22=0.006682202, n=0.001664992, R1=6370228.665305, R2=6370227.101785341, R3=6370220.819516171, Rr=6366699.578394239, Rs=6366686.341077896) 

@var Ellipsoids.Fisher1968: Ellipsoid(name='Fisher1968', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, e=0.081813334, e2=0.006693422, e12=0.993306578, e22=0.006738525, n=0.001678979, R1=6371022.77908146, R2=6371021.189037352, R3=6371014.799950343, Rr=6367463.656043012, Rs=6367450.193774211) 

@var Ellipsoids.GEM10C: Ellipsoid(name='GEM10C', a=6378137, b=6356752.31424783, f_=298.2572236, f=0.003352811, e=0.081819191, e2=0.00669438, e12=0.99330562, e22=0.006739497, n=0.00167922, R1=6371008.771415944, R2=6371007.180919358, R3=6371000.790010039, Rr=6367449.145823942, Rs=6367435.679717519) 

@var Ellipsoids.GRS67: Ellipsoid(name='GRS67', a=6378160, b=6356774.516, f_=298.247167427, f=0.003352924, e=0.081820568, e2=0.006694605, e12=0.993305395, e22=0.006739725, n=0.001679277, R1=6371031.505333333, R2=6371029.914708731, R3=6371023.523359839, Rr=6367471.747019209, Rs=6367458.279955242) 

@var Ellipsoids.GRS80: Ellipsoid(name='GRS80', a=6378137, b=6356752.314140347, f_=298.257222101, f=0.003352811, e=0.081819191, e2=0.00669438, e12=0.99330562, e22=0.006739497, n=0.00167922, R1=6371008.771380115, R2=6371007.180883513, R3=6371000.789974131, Rr=6367449.145770246, Rs=6367435.679663688) 

@var Ellipsoids.Helmert1906: Ellipsoid(name='Helmert1906', a=6378200, b=6356818.16962789, f_=298.3, f=0.00335233, e=0.081813334, e2=0.006693422, e12=0.993306578, e22=0.006738525, n=0.001678979, R1=6371072.723209296, R2=6371071.133152724, R3=6371064.744015628, Rr=6367513.572269944, Rs=6367500.10989561) 

@var Ellipsoids.IERS1989: Ellipsoid(name='IERS1989', a=6378136, b=6356751.302, f_=298.257, f=0.003352813, e=0.081819221, e2=0.006694385, e12=0.993305615, e22=0.006739502, n=0.001679222, R1=6371007.767333333, R2=6371006.176906484, R3=6370999.785917024, Rr=6367448.139705879, Rs=6367434.673581903) 

@var Ellipsoids.IERS1992TOPEX: Ellipsoid(name='IERS1992TOPEX', a=6378136.3, b=6356751.61659215, f_=298.257223563, f=0.003352811, e=0.081819191, e2=0.00669438, e12=0.99330562, e22=0.006739497, n=0.00167922, R1=6371008.072197382, R2=6371006.481700972, R3=6371000.090792353, Rr=6367448.446995611, Rs=6367434.980890662) 

@var Ellipsoids.IERS2003: Ellipsoid(name='IERS2003', a=6378136.6, b=6356751.85797165, f_=298.25642, f=0.00335282, e=0.081819301, e2=0.006694398, e12=0.993305602, e22=0.006739515, n=0.001679225, R1=6371008.352657217, R2=6371006.762152168, R3=6371000.371208764, Rr=6367448.71770978, Rs=6367435.251531576) 

@var Ellipsoids.Intl1924: Ellipsoid(name='Intl1924', a=6378388, b=6356911.946, f_=297, f=0.003367003, e=0.08199189, e2=0.00672267, e12=0.99327733, e22=0.00676817, n=0.001686341, R1=6371229.315333334, R2=6371227.711270464, R3=6371221.265832124, Rr=6367654.499992855, Rs=6367640.919007843) 

@var Ellipsoids.Intl1967: Ellipsoid(name='Intl1967', a=6378157.5, b=6356772.2, f_=298.24961539, f=0.003352896, e=0.081820233, e2=0.00669455, e12=0.99330545, e22=0.00673967, n=0.001679263, R1=6371029.066666666, R2=6371027.476083895, R3=6371021.084827519, Rr=6367469.33894366, Rs=6367455.872106339) 

@var Ellipsoids.Krassovski1940: Ellipsoid(name='Krassovski1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, e=0.081813334, e2=0.006693422, e12=0.993306578, e22=0.006738525, n=0.001678979, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.693674386, Rr=6367558.496874185, Rs=6367545.034404869) 

@var Ellipsoids.Krassowsky1940: Ellipsoid(name='Krassowsky1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, e=0.081813334, e2=0.006693422, e12=0.993306578, e22=0.006738525, n=0.001678979, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.693674386, Rr=6367558.496874185, Rs=6367545.034404869) 

@var Ellipsoids.Maupertuis1738: Ellipsoid(name='Maupertuis1738', a=6397300, b=6363806.28272251, f_=191, f=0.005235602, e=0.102194876, e2=0.010443793, e12=0.989556207, e22=0.010554017, n=0.002624672, R1=6386135.42757417, R2=6386131.541448465, R3=6386115.886282292, Rr=6380564.130113637, Rs=6380531.163818629) 

@var Ellipsoids.Mercury1960: Ellipsoid(name='Mercury1960', a=6378166, b=6356784.28360711, f_=298.3, f=0.00335233, e=0.081813334, e2=0.006693422, e12=0.993306578, e22=0.006738525, n=0.001678979, R1=6371038.76120237, R2=6371037.171154276, R3=6371030.782051236, Rr=6367479.629235634, Rs=6367466.166933062) 

@var Ellipsoids.Mercury1968Mod: Ellipsoid(name='Mercury1968Mod', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, e=0.081813334, e2=0.006693422, e12=0.993306578, e22=0.006738525, n=0.001678979, R1=6371022.77908146, R2=6371021.189037352, R3=6371014.799950343, Rr=6367463.656043012, Rs=6367450.193774211) 

@var Ellipsoids.NWL1965: Ellipsoid(name='NWL1965', a=6378145, b=6356759.76948868, f_=298.25, f=0.003352892, e=0.08182018, e2=0.006694542, e12=0.993305458, e22=0.006739661, n=0.001679261, R1=6371016.58982956, R2=6371014.999254003, R3=6371008.60802666, Rr=6367456.873667615, Rs=6367443.406891448) 

@var Ellipsoids.OSU86F: Ellipsoid(name='OSU86F', a=6378136.2, b=6356751.51693008, f_=298.2572236, f=0.003352811, e=0.081819191, e2=0.00669438, e12=0.99330562, e22=0.006739497, n=0.00167922, R1=6371007.972310026, R2=6371006.381813641, R3=6370999.990905124, Rr=6367448.347164505, Rs=6367434.88105977) 

@var Ellipsoids.OSU91A: Ellipsoid(name='OSU91A', a=6378136.3, b=6356751.6165948, f_=298.2572236, f=0.003352811, e=0.081819191, e2=0.00669438, e12=0.99330562, e22=0.006739497, n=0.00167922, R1=6371008.072198267, R2=6371006.481701856, R3=6371000.090793238, Rr=6367448.446996935, Rs=6367434.98089199) 

