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# -*- coding: utf-8 -*- 

 

u'''Ellipsoidal and spherical earth models. 

 

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

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

I{flattenings} and I{inverse flattening}. 

 

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

 

@var Ellipsoids.ATS1977: Ellipsoid(name='ATS1977', a=6378135, b=6356750.30492159, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367447.14116695, L=10001962.58040571, R1=6371006.7683072, R2=6371005.17780873, R3=6370998.78689182, Rbiaxial=6367451.62986519, Rtriaxial=6372795.55363648) 

@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, A=6366914.60892522, L=10001126.0807165, R1=6370461.23374576, R2=6370459.65470808, R3=6370453.30994572, Rbiaxial=6366919.065224, Rtriaxial=6372243.45317691) 

@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, A=6366691.77461988, L=10000776.05340819, R1=6370238.27531284, R2=6370236.69633043, R3=6370230.35179013, Rbiaxial=6366696.2307627, Rtriaxial=6372020.43236847) 

@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, A=6367471.84853228, L=10002001.39064442, R1=6371031.5730651, R2=6371029.9824858, R3=6371023.59124344, Rbiaxial=6367476.337459, Rtriaxial=6372820.40754721) 

@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, A=6366742.52023395, L=10000855.76443237, R1=6370291.09093933, R2=6370289.51012659, R3=6370283.15821523, Rbiaxial=6366746.98155108, Rtriaxial=6372074.29334012) 

@var Ellipsoids.BesselModified: Ellipsoid(name='BesselModified', a=6377492.018, b=6356173.5087127, f_=299.1528128, f=0.00334277, f2=0.00335398, n=0.00167418, e=0.08169683, e2=0.00667437, e22=0.00671922, e32=0.00334836, A=6366837.22474766, L=10001004.52593463, R1=6370385.84823756, R2=6370384.26740131, R3=6370377.91539546, Rbiaxial=6366841.68613115, Rtriaxial=6372169.07716325) 

@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, A=6366208.88184784, L=10000017.52721564, R1=6369383.10852498, R2=6369381.8434158, R3=6369376.76247022, Rbiaxial=6366212.45090321, Rtriaxial=6370977.3559758) 

@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, A=6367399.68916978, L=10001888.04298286, R1=6370998.86666667, R2=6370997.240633, R3=6370990.70659881, Rbiaxial=6367404.2783313, Rtriaxial=6372807.62791066) 

@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, A=6367386.64398051, L=10001867.55164747, R1=6371004.38651659, R2=6371002.74366963, R3=6370996.1419165, Rbiaxial=6367391.2806777, Rtriaxial=6372822.52526083) 

@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, A=6367386.73667336, L=10001867.69724907, R1=6371004.46666667, R2=6371002.82383112, R3=6370996.22212395, Rbiaxial=6367391.37333829, Rtriaxial=6372822.59907505) 

@var Ellipsoids.Clarke1880Mod: Ellipsoid(name='Clarke1880Mod', a=6378249.145, b=6356514.96639549, f_=293.46630766, f=0.00340755, f2=0.0034192, n=0.00170668, e=0.08248322, e2=0.00680348, e22=0.00685009, e32=0.00341335, A=6367386.69236201, L=10001867.62764495, R1=6371004.4187985, R2=6371002.77596616, R3=6370996.17427195, Rbiaxial=6367391.32901784, Rtriaxial=6372822.5494103) 

@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, A=6366197.07684334, L=9999998.98395793, R1=6369604.64205457, R2=6369603.18419749, R3=6369597.32739068, Rbiaxial=6366201.19059818, Rtriaxial=6371316.64722284) 

@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, A=6367448.17507971, L=10001964.20447208, R1=6371007.80757385, R2=6371006.21707085, R3=6370999.82613573, Rbiaxial=6367452.66379074, Rtriaxial=6372796.59560563) 

@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, A=6366699.57839501, L=10000788.3115495, R1=6370228.665305, R2=6370227.10178537, R3=6370220.81951618, Rbiaxial=6366703.99082487, Rtriaxial=6372002.02812501) 