@var Ellipsoids.Plessis1817: Ellipsoid(name='Plessis1817', a=6376523, b=6355862.93325557, f_=308.64, f=0.003240021, e=0.080433474, e2=0.006469544, e12=0.993530456, e22=0.006511671, n=0.001622639, R1=6369636.31108519, R2=6369634.826085826, R3=6369628.859996674, Rr=6366197.15710669, Rs=6366184.585664447) 

@var Ellipsoids.SGS85: Ellipsoid(name='SGS85', a=6378136, b=6356751.30156878, f_=298.257, f=0.003352813, e=0.081819221, e2=0.006694385, e12=0.993305615, e22=0.006739502, n=0.001679222, R1=6371007.767189593, R2=6371006.176690875, R3=6370999.785772962, Rr=6367448.13949045, Rs=6367434.673365931) 

@var Ellipsoids.SoAmerican1969: Ellipsoid(name='SoAmerican1969', a=6378160, b=6356774.71919531, f_=298.25, f=0.003352892, e=0.08182018, e2=0.006694542, e12=0.993305458, e22=0.006739661, n=0.001679261, R1=6371031.573065103, R2=6371029.982485807, R3=6371023.591243432, Rr=6367471.848531487, Rs=6367458.38172365) 

@var Ellipsoids.Sphere: Ellipsoid(name='Sphere', a=6371008.771415, b=6371008.771415, f_=0, f=0, e=0, e2=0, e12=1, e22=0, n=0, R1=6371008.771415, R2=6371008.771415, R3=6371008.771415, Rr=6371008.771415, Rs=6371008.771415) 

@var Ellipsoids.SphereAuthalic: Ellipsoid(name='SphereAuthalic', a=6371000, b=6371000, f_=0, f=0, e=0, e2=0, e12=1, e22=0, n=0, R1=6371000, R2=6371000, R3=6371000, Rr=6371000, Rs=6371000) 

@var Ellipsoids.SpherePopular: Ellipsoid(name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, e=0, e2=0, e12=1, e22=0, n=0, R1=6378137, R2=6378137, R3=6378137, Rr=6378137, Rs=6378137) 

@var Ellipsoids.Struve1860: Ellipsoid(name='Struve1860', a=6378298.3, b=6356657.14266956, f_=294.73, f=0.003392936, e=0.082306499, e2=0.00677436, e12=0.99322564, e22=0.006820565, n=0.001699351, R1=6371084.580889854, R2=6371082.952089875, R3=6371076.406914177, Rr=6367482.318324657, Rs=6367468.527348378) 

@var Ellipsoids.WGS60: Ellipsoid(name='WGS60', a=6378165, b=6356783.28695944, f_=298.3, f=0.00335233, e=0.081813334, e2=0.006693422, e12=0.993306578, e22=0.006738525, n=0.001678979, R1=6371037.762319813, R2=6371036.172271968, R3=6371029.783169931, Rr=6367478.630911095, Rs=6367465.168610634) 

@var Ellipsoids.WGS66: Ellipsoid(name='WGS66', a=6378145, b=6356759.76948868, f_=298.25, f=0.003352892, e=0.08182018, e2=0.006694542, e12=0.993305458, e22=0.006739661, n=0.001679261, R1=6371016.58982956, R2=6371014.999254003, R3=6371008.60802666, Rr=6367456.873667615, Rs=6367443.406891448) 

@var Ellipsoids.WGS72: Ellipsoid(name='WGS72', a=6378135, b=6356750.52, f_=298.26, f=0.003352779, e=0.081818811, e2=0.006694318, e12=0.993305682, e22=0.006739434, n=0.001679205, R1=6371006.84, R2=6371005.249530816, R3=6370998.858745317, Rr=6367447.24861499, Rs=6367433.78276368) 

@var Ellipsoids.WGS84: Ellipsoid(name='WGS84', a=6378137, b=6356752.31425, f_=298.257223563, f=0.003352811, e=0.081819191, e2=0.00669438, e12=0.99330562, e22=0.006739497, n=0.00167922, R1=6371008.771416667, R2=6371007.180920884, R3=6371000.790010764, Rr=6367449.145825027, Rs=6367435.679718607) 

 

@var Transforms.BD72: Transform(name='BD72', tx=106.86863, ty=-52.29778, tz=103.72389, rx=-0, ry=-0, rz=-0.00001, s=1.2727, s1=1, sx=-0.33657, sy=-0.45696, sz=-1.84218) 

@var Transforms.Bessel1841: Transform(name='Bessel1841', tx=-582, ty=-105, tz=-414, rx=-0.00001, ry=-0, rz=0.00001, s=-8.3, s1=0.99999, sx=-1.04, sy=-0.35, sz=3.08) 

@var Transforms.Clarke1866: Transform(name='Clarke1866', tx=8, ty=-160, tz=-176, rx=0, ry=0, rz=0, s=0, s1=1, sx=0, sy=0, sz=0) 

@var Transforms.DHDN: Transform(name='DHDN', tx=-591.28, ty=-81.35, tz=-396.39, rx=0.00001, ry=-0, rz=-0.00001, s=-9.82, s1=0.99999, sx=1.477, sy=-0.0736, sz=-1.458) 

@var Transforms.ED50: Transform(name='ED50', tx=89.5, ty=93.8, tz=123.1, rx=0, ry=0, rz=0, s=-1.2, s1=1, sx=0, sy=0, sz=0.156) 

@var Transforms.Irl1965: Transform(name='Irl1965', tx=-482.53, ty=130.596, tz=-564.557, rx=0.00001, ry=0, rz=0, s=-8.15, s1=0.99999, sx=1.042, sy=0.214, sz=0.631) 

@var Transforms.Irl1975: Transform(name='Irl1975', tx=-482.53, ty=130.596, tz=-564.557, rx=-0.00001, ry=-0, rz=-0, s=-1.1, s1=1, sx=-1.042, sy=-0.214, sz=-0.631) 

@var Transforms.Krassovski1940: Transform(name='Krassovski1940', tx=-24, ty=123, tz=94, rx=-0, ry=0, rz=0, s=-2.423, s1=1, sx=-0.02, sy=0.26, sz=0.13) 

@var Transforms.Krassowsky1940: Transform(name='Krassowsky1940', tx=-24, ty=123, tz=94, rx=-0, ry=0, rz=0, s=-2.423, s1=1, sx=-0.02, sy=0.26, sz=0.13) 

@var Transforms.MGI: Transform(name='MGI', tx=-577.326, ty=-90.129, tz=-463.92, rx=0.00002, ry=0.00001, rz=0.00003, s=-2.423, s1=1, sx=5.137, sy=1.474, sz=5.297) 

@var Transforms.NAD27: Transform(name='NAD27', tx=8, ty=-160, tz=-176, rx=0, ry=0, rz=0, s=0, s1=1, sx=0, sy=0, sz=0) 

@var Transforms.NAD83: Transform(name='NAD83', tx=1.004, ty=-1.91, tz=-0.515, rx=0, ry=0, rz=0, s=-0.0015, s1=1, sx=0.0267, sy=0.00034, sz=0.011) 

@var Transforms.NTF: Transform(name='NTF', tx=-168, ty=-60, tz=320, rx=0, ry=0, rz=0, s=0, s1=1, sx=0, sy=0, sz=0) 

@var Transforms.OSGB36: Transform(name='OSGB36', tx=-446.448, ty=125.157, tz=-542.06, rx=-0, ry=-0, rz=-0, s=20.4894, s1=1.00002, sx=-0.1502, sy=-0.247, sz=-0.8421) 

@var Transforms.TokyoJapan: Transform(name='TokyoJapan', tx=148, ty=-507, tz=-685, rx=0, ry=0, rz=0, s=0, s1=1, sx=0, sy=0, sz=0) 

@var Transforms.WGS72: Transform(name='WGS72', tx=0, ty=0, tz=-4.5, rx=0, ry=0, rz=0, s=-0.22, s1=1, sx=0, sy=0, sz=0.554) 

@var Transforms.WGS84: Transform(name='WGS84', tx=0, ty=0, tz=0, rx=0, ry=0, rz=0, s=0, s1=1, sx=0, sy=0, sz=0) 

''' 

 

# make sure int/int division yields float quotient 

from __future__ import division 

division = 1 / 2 # double check int division, see .fmath.py, .utily.py 

if not division: 

raise ImportError('%s 1/2 == %d' % ('division', division)) 

del division 

 

from pygeodesy.fmath import _2_3rd, EPS, EPS1, cbrt, cbrt2, fdot, \ 

fpowers, Fsum, fsum_, hypot1, sqrt3 

from pygeodesy.lazily import _ALL_LAZY 

from pygeodesy.named import Curvature2Tuple, Distance2Tuple, inStr, \ 

_NamedEnum, _NamedEnumItem, Vector3Tuple 

from pygeodesy.utily import PI2, R_M, degrees360, m2degrees, m2km, \ 

m2NM, m2SM, property_RO, _TypeError 

 

from math import atan, atan2, atanh, copysign, cos, exp, hypot, \ 

radians, sin, sinh, sqrt 

 

R_M = R_M #: Mean (spherical) earth radius (C{meter}). 

R_MA = 6378137.0 #: Major (equatorial) earth radius (C{meter}) WGS84, EPSG:3785. 

R_MB = 6356752.0 #: Minor (polar) earth radius (C{meter}) WGS84, EPSG:3785. 

R_KM = m2km(R_M) #: Mean (spherical) earth radius (C{KM}, kilo meter). 

R_NM = m2NM(R_M) #: Mean (spherical) earth radius (C{NM}, nautical miles). 

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 = 6371000.0 #: Former FAI Sphere earth radius (C{meter}). 

R_VM = 6366707.0194937 #: Aviation/Navigation earth radius (C{meter}). 

# R_ = 6372797.560856 #: XXX some other earth radius??? 

 

# all public contants, classes and functions 

__all__ = _ALL_LAZY.datum 

__version__ = '20.01.22' 

 

_TOL = sqrt(EPS * 0.1) # for Ellipsoid.estauf, imported by .ups 

 

 

def _4Ecef(this, Ecef): 

'''Return an ECEF converter. 

''' 

from pygeodesy.ecef import EcefKarney, EcefVeness, EcefYou 

 

if Ecef is None: 

Ecef = EcefKarney 

else: 

_TypeError(EcefKarney, EcefVeness, EcefYou, Ecef=Ecef) 

return Ecef(this, name=this.name) # datum or ellipsoid 

 

 

class Ellipsoid(_NamedEnumItem): 

'''Ellipsoid with major and minor radius, semi-axis, (inverse) 

flattening and other pre-computed, frequently used attributes. 

''' 

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

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

_f = 0 #: Flattening: (a - b) / a 

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

 

_f2 = None #: 2nd Flattening: (a - b) / b # un-/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 

 

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

_a2_b = None #: (INTERNAL) Meridional radius of curvature at poles: a**2 / b (C{meter}) 

_b2_a = None #: (INTERNAL) Meridional radius of curvature at equator: b**2 / a (C{meter}) 

 

# eccentricities 

_e = None #: 1st Eccentricity: sqrt(1 - (b / a)**2)) # for utm 

_es_c = None #: M{(1 - f) * exp(es_atanh(1))} 

_e2 = None #: 1st Eccentricity squared: f * (2 - f) = 1 - (b / a)**2 

_e4 = None #: e2**2 # for ellipsoidalNvector.Cartesian.toNvector, ecef 

_e12 = None #: 1 - e2 = (1 - f)**2 # for ellipsoidalNvector.Cartesian.toNvector, ecef utm 

_e22 = None #: 2nd Eccentricity squared: e2 / (1 - e2) = (a / b)**2 - 1 

 

# fixed earth radii from <https://WikiPedia.org/wiki/Earth_radius> 

_R1 = None #: (INTERNAL) Mean earth radius: (2 * a + b) / 3 per IUGG definition (C{meter}) 

_R2 = None #: (INTERNAL) Authalic radius: sqrt((a**2 + b**2 * atanh(e) / e) / 2) (C{meter}) 

# _c = None #: (INTERNAL) Authalic radius: equ (60) in Karney's "Algorithms for Geodesics" 

_R3 = None #: (INTERNAL) Volumetric radius: (a * a * b)**1/3 (C{meter}) 

_Rr = None #: (INTERNAL) Rectifying radius: ((a**3/2 + b**3/2) / 2)**2/3 (C{meter}) 

_Rs = None #: (INTERNAL) Mean earth radius: sqrt(a * b) (C{meter}) 

 

_ab_90 = None #: (a - b) / 90 # for .Rlat below 

_area = None #: (INTERNAL) Surface area: 4 * PI * R2**2 

_volume = None #: (INTERNAL) Volume: 4 / 3 * PI * a**2 * b 

 

_A = None #: (INTERNAL) Meridional radius 

_AlphaKs = None #: (INTERNAL) Up to 8th-order Krüger Alpha series 

_BetaKs = None #: (INTERNAL) Up to 8th-order Krüger Beta series 

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

_Mabcd = None #: (INTERNAL) OSGB meridional coefficients 

 

_geodesic = None #: (INTERNAL) Cached C{geographiclib.geodesic.Geodesic} 

 

def __init__(self, a, b, f_, name=''): 

'''New L{Ellipsoid}. 

 

@param a: Semi-major, equatorial axis (C{meter}). 

@param b: Semi-minor, polar axis (C{meter}). 

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

@keyword name: Optional, unique name (C{str}). 

 

@raise NameError: Ellipsoid with that B{C{name}} already exists. 

''' 

self._a = a = float(a) # major half-axis in meter 

if not b: # get b from a and f_, minor half-axis in meter 

self._b = b = a * (f_ - 1) / float(f_) 

else: # get f_ from a and b if not spherical 

self._b = b = float(b) 

if not f_ and a > b: 

f_ = a / (a - b) 

 

if f_ > 0 and a > b > 0: 

self._f_ = f_ = float(f_) # inverse flattening 

self._f = f = 1 / f_ # flattening 

elif a > 0: # sphere 

self._b = b = self._a2b = self._b2a = a 

self._f2 = self._n = f_ = f = 0 

self._a_b = 1 

self._R1 = self._R2 = self._R3 = self._Rr = self._Rs = a 

else: 

raise ValueError('%s invalid: %s' % ('ellipsoid', 

inStr(self, a, b, f_, name=name))) 

 

d = a - b 

# some sanity checks to catch mistakes 

if d < 0 or min(a, b) < 1: 

raise AssertionError('%s: %s=%0.9f vs %s=%0.9f' % (name, 

'a', a, 'b', b)) 

t = d / a 

if abs(f - t) > 1e-8: 

raise AssertionError('%s: %s=%.9e vs %s=%.9e' % (name, 

'f', f, '(a-b)/a', t)) 

t = (1 - self.f)**2 

if abs(self.e12 - t) > EPS: 

raise AssertionError('%s: %s=%.9e vs %s=%.9e' % (name, 

'e12', self.e12, '(1-f)**2', t)) 

 

self._register(Ellipsoids, name) 

 

def __eq__(self, other): 

'''Compare this and an other ellipsoid. 