@var Ellipsoids.Everest1975: Ellipsoid(name='Everest1975', a=6377299.151, b=6356098.14512013, f_=300.8017255, f=0.00332445, f2=0.00333554, n=0.00166499, e=0.08147298, e2=0.00663785, e22=0.0066822, e32=0.00332997, A=6366703.06049924, L=10000793.78122603, R1=6370232.14904004, R2=6370230.58551983, R3=6370224.30324826, Rbiaxial=6366707.47293076, Rtriaxial=6372005.51267879) 

@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, A=6367463.65604381, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995035, Rbiaxial=6367468.14345752, Rtriaxial=6372811.30979281) 

@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, A=6367449.14582474, L=10001965.7293148, R1=6371008.77141594, R2=6371007.18091936, R3=6371000.79001005, Rbiaxial=6367453.63451765, Rtriaxial=6372797.55596006) 

@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, A=6367471.74706533, L=10002001.2312605, R1=6371031.50536357, R2=6371029.91475409, R3=6371023.52339015, Rbiaxial=6367476.23607738, Rtriaxial=6372820.3568989) 

@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, A=6367449.14577104, L=10001965.72923046, R1=6371008.77138012, R2=6371007.18088351, R3=6371000.78997414, Rbiaxial=6367453.634464, Rtriaxial=6372797.55593326) 

@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, A=6367513.57227074, L=10002066.93013953, R1=6371072.7232093, R2=6371071.13315272, R3=6371064.74401563, Rbiaxial=6367518.05971963, Rtriaxial=6372861.26794141) 

@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, A=6367448.13949125, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669088, R3=6370999.78577297, Rbiaxial=6367452.62819019, Rtriaxial=6372796.55279934) 

@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, A=6367448.44699641, L=10001964.63159782, R1=6371008.07219738, R2=6371006.48170097, R3=6371000.09079236, Rbiaxial=6367452.93568883, Rtriaxial=6372796.85654541) 

@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, A=6367448.71771058, L=10001965.05683465, R1=6371008.35265722, R2=6371006.76215217, R3=6371000.37120877, Rbiaxial=6367453.20642742, Rtriaxial=6372797.14192686) 

@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, A=6367654.50005758, L=10002288.29898944, R1=6371229.31537598, R2=6371227.71133444, R3=6371221.26587487, Rbiaxial=6367659.02704315, Rtriaxial=6373025.77129687) 

@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, A=6367469.33894446, L=10001997.44859308, R1=6371029.06666667, R2=6371027.47608389, R3=6371021.08482752, Rbiaxial=6367473.827881, Rtriaxial=6372817.9027631) 

@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, A=6367558.49687498, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439, Rbiaxial=6367562.98435553, Rtriaxial=6372906.23027515) 

@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, A=6367558.49687498, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439, Rbiaxial=6367562.98435553, Rtriaxial=6372906.23027515) 

@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, A=6380564.13011837, L=10022566.69846922, R1=6386135.42757417, R2=6386131.54144847, R3=6386115.8862823, Rbiaxial=6380575.11882818, Rtriaxial=6388943.03218495) 

@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, A=6367479.62923643, L=10002013.61254591, R1=6371038.76120237, R2=6371037.17115427, R3=6371030.78205124, Rbiaxial=6367484.1166614, Rtriaxial=6372827.29640037) 

@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, A=6367463.65604381, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995035, Rbiaxial=6367468.14345752, Rtriaxial=6372811.30979281) 

@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, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802667, Rbiaxial=6367461.36258457, Rtriaxial=6372805.42010473) 

@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, A=6367448.3471653, L=10001964.47478349, R1=6371007.97231003, R2=6371006.38181364, R3=6370999.99090513, Rbiaxial=6367452.83585765, Rtriaxial=6372796.75662978) 

@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, A=6367448.44699773, L=10001964.63159991, R1=6371008.07219827, R2=6371006.48170186, R3=6371000.09079324, Rbiaxial=6367452.93569015, Rtriaxial=6372796.85654607) 