 

@param other: The other ellipsoid (L{Ellipsoid}). 

 

@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 Krüger Alpha 

or Beta series coefficients per 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>}. 

 

@param AB8Ks: 8-Tuple of 8-th order Krüger Alpha or Beta 

series coefficient tuples. 

 

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

 

@see: The 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 major (equatorial) radius, semi-axis (C{meter}). 

''' 

return self._a 

 

@property_RO 

def a2(self): 

'''Get the major radius I{squared} (C{float}), M{a**2}. 

''' 

if self._a2 is None: 

self._a2 = self.a**2 

return self._a2 

 

@property_RO 

def a2_(self): 

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

''' 

if self._a2_ is None: 

self._a2_ = 1 / 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 = self.a / self.b 

return self._a_b 

 

@property_RO 

def a2_b(self): 

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

 

@see: U{Radii of Curvature 

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

''' 

if self._a2_b is None: 

self._a2_b = self.a2 / self.b 

return self._a2_b 

 

@property_RO 

def area(self): 

'''Get the ellipsoid's surface area (C{meter**2}), M{4 * PI * R2**2}. 

''' 

if self._area is None: 

self._area = 2 * PI2 * self.R2**2 

return self._area 

 

@property_RO 

def A(self): 

'''Get the UTM meridional radius (C{meter}). 

''' 

if self._A is None: 

n = self.n 

# <https://GeographicLib.SourceForge.io/html/transversemercator.html> 

self._A = self.a / (1 + 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> 

# self._A = self.a / (1 + n) * (fhorner(n**2, 16384, 4096, 256, 64, 25) / 16384) 

return self._A 

 

@property_RO 

def AlphaKs(self): 

'''Get the U{Krüger 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 

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

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

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

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

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

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

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

(1424729850961/743921418240,)) # PYCHOK unaligned 

return self._AlphaKs 

 

@property_RO 

def b(self): 

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

''' 

return self._b 

 

@property_RO 

def b2(self): 

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

''' 

if self._b2 is None: 

self._b2 = self.b**2 

return self._b2 

 

@property_RO 

def b2_a(self): 

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

 

@see: U{Radii of Curvature 

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

''' 

if self._b2_a is None: 

self._b2_a = self.b2 / self.a 

return self._b2_a 

 

@property_RO 

def BetaKs(self): 

'''Get the U{Krüger 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 

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

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

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

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

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

(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligned 

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

(191773887257/3719607091200,)) # PYCHOK unaligned 

return self._BetaKs 

 

@property_RO 

def c(self): 

'''Get the authalic earth radius (C{meter}), see C{R2}. 

 

@see: Symbol C{c} in U{equation 60 

<https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>}. 

''' 

return self.R2 

 

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

'''Approximate the distance and (initial) bearing between two 

points based on the radii of curvature. 

 

Suitable only for short distances up to a few hundred Km 

or Miles and only between points not near-polar. 

 

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

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

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

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

 

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

 

@see: U{Local, flat earth approximation 

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

''' 

m, n = self.roc2(lat0) 

m *= radians(lat1 - lat0) 

n *= radians(lon1 - lon0) * cos(radians(lat0)) 

return Distance2Tuple(hypot(m, n), degrees360(atan2(n, m))) 

 

@property_RO 

def e(self): 

'''Get the (1st) Eccentricity (C{float}), M{sqrt(1 - (b / a)**2))}. 

''' 

if self._e is None: 

self._e = sqrt(abs(self.e2)) 

return self._e 

 

@property_RO 

def e12(self): 

'''Get M{1 - e**2} (C{float}). 

''' 

if self._e12 is None: 

self._e12 = 1 - self.e2 

return self._e12 # 1 - e2 

 

@property_RO 

def e2(self): 

'''Get the (1st) Eccentricity squared (C{float}), M{f * (2 - f) == 1 - (b / a)**2}. 

''' 

if self._e2 is None: 

self._e2 = e2 = self.f * (2 - self.f) 

t = 1 - (self.b / self.a)**2 

if abs(e2 - t) > 1e-9: 

raise AssertionError('%s: %s=%.9e vs %s=%.9e' % (self.name, 

'e2', e2, '1-(b/a)**2', t)) 

return self._e2 

 

@property_RO 

def e22(self): 

'''Get the 2nd Eccentricity I{squared} (C{float}), M{e2 / (1 - e2) == (a / b)**2 - 1}. 

''' 

if self._e22 is None: 

self._e22 = e22 = self.e2 / (1 - self.e2) 

t = self.a_b**2 - 1 

if abs(e22 - t) > 1e-9: 

raise AssertionError('%s: %s=%.9e vs %s=%.9e' % (self.name, 

'e22', e22, '(a/b)**2-1', t)) 

return self._e22 

 

@property_RO 

def e4(self): 

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

''' 

if self._e4 is None: 

self._e4 = self.e2**2 

return self._e4 

 

def ecef(self, Ecef=None): 

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

 

@keyword 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) 

 

def e2s(self, s): 

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

 

@param s: S value (C{scalar}). 

 

@return: Norm (C{float}). 

 

@raise ValueError: Invalid B{C{s}}. 

''' 

try: 

return sqrt(self.e2s2(s)) 

except (TypeError, ValueError): 

raise ValueError('%s.%s invalid: %r' % (self.name, 'e2s', s)) 

 

def e2s2(self, s): 

'''Compute M{1 - e2 * s**2}. 

 

@param s: S value (C{scalar}). 

 

@return: Result (C{float}). 

 

@raise ValueError: Invalid B{C{s}}. 

''' 

try: 

r = 1 - self.e2 * s**2 

if r >= 0: 

return r 

except (TypeError, ValueError): 

pass 

raise ValueError('%s.%s invalid: %r' % (self.name, 'e2s2', s)) 

 

@property_RO 

def es(self): 

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

''' 

# note, self.e is always non-negative 

return copysign(self.e, self.f) # see .ups.py 

 

def es_atanh(self, x): 

'''Compute M{es * atanh(es * x)} where I{es} is the I{signed} 

1st Eccentricity. 

 

@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: 

r = self.e * atanh(self.e * x) 

elif self.f < 0: 

r = self.e * atan(-self.e * x) 

else: 

r = 0 

return r 

 

@property_RO 

def es_c(self): 

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

''' 

if self._es_c is None: 

self._es_c = (1 - self.f) * exp(self.es_atanh(1.0)) 

return self._es_c 

 

def es_tauf(self, taup): 

'''Compute U{Karney's<https://ArXiv.org/abs/1002.1417>} 

equations (19), (20) and (21). 

 

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

html/classGeographicLib_1_1Math.html>}. 

''' 

tol = max(abs(taup), 1) * _TOL 

e2m = 1 - abs(self.e2) # == self.e**2 

t = taup / e2m 

T = Fsum(t) 

for _ in range(5): 

a = self.es_taupf(t) 

d = (taup - a) * (1 + e2m * t**2) / (e2m * hypot1(t) * hypot1(a)) 

t, d = T.fsum2_(d) 

if abs(d) < tol: 

break 

return t 

 

def es_taupf(self, tau): 

'''Compute U{Karney's<https://ArXiv.org/abs/1002.1417>} 

equations (7), (8) and (9). 

 

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

html/classGeographicLib_1_1Math.html>}. 

''' 

t = hypot1(tau) 

s = sinh(self.es_atanh(tau / t)) 

return hypot1(s) * tau - s * t 

 

@property_RO 

def f(self): 

'''Get the Flattening (C{float}), M{(a - b) / a}. 

''' 

return self._f 

 

@property_RO 

def f_(self): 

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

''' 

return self._f_ 

 

@property_RO 

def f2(self): 

'''Get the 2nd Flattening (C{float}), M{(a - b) / b}. 

''' 

if self._f2 is None: 

self._f2 = (self.a - self.b) / self.b 

return self._f2 

 

@property_RO 

def geodesic(self): 

'''Get this ellipsoid's U{Karney 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: 

try: 

from geographiclib.geodesic import Geodesic 

self._geodesic = Geodesic(self.a, self.f) 

except ImportError: 

raise # ImportError('no %s' % ('geographiclib',)) 

return self._geodesic 

 

@property_RO 

def _geodesic2(self): 

'''(INTERNAL) Get this ellipsoid's C{Geodesic} and C{Math} module. 

''' 

g = self.geodesic 

from geographiclib.geomath import Math 

return g, Math 

 

@property_RO 

def isEllipsoidal(self): 

'''Check whether this model is ellipsoidal (C{bool}). 

''' 

return self.a > self.R1 > self.b 

 

@property_RO 

def isSpherical(self): 

'''Check whether this model is spherical (C{bool}). 

''' 

return self.a == self.R1 == self.b 

 

@property 

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 Krüger series order. 

 

@param order: New Krüger series order (C{int} 4, 6 or 8). 

 

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

''' 

if order not in (4, 6, 8): 

raise ValueError('%s invalid: %r' % ('order', order)) 

if order != self._KsOrder: 

self._KsOrder = order 

self._AlphaKs = self._BetaKs = None 

 

def m2degrees(self, meter): 

'''Convert distance to angle along equator. 

 

@param meter: Distance (C{meter}). 

 

@return: Angle (C{degrees}). 

''' 

return m2degrees(meter, self.a) 

 

@property_RO 

def Mabcd(self): 

'''Get the OSGR meridional coefficients (4-tuple), 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, 

fdot( (n, n2, n3), 24, 24, 21) / 8, 

fdot( (n2, n3), 15, 15) / 8, 

35 * n3 / 24) 

return self._Mabcd 

 

@property_RO 

def n(self): 

'''Get the 3rd Flattening (C{float}), M{f / (2 - f) == (a - b) / (a + b)}. 

''' 

if self._n is None: 

self._n = n = self.f / (2 - self.f) 

t = (self.a - self.b) / (self.a + self.b) 

if abs(n - t) > 1e-9: 

raise AssertionError('%s: %s=%.9e vs %s=%.9e' % (self.name, 

'n', n, '(a-b)/(a+b)', t)) 

return self._n 

 

@property_RO 

def R1(self): 

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

 

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

''' 

if self._R1 is None: 

self._R1 = (self.a * 2 + self.b) / 3 

return self._R1 

 

@property_RO 

def R2(self): 

'''Get the 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>} and 

U{c<https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>}. 

''' 

if self._R2 is None: 

if self.e2 > 0: 

r = atanh(self.e) / self.e 

elif self.e2 < 0: # GeographicLib Ellipsoid Area 

r = atan(self.e) / self.e 

else: 

r = 1 

self._R2 = sqrt((self.a2 + self.b2 * r) * 0.5) 

return self._R2 

 

@property_RO 

def R3(self): 

'''Get the 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: 

self._R3 = cbrt(self.a2 * self.b) 

return self._R3 

 

def Rgeocentric(self, lat): 

'''Compute the geocentric earth radius at the given latitude. 

 

@param lat: Latitude (C{degrees90}). 

 

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

 

@see: U{Geocentric Radius 

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

''' 

a2 = self.a2 

b2 = self.b2 

c2 = cos(radians(lat))**2 

s2 = 1 - c2 

return sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2)) 

 

@property_RO 

def Rr(self): 

'''Get the 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: 

self._Rr = cbrt2((sqrt3(self.a) + sqrt3(self.b)) * 0.5) 

return self._Rr 

 

@property_RO 

def Rs(self): 

'''Get another mean earth radius (C{meter}), M{sqrt(a * b)}. 

''' 

if self._Rs is None: 

self._Rs = sqrt(self.a * self.b) 

return self._Rs 

 

def Rlat(self, lat): 

'''Approximate the earth radius at the given latitude. 

 

@param lat: Latitude (C{degrees90}). 

 

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

''' 

if self._ab_90 is None: 

self._ab_90 = (self.a - self.b) / 90.0 

# r = major - (major - minor) * |lat| / 90 

return self.a - self._ab_90 * min(abs(lat), 90) 

 

def roc2(self, lat): 

'''Compute the meridional and prime-vertical radii of curvature 

at the given latitude. 

 

@param lat: Latitude (C{degrees90}). 

 

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

radii of curvature. 

 

@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>}. 

''' 

r = self.e2s2(sin(radians(lat))) 

if r < EPS: 

m = n = 0 

elif r < EPS1: 

n = self.a / sqrt(r) 

m = n * self.e12 / r 

else: 

n = self.a 

m = n * self.e12 

return Curvature2Tuple(m, n) 

 

def rocBearing(self, lat, bearing): 

'''Compute the directional radius of curvature at the 

given latitude and compass direction. 

 

@param lat: Latitude (C{degrees90}). 

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

 

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

 

@see: U{Radii of Curvature 

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

''' 

c2 = cos(radians(bearing))**2 

s2 = 1 - c2 

m, n = self.roc2(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 n / (c2 + s2) # == 1 / (c2 / m + s2 / n) 

 

def rocGauss(self, lat): 

'''Compute the Gaussian radius of curvature at the given latitude. 

 

@param lat: Latitude (C{degrees90}). 

 

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

 

@see: U{Radii of Curvature 

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

''' 

# using ... 

# m, n = self.roc2(lat) 

# return sqrt(m * n) 

# ... requires 1 or 2 sqrt 

a2, c2 = self.a2, cos(radians(lat))**2 

return a2 * self.b / (a2 * c2 + self.b2 * (1 - c2)) 

 

def rocMean(self, lat): 

'''Compute the mean radius of curvature at the given latitude. 

 

@param lat: Latitude (C{degrees90}). 