@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, A=6366197.15710739, L=9999999.11003639, R1=6369636.31108519, R2=6369634.82608583, R3=6369628.85999668, Rbiaxial=6366201.34758009, Rtriaxial=6371364.26393357) 

@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, A=6367448.13949125, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669087, R3=6370999.78577297, Rbiaxial=6367452.62819019, Rtriaxial=6372796.55279934) 

@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, A=6367471.84853228, L=10002001.39064442, R1=6371031.5730651, R2=6371029.98248581, R3=6371023.59124344, Rbiaxial=6367476.337459, Rtriaxial=6372820.40754721) 

@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, A=6371008.771415, L=10007557.17611675, R1=6371008.771415, R2=6371008.771415, R3=6371008.771415, Rbiaxial=6371008.771415, Rtriaxial=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, A=6371000, L=10007543.39801029, R1=6371000, R2=6371000, R3=6371000, Rbiaxial=6371000, Rtriaxial=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, A=6378137, L=10018754.17139462, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=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, A=6367482.31832549, L=10002017.83655714, R1=6371084.58088985, R2=6371082.95208988, R3=6371076.40691418, Rbiaxial=6367486.91530791, Rtriaxial=6372894.90029454) 

@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, A=6367478.63091189, L=10002012.0443814, R1=6371037.76231981, R2=6371036.17227197, R3=6371029.78316994, Rbiaxial=6367483.11833616, Rtriaxial=6372826.29723739) 

@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, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802667, Rbiaxial=6367461.36258457, Rtriaxial=6372805.42010473) 

@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, A=6367447.24862383, L=10001962.74919858, R1=6371006.84000536, R2=6371005.24953886, R3=6370998.8587507, Rbiaxial=6367451.7372317, Rtriaxial=6372795.60727472) 

@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, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594) 

''' 

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

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

 

from pygeodesy.basics import copysign0, _isfinite, isint 

from pygeodesy.errors import _AssertionError, _ValueError, _xkwds_not 

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

hypot1, hypot2, sqrt3 

# from pygeodesy.fsums import Fsum # from .fmath 

from pygeodesy.interns import EPS, EPS0, EPS02, EPS1, INF, NINF, NN, PI4, PI_2, R_M, _a_, \ 

_Airy1830_, _AiryModified_, _b_, _Bessel1841_, _Clarke1866_, \ 

_Clarke1880IGN_, _DOT_, _1_EPS, _EPSqrt, _EPStol as _TOL, _f_, \ 

_float as _F, _floatuple as _T, _GRS80_, _height_, _Intl1924_, \ 

_incompatible_, _invalid_, _Krassovski1940_, _Krassowsky1940_, \ 

_meridional_, _lat_, _negative_, _not_finite_, _null_, _vs_, \ 

_prime_vertical_, _radius_, _Sphere_, _SPACE_, _SQRT2_2, \ 

_WGS72_, _WGS84_, _0_0, _0_5, _1_0, _2_0, _4_0, _90_0 

from pygeodesy.interns import _0_25, _3_0, _8_0, _24_0 # PYCHOK used! 

from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS 

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

_NamedEnumItem, _NamedTuple, _Pass 

from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple 

from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_ 

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

from pygeodesy.units import Bearing_, Distance, Float, Float_, Height, Lam_, Lat, Meter, \ 

Meter2, Meter3, Phi, Phi_, Radius, Radius_, Scalar 

from pygeodesy.utily import atand, atan2b, atan2d, degrees90, m2km, m2NM, m2SM, \ 

m2radians, radians2m, sincos2d 

 

from math import asinh, atan, atanh, cos, degrees, exp, radians, sin, sinh, sqrt, tan 

 

R_M = Radius(R_M =R_M) # mean (spherical) earth radius (C{meter}) 

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

R_MB = Radius(R_MB=_F(6356752.3)) # polar earth radius (C{meter}), WGS84, EPSG:3785 

R_KM = Radius(R_KM=_F(m2km(R_M))) # mean (spherical) earth radius (C{KM}, kilo meter) 