 

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

 

@see: U{Radii of Curvature 

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

''' 

m, n = self.roc2(lat) 

return 2 * m * n / (m + n) # == 2 / (1 / m + 1 / n) 

 

def rocMeridional(self, lat): 

'''Compute the meridional radius of curvature at the given latitude. 

 

@param lat: Latitude (C{degrees90}). 

 

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

 

@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(lat).meridional 

 

def rocPrimeVertical(self, lat): 

'''Compute the prime-vertical radius of curvature at the given latitude. 

 

@param lat: Latitude (C{degrees90}). 

 

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

 

@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(lat).prime_vertical 

 

def toStr(self, prec=9): # PYCHOK expected 

'''Return this ellipsoid as a text string. 

 

@keyword prec: Optional number of decimals, unstripped (C{int}). 

 

@return: Ellipsoid attributes (C{str}). 

''' 

return self._fStr(prec, 'a', 'b', 'f_', 'f', 'e', 'e2', 'e12', 'e22', 

'n', 'R1', 'R2', 'R3', 'Rr', 'Rs') 

 

@property_RO 

def volume(self): 

'''Get the ellipsoid's volume (C{meter**3}), M{4 / 3 * PI * a**2 * b}. 

''' 

if self._volume is None: 

self._volume = PI2 * _2_3rd * self.a2 * self.b 

return self._volume 

 

 

Ellipsoids = _NamedEnum('Ellipsoids', Ellipsoid) #: Registered ellipsoids. 

# <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, 6356256.909, 299.3249646, 'Airy1830'), 

AiryModified = Ellipsoid(6377340.189, 6356034.448, 299.3249646, 'AiryModified'), 

Australia1966 = Ellipsoid(6378160.0, 6356774.719, 298.25, 'Australia1966'), 

# 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, 6356774.516, 298.247167427, 'GRS67'), # Lucerne 

GRS80 = Ellipsoid(6378137.0, 6356752.314140347, 298.257222101, 'GRS80'), # ITRS, ETRS89 

Helmert1906 = Ellipsoid(6378200.0, 6356818.16962789, 298.3, 'Helmert1906'), 

IERS1989 = Ellipsoid(6378136.0, 6356751.302, 298.257, 'IERS1989'), 

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, 6356911.946, 297.0, 'Intl1924'), # aka Hayford 

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, 6356750.52, 298.26, 'WGS72'), 

WGS84 = Ellipsoid(6378137.0, 6356752.31425, 298.257223563, 'WGS84'), # GPS 

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 

) 

 

 

class Transform(_NamedEnumItem): 

'''Helmert transformation. 

''' 

tx = 0 #: X translation (C{meter}). 

ty = 0 #: Y translation (C{meter}). 

tz = 0 #: Z translation (C{meter}). 

 

rx = 0 #: X rotation (C{radians}). 

ry = 0 #: Y rotation (C{radians}). 

rz = 0 #: Z rotation (C{radians}). 

 

s = 0 #: Scale ppm (C{float}). 

s1 = 1 #: Scale + 1 (C{float}). 

 

sx = 0 #: X rotation (degree seconds). 

sy = 0 #: Y rotation (degree seconds). 

sz = 0 #: Z rotation (degree seconds). 

 

def __init__(self, name='', tx=0, ty=0, tz=0, 

sx=0, sy=0, sz=0, s=0): 

'''New L{Transform}. 

 

@keyword name: Optional, unique name (C{str}). 

@keyword tx: Optional X translation (C{meter}). 

@keyword ty: Optional Y translation (C{meter}). 

@keyword tz: Optional Z translation (C{meter}). 

@keyword s: Optional scale ppm (C{float}). 

@keyword sx: Optional X rotation (C{degree seconds}). 

@keyword sy: Optional Y rotation (C{degree seconds}). 

@keyword sz: Optional Z rotation (C{degree seconds}). 

 

@raise NameError: Transform with that B{C{name}} already exists. 

''' 

if tx: 

self.tx = float(tx) 

if ty: 

self.ty = float(ty) 

if tz: 

self.tz = float(tz) 

if sx: # secs to rads 

self.rx = radians(sx / 3600.0) 

self.sx = sx 

if sy: 

self.ry = radians(sy / 3600.0) 

self.sy = sy 

if sz: 

self.rz = radians(sz / 3600.0) 

self.sz = sz 

if s: 

self.s = float(s) 

self.s1 = s * 1.e-6 + 1 # normalize ppm to (s + 1) 

 

self._register(Transforms, name) 

 

def __eq__(self, other): 

'''Compare this and an other transform. 

 

@param other: The other transform (L{Transform}). 

 

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

''' 

return self is other or (isinstance(other, Transform) and 

self.tx == other.tx and 

self.ty == other.ty and 

self.tz == other.tz and 

self.rx == other.rx and 

self.ry == other.ry and 

self.rz == other.rz and 

self.s == other.s) 

 

def inverse(self, name=''): 

'''Return the inverse of this transform. 

 

@keyword name: Optional, unique name (C{str}). 

 

@return: Inverse (Transform). 

 

@raise NameError: Transform with that B{C{name}} already exists. 

''' 

return Transform(name=name or (self.name + 'Inverse'), 

tx=-self.tx, ty=-self.ty, tz=-self.tz, 

sx=-self.sx, sy=-self.sy, sz=-self.sz, s=-self.s) 

 

def toStr(self, prec=5): # PYCHOK expected 

'''Return this transform as a string. 

 

@keyword prec: Optional number of decimals, unstripped (C{int}). 

 

@return: Transform attributes (C{str}). 

''' 

return self._fStr(prec, 'tx', 'ty', 'tz', 

'rx', 'ry', 'rz', 's', 's1', 

'sx', 'sy', 'sz') 

 

def transform(self, x, y, z, inverse=False): 

'''Transform a (geocentric) Cartesian point, forward or inverse. 

 

@param x: X coordinate (C{meter}). 

@param y: Y coordinate (C{meter}). 

@param z: Z coordinate (C{meter}). 

@keyword inverse: Optional direction, forward or inverse (C{bool}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}, transformed. 

''' 

if inverse: 

xyz = -1, -x, -y, -z 

_s1 = self.s1 - 2 # negative inverse: -(1 - s * 1.e-6) 

else: 

xyz = 1, x, y, z 

_s1 = self.s1 

# x', y', z' = (.tx + x * .s1 - y * .rz + z * .ry, 

# .ty + x * .rz + y * .s1 - z * .rx, 

# .tz - x * .ry + y * .rx + z * .s1) 

r = Vector3Tuple(fdot(xyz, self.tx, _s1, -self.rz, self.ry), 

fdot(xyz, self.ty, self.rz, _s1, -self.rx), 

fdot(xyz, self.tz, -self.ry, self.rx, _s1)) 

return self._xnamed(r) 

 

 

Transforms = _NamedEnum('Transforms', Transform) #: Registered transforms. 

# <https://WikiPedia.org/wiki/Helmert_transformation> from WGS84 

Transforms._assert( 

BD72 = Transform('BD72', tx=106.868628, ty=-52.297783, tz=103.723893, 

# <https://www.NGI.BE/FR/FR4-4.shtm> ETRS89 == WG84 

# <https://GeoRepository.com/transformation_15929/BD72-to-WGS-84-3.html> 

sx=-0.33657, sy= -0.456955, sz= -1.84218, 

s= 1.2727), 

Bessel1841 = Transform('Bessel1841', tx=-582.0, ty=-105.0, tz=-414.0, 

sx= -1.04, sy= -0.35, sz= 3.08, 

s= -8.3), 

Clarke1866 = Transform('Clarke1866', tx=8, ty=-160, tz=-176), 

DHDN = Transform('DHDN', tx=-591.28, ty=-81.35, tz=-396.39, 

sx= 1.477, sy= -0.0736, sz= -1.458, 

s= -9.82), # Germany 

ED50 = Transform('ED50', tx=89.5, ty=93.8, tz=123.1, 

# <https://GeoNet.ESRI.com/thread/36583> sz=-0.156 

# <https://GitHub.com/ChrisVeness/geodesy/blob/master/latlon-ellipsoidal.js> 

# <https://www.Gov.UK/guidance/oil-and-gas-petroleum-operations-notices#pon-4> 

sz= 0.156, s=-1.2), 

Irl1965 = Transform('Irl1965', tx=-482.530, ty=130.596, tz=-564.557, 

sx= 1.042, sy= 0.214, sz= 0.631, 

s= -8.15), 

Irl1975 = Transform('Irl1975', tx=-482.530, ty=130.596, tz=-564.557, 

# XXX rotation signs may be opposite, to be checked 

sx= -1.042, sy= -0.214, sz= -0.631, 

s= -1.1), 

Krassovski1940 = Transform('Krassovski1940', tx=-24.0, ty=123.0, tz=94.0, 

sx= -0.02, sy= 0.26, sz= 0.13, 

s= -2.423), # spelling 

Krassowsky1940 = Transform('Krassowsky1940', tx=-24.0, ty=123.0, tz=94.0, 

sx= -0.02, sy= 0.26, sz= 0.13, 

s= -2.423), # spelling 

MGI = Transform('MGI', tx=-577.326, ty=-90.129, tz=-463.920, 

sx= 5.137, sy= 1.474, sz= 5.297, 

s= -2.423), # Austria 

NAD27 = Transform('NAD27', tx=8, ty=-160, tz=-176), 

NAD83 = Transform('NAD83', tx= 1.004, ty=-1.910, tz=-0.515, 

sx= 0.0267, sy= 0.00034, sz= 0.011, 

s=-0.0015), 

NTF = Transform('NTF', tx=-168, ty= -60, tz=320), # XXX verify 

OSGB36 = Transform('OSGB36', tx=-446.448, ty=125.157, tz=-542.060, 

sx= -0.1502, sy= -0.2470, sz= -0.8421, 

s= 20.4894), 

TokyoJapan = Transform('TokyoJapan', tx=148, ty=-507, tz=-685), 

WGS72 = Transform('WGS72', tz=-4.5, sz=0.554, s=-0.22), 

WGS84 = Transform('WGS84'), # unity 

) 

 

 

class Datum(_NamedEnumItem): 

'''Ellipsoid and transform parameters for an earth model. 

''' 

_ellipsoid = Ellipsoids.WGS84 #: (INTERNAL) Default ellipsoid (L{Ellipsoid}). 

_exactTM = None #: (INTERNAL) L{ExactTransverseMercator} projection. 

_transform = Transforms.WGS84 #: (INTERNAL) Default transform (L{Transform}). 

 

def __init__(self, ellipsoid, transform=None, name=''): 

'''New L{Datum}. 

 

@param ellipsoid: The ellipsoid (L{Ellipsoid}). 

@keyword transform: Optional transform (L{Transform}). 

@keyword name: Optional, unique name (C{str}). 

 

@raise NameError: Datum with that B{C{name}} already exists. 

 

@raise TypeError: If B{C{ellipsoid}} is not an L{Ellipsoid} 

or B{C{transform}} is not a L{Transform}. 

''' 

self._ellipsoid = ellipsoid or Datum._ellipsoid 

_TypeError(Ellipsoid, ellipsoid=self.ellipsoid) 

 

self._transform = transform or Datum._transform 

_TypeError(Transform, transform=self.transform) 

 

self._register(Datums, name or self.transform.name or self.ellipsoid.name) 

 

def __eq__(self, other): 

'''Compare this and an other datum. 

 

@param other: The other datum (L{Datum}). 

 

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

''' 

return self is other or (isinstance(other, Datum) and 

self.ellipsoid == other.ellipsoid and 

self.transform == other.transform) 

 

def ecef(self, Ecef=None): 

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

 

@keyword Ecef: ECEF class to use (L{EcefKarney}, L{EcefVeness} 

or L{EcefYou}). 

 

@return: An ECEF converter for this C{datum} (L{EcefKarney}, 

L{EcefVeness} or L{EcefYou}). 

 

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

''' 

return _4Ecef(self, Ecef) 

 

@property_RO 

def ellipsoid(self): 

'''Get this datum's ellipsoid (L{Ellipsoid}). 

''' 

return self._ellipsoid 

 

@property_RO 

def exactTM(self): 

'''Get the C{ExactTM} projection (L{ExactTransverseMercator}). 

''' 

if self._exactTM is None: 

from pygeodesy.etm import ExactTransverseMercator 

self._exactTM = ExactTransverseMercator(datum=self) 

return self._exactTM 

 

@property_RO 

def isEllipsoidal(self): 

'''Check whether this datum is ellipsoidal (C{bool}). 

''' 

return self._ellipsoid.isEllipsoidal 

 

@property_RO 

def isSpherical(self): 

'''Check whether this datum is spherical (C{bool}). 

''' 

return self._ellipsoid.isSpherical 

 

def toStr(self, **unused): # PYCHOK expected 

'''Return this datum as a string. 

 

@return: Datum attributes (C{str}). 

''' 

t = [] 

for a in ('ellipsoid', 'transform'): 

v = getattr(self, a) 

t.append('%s=%ss.%s' % (a, v.classname, v.name)) 

return ', '.join(['name=%r' % (self.named,)] + t) 

 

@property_RO 

def transform(self): 

'''Get this datum's transform (L{Transform}). 

''' 

return self._transform 

 

 

Datums = _NamedEnum('Datums', Datum) #: Registered datums. 

# Datums with associated ellipsoid and Helmert transform parameters 

# to convert from WGS84 into the given datum. More are available at 

# <https://Earth-Info.NGA.mil/GandG/coordsys/datums/NATO_DT.pdf> and 

# <XXX://www.FieldenMaps.info/cconv/web/cconv_params.js>. 

Datums._assert( 

# Belgian Datum 1972, based on Hayford ellipsoid. 

# <https://NL.WikiPedia.org/wiki/Belgian_Datum_1972> 

# <https://SpatialReference.org/ref/sr-org/7718/html/> 

BD72 = Datum(Ellipsoids.Intl1924, Transforms.BD72), 

 

# Germany <https://WikiPedia.org/wiki/Bessel-Ellipsoid> 

# <https://WikiPedia.org/wiki/Helmert_transformation> 

DHDN = Datum(Ellipsoids.Bessel1841, Transforms.DHDN), 

 

# <https://www.Gov.UK/guidance/oil-and-gas-petroleum-operations-notices#pon-4> 

ED50 = Datum(Ellipsoids.Intl1924, Transforms.ED50), 

 

# Australia <https://ICSM.Gov.AU/datum/gda2020-and-gda94-technical-manuals> 

# GDA94 = Datum(Ellipsoids.GRS80, Transforms.WGS84, name='GDA94'), 

GDA2020 = Datum(Ellipsoids.GRS80, Transforms.WGS84, name='GDA2020'), # XXX Transform? 