R_NM = Radius(R_NM=_F(m2NM(R_M))) # mean (spherical) earth radius (C{NM}, nautical miles) 

R_SM = Radius(R_SM=_F(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=_F(6371000.0)) # former FAI Sphere earth radius (C{meter}) 

R_GM = Radius(R_GM=_F(6371230.0)) # Avg. radius, distance to geoid surface (C{meter}) 

R_VM = Radius(R_VM=_F(6366707.0194937)) # Aviation/Navigation earth radius (C{meter}) 

# R_ = Radius(R_ =_F(6372797.560856)) # XXX some other earth radius??? 

 

__all__ = _ALL_LAZY.ellipsoids 

__version__ = '22.05.04' 

 

_f_0_0 = Float(f =_0_0) # zero flattening 

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

# see U{WGS84_f()<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Constants.html>} 

_f__WGS84 = Float(f_=_1_0 / (1000000000 / 298257223563)) # 298.25722356299997 vs 298.257223563 

 

 

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

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

''' 

return Lat(lat, clip=clip, name=_lat_ if inverse else auxLat.__name__) 

 

 

def _s2_c2(phi): 

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

''' 

if phi: 

s2 = sin(phi)**2 

if s2 > 0: 

c2 = _1_0 - s2 

if c2 > 0: 

if c2 < _1_0: 

return s2, c2 

else: 

return _1_0, _0_0 # phi == PI_2 

return _0_0, _1_0 # phi == 0 

 

 

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_) # name 'f' not 'f_' 

_Units_ = (_Pass, _Pass) 

 

def __new__(cls, a, f, **name): 

'''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) # low=EPS, high=None 

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

if abs(f) < EPS: # force spherical 

f = _f_0_0 

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

 

@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 

 

def ellipsoid(self, name=NN): 

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

 

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

same B{C{name}} already exists. 

''' 

return Ellipsoid(self.a, f=self.f, name=name or self.name) # 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 Circle4Tuple(_NamedTuple): 

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

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

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

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

 

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

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

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

equals C{lat}. 

''' 

_Names_ = (_radius_, _height_, _lat_, 'beta') 

_Units_ = ( Radius, Height, Lat, Lat) 

 

 

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} radii, I{flattening}, I{inverse 

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

I{oblate} and I{prolate} ellipsoidal and I{spherical} earth 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) 

 

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

 

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

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

I{polar} radius or I{inverse flattening} or I{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). 

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

spherical). 

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

 

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

 

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

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

 

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

to M{1 / f_ = 0}, spherical. 

''' 

ff_ = f, f_ # assertion below 

try: 

a = Radius_(a=a) # low=EPS 

if not _isfinite(a): 

raise ValueError(_SPACE_(_a_, _not_finite_)) 

 

if b: # not in (_0_0, None) 

b = Radius_(b=b) # low=EPS 

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

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

elif f is not None: 

f = Float(f=f) 

b = a_f2b(a, f) 

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

elif f_: 

f_ = Float(f_=f_) 

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

f = f_2f(f_) 

else: # only a, spherical 

f = f_ = 0 

b = a # superfluous 

 

if not _isfinite(b): 

raise ValueError(_SPACE_(_b_, _not_finite_)) 

 

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

b = a 

f = _f_0_0 

f_ = _f__0_0 

elif f > EPS1: # sanity check, see .ecef.Ecef.__init__ 

raise ValueError(_SPACE_(_f_, _invalid_)) 

 

except (TypeError, ValueError) as x: 

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

t = instr(self, a=a, name=name, **d) 

raise _ValueError(t, txt=str(x)) 

 

self._a = a 

self._b = b 

self._f = f 

self._f_ = f_ 

 

self._register(Ellipsoids, name) 

 

if f and f_: # see .test/testEllipsoidal.py 

d = dict(eps=_TOL) 

if None in ff_: # both f_ and f given 

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

self._assert(_1_0 / f, f_=f_, **d) 

self._assert(_1_0 / f_, f =f, **d) 

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)) 

 

@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{meter**2}), M{a**2}. 

''' 

return Meter2(a2=self.a**2) 

 

@Property_RO 

def a2_(self): 

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

''' 

return Float(a2_=_1_0 / self.a2) 

 

@Property_RO 

def a_b(self): 

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

''' 

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

 

@Property_RO 

def a2_b(self): 

'''Get the I{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>} 

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>}, see L{c2} 

and L{R2} or L{Rauthalic}. 

''' 

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

 

@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)} == M{1 / (1 - e2)} == M{1 / e12}. 

''' 

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

 

@Property_RO 

def a_f(self): 

'''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}). 

''' 

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

 

@Property_RO 

def A(self): 

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

''' 

A, n = self.a, self.n 

if n: 

n1_A = (_1_0 + n) * 1048576 / A 

if n1_A: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8 

# <https://GeographicLib.SourceForge.io/html/tmseries30.html> 

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

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

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

A = Radius(A=Fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441).fover(n1_A)) 

return A 

 

@Property_RO 

def _albersCyl(self): 

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

''' 

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

 

@Property_RO 

def AlphaKs(self): 

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

''' 

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

# 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 

 

@Property_RO 

def area(self): 

'''Get the ellipsoid's surface area (C{meter**2}), M{4 * PI * c2}. 

 

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

''' 

return Meter2(area=self.c2 * PI4) 

 

@Property_RO 

def areax(self): 

'''Get the ellipsoid's surface area (C{meter**2}), M{4 * PI * c2x}, 

more accurate for very I{oblate} ellipsoids. 

 

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

''' 

return Meter2(areax=self.c2x * PI4) 

 

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

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

''' 

for n, v in name_value.items(): 

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

t = (v, _vs_, val) 

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

t = Fmt.EQUAL(self._DOT_(n), t) 

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

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

raise Error(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.USGS.gov/pp/1395/report.pdf>}, p 16. 

''' 

if self.f: 

f = self._albersCyl._tanf if inverse else self._albersCyl._txif # PYCHOK attr 

lat = atand(f(tan(Phi_(lat)))) # PYCHOK attr 

return _aux(lat, inverse, Ellipsoid.auxAuthalic) 

 

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.USGS.gov/pp/1395/report.pdf>}, pp 15-16. 

''' 

if self.f: 

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) 

 

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.USGS.gov/pp/1395/report.pdf>}, pp 17-18. 

''' 

if self.f: 

f = self.a2_b2 if inverse else self.b2_a2 

lat = atand(f * tan(Phi_(lat))) 

return _aux(lat, inverse, Ellipsoid.auxGeocentric) 

 

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.USGS.gov/pp/1395/report.pdf>}, pp 15-16. 

''' 

if self.f: 

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, 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.USGS.gov/pp/1395/report.pdf>}, p 18. 

''' 

if self.f: 

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

return _aux(lat, inverse, Ellipsoid.auxParametric) 

 

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.USGS.gov/pp/1395/report.pdf>}, pp 16-17. 

''' 

if self.f: 

lat = Lat(lat) 

if 0 < abs(lat) < _90_0: 

if inverse: 

e = self._elliptic_e22 

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

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

else: 

lat = _90_0 * self.Llat(lat) / self.L 

return _aux(lat, inverse, Ellipsoid.auxRectifying) 

 

@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 == f1 == 1 - f}. 

 

@see: Property L{f1}. 

''' 

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

 

@Property_RO 

def b2(self): 

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

''' 

return Meter2(b2=self.b**2) 

 

@Property_RO 

def b2_a(self): 

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

 

@see: U{Radii of Curvature 

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

''' 

return Radius(b2_a=self.b2 / self.a if self.f else self.b) 

 

@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} == C{e12}. 

''' 

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

 

def _beta(self, lat, inverse=False): 

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

''' 

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

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

return atan2d(s, c) # == atand(s / c) if c else copysign0(_90_0, lat) 

 

@Property_RO 

def BetaKs(self): 

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

''' 

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

# 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 

 

@deprecated_Property_RO 

def c(self): # PYCHOK no cover 

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

return self.R2 

 

@Property_RO 

def c2(self): 

'''Get the I{authalic} earth radius I{squared} (C{meter**2}). 