 

# <https://WikiPedia.org/wiki/GRS_80> 

GRS80 = Datum(Ellipsoids.GRS80, Transforms.WGS84, name='GRS80'), 

 

# <https://OSI.IE/OSI/media/OSI/Content/Publications/transformations_booklet.pdf> 

Irl1975 = Datum(Ellipsoids.AiryModified, Transforms.Irl1975), 

 

# Germany <https://WikiPedia.org/wiki/Helmert_transformation> 

Krassovski1940 = Datum(Ellipsoids.Krassovski1940, Transforms.Krassovski1940), # spelling 

Krassowsky1940 = Datum(Ellipsoids.Krassowsky1940, Transforms.Krassowsky1940), # spelling 

 

# Austria <https://DE.WikiPedia.org/wiki/Datum_Austria> 

MGI = Datum(Ellipsoids.Bessel1841, Transforms.MGI), 

 

# <https://WikiPedia.org/wiki/Helmert_transformation> 

NAD27 = Datum(Ellipsoids.Clarke1866, Transforms.NAD27), 

 

# NAD83 (2009) == WGS84 - <https://www.UVM.edu/giv/resources/WGS84_NAD83.pdf> 

# (If you *really* must convert WGS84<->NAD83, you need more than this!) 

NAD83 = Datum(Ellipsoids.GRS80, Transforms.NAD83), 

 

# Nouvelle Triangulation Francaise (Paris) XXX verify 

NTF = Datum(Ellipsoids.Clarke1880IGN, Transforms.NTF), 

 

# <https://www.OrdnanceSurvey.co.UK/docs/support/guide-coordinate-systems-great-britain.pdf> 

OSGB36 = Datum(Ellipsoids.Airy1830, Transforms.OSGB36), 

 

# Germany <https://WikiPedia.org/wiki/Helmert_transformation> 

Potsdam = Datum(Ellipsoids.Bessel1841, Transforms.Bessel1841, name='Potsdam'), 

 

# XXX psuedo-ellipsoids for spherical LatLon 

Sphere = Datum(Ellipsoids.Sphere, Transforms.WGS84, name='Sphere'), 

 

# <https://www.GeoCachingToolbox.com?page=datumEllipsoidDetails> 

TokyoJapan = Datum(Ellipsoids.Bessel1841, Transforms.TokyoJapan), 

 

# <https://www.ICAO.int/safety/pbn/documentation/eurocontrol/eurocontrol%20wgs%2084%20implementation%20manual.pdf> 

WGS72 = Datum(Ellipsoids.WGS72, Transforms.WGS72), 

 

WGS84 = Datum(Ellipsoids.WGS84, Transforms.WGS84), 

) 

 

 

if __name__ == '__main__': 

 

for E in (Datums.WGS84.ellipsoid, Datums.NAD83.ellipsoid, 

Ellipsoids.Sphere, Ellipsoids.SpherePopular): 

e = (E.a - E.b) / (E.a + E.b) - E.n 

if E.f: 

f_ = 'f_=1/%.10f' % (1 / E.f,) 

else: 

f_ = 'f_=N/A' 

t = (E.toStr(prec=10), 

'A=%r, e=%.13e, %s, n=%.10f(%.10e)' % (E.A, E.e, f_, E.n, e), 

'AlphaKs=%r' % (E.AlphaKs,), 

'BetaKs= %r' % (E.BetaKs,), 

'KsOrder=%r' % (E.KsOrder,), 

'Mabcd=%r' % (E.Mabcd,)) 

print('\nEllipsoid.%s: %s' % (E.name, ',\n '.join(t))) 

 

# __doc__ of this file 

for e in (Datums, Ellipsoids, Transforms): 

t = [''] + 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.datum 

 

# Ellipsoid.WGS84: name='WGS84', a=6378137, b=6356752.3142499998, f_=298.257223563, f=0.0033528107, e=0.0818191908, e2=0.00669438, e12=0.99330562, e22=0.0067394967, n=0.0016792204, R1=6371008.7714166669, R2=6371007.180920884, R3=6371000.7900107643, Rr=6367449.1458250266, Rs=6367435.6797186071, 

# A=6367449.145823414, e=8.1819190842621e-02, f_=1/298.2572235630, n=0.0016792204(-3.7914875232e-13), 

# AlphaKs=(0.0008377318206244698, 7.608527773572489e-07, 1.1976455032424919e-09, 2.4291706803970904e-12, 5.711818370428019e-15, 1.4799979313796632e-17, 4.1076241093707195e-20, 1.2107850389225785e-22), 

# BetaKs= (0.0008377321640579486, 5.905870152220365e-08, 1.6734826653438247e-10, 2.164798110490642e-13, 3.78793096862602e-16, 7.236769021815623e-19, 1.4934798247781072e-21, 3.2595225458381582e-24), 

# KsOrder=8, 

# Mabcd=(1.0016827510315587, 0.005046132931933289, 5.2959677624344715e-06, 6.90525779768578e-09) 

 

# Ellipsoid.GRS80: name='GRS80', a=6378137, b=6356752.3141403468, f_=298.257222101, f=0.0033528107, e=0.081819191, e2=0.00669438, e12=0.99330562, e22=0.0067394968, n=0.0016792204, R1=6371008.7713801153, R2=6371007.1808835128, R3=6371000.7899741307, Rr=6367449.1457702462, Rs=6367435.6796636879, 

# A=6367449.145771047, e=8.1819191042816e-02, f_=1/298.2572221010, n=0.0016792204(7.0906822081e-16), 

# AlphaKs=(0.0008377318247285514, 7.608527848149656e-07, 1.1976455208553066e-09, 2.4291707280367475e-12, 5.7118185104663376e-15, 1.4799979749262572e-17, 4.107624250383829e-20, 1.2107850864826076e-22), 

# BetaKs= (0.0008377321681620353, 5.905870210369122e-08, 1.6734826899771703e-10, 2.1647981529930687e-13, 3.7879310615900127e-16, 7.236769234953247e-19, 1.4934798760970368e-21, 3.2595226738732615e-24), 

# KsOrder=8, 

# Mabcd=(1.0016827510398385, 0.005046132956751664, 5.295967814484897e-06, 6.9052578994010674e-09) 

 

# Ellipsoid.Sphere: name='Sphere', a=6371008.7714149999, b=6371008.7714149999, f_=0, f=0, e=0, e2=0, e12=1, e22=0, n=0, R1=6371008.7714149999, R2=6371008.7714149999, R3=6371008.7714149999, Rr=6371008.7714149999, Rs=6371008.7714149999, 

# A=6371008.771415, e=0.0000000000000e+00, f_=N/A, n=0.0000000000(0.0000000000e+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, e=0, e2=0, e12=1, e22=0, n=0, R1=6378137, R2=6378137, R3=6378137, Rr=6378137, Rs=6378137, 

# A=6378137.0, e=0.0000000000000e+00, f_=N/A, n=0.0000000000(0.0000000000e+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)