 

 

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

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

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Ellipsoid.html>}, 

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

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

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

''' 

return self._c2f(False) 

 

@Property_RO 

def c2x(self): 

'''Get the I{authalic} earth radius I{squared} (C{meter**2}), more accurate for very I{oblate} 

ellipsoids. 

 

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

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

''' 

return self._c2f(True) 

 

def _c2f(self, c2x): 

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

''' 

f = self.f 

if f: 

c2, e = self.b2, self.e 

if e > EPS0: 

if f > 0: # .isOblate 

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

elif f < 0: # .isProlate 

c2 *= atan(e) / e # XXX asin? 

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

else: 

c2 = self.a2 

return c2 

 

def circle4(self, lat): 

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

 

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

 

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

instance. 

 

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

L{pygeodesy.rangerrors} set to C{True}. 

 

@raise TypeError: Invalid B{C{lat}}. 

 

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

 

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

reduced) latitude}<https://WikiPedia.org/wiki/Latitude>} 

and I{Karney's} C++ U{CircleRadius and CircleHeight 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Ellipsoid.html>}. 

''' 

lat = Lat(lat) 

if lat: 

b = lat 

if abs(lat) < _90_0: 

if self.f: 

b = self._beta(lat) 

z, r = sincos2d(b) 

r *= self.a 

z *= self.b 

else: # near-polar 

r, z = _0_0, copysign0(self.b, lat) 

else: # equator 

r = self.a 

z = lat = b = _0_0 

return Circle4Tuple(r, z, lat, b) 

 

def degrees2m(self, deg, lat=0): 

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

along a parallel of (geodetic) latitude. 

 

@arg deg: The angle (C{degrees}). 

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

 

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

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

 

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

L{pygeodesy.rangerrors} set to C{True}. 

 

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

''' 

return self.radians2m(radians(deg), 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{pygeodesy.flatLocal}/L{pygeodesy.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{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}. 

 

@see: Property L{es}. 

''' 

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

 

@Property_RO 

def e2(self): 

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

== 1 - (b / a)**2}, see C{a_b2e2}. 

''' 

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

 

@Property_RO 

def e2abs(self): 

'''Get the I{unsigned, (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 C{b2_a2}. 

''' 

return self._assert((_1_0 - self.f)**2, e12=_1_0 - self.e2, f0=_1_0) # 1 - e2 

 

_1_e12 = a2_b2 # M{1 / e12} == M{1 / (1 - e**2)} 

 

@Property_RO 

def e22(self): 

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

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

''' 

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

 

@Property_RO 

def e22abs(self): 

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

''' 

return abs(self.e22) 

 

@Property_RO 

def e32(self): 

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

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

''' 

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

 

@Property_RO 

def e32abs(self): 

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

''' 

return abs(self.e32) 

 

@Property_RO 

def e4(self): 

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

''' 

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

 

eccentricity = e # eccentricity 

# eccentricity2 = e2 # eccentricity squared 

eccentricity1st2 = e2 # first eccentricity squared 

eccentricity2nd2 = e22 # second eccentricity squared 

eccentricity3rd2 = e32 # third eccentricity squared 

 

def ecef(self, Ecef=None): 

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

 

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

 

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

 

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

 

@see: Module L{pygeodesy.ecef}. 

''' 

return _MODS.ecef._4Ecef(self, Ecef) 

 

@Property_RO 

def _elliptic_e22(self): 

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

''' 

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

 

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)) if self.e2 else _1_0 

 

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}}. 

''' 

r = _1_0 

if self.e2: 

try: 

r -= self.e2 * Scalar(s=s)**2 

if r < 0: 

raise ValueError(_negative_) 

except (TypeError, ValueError) as x: 

t = self._DOT_(Ellipsoid.e2s2.__name__) 

raise _ValueError(t, s, txt=str(x)) 

return r 

 

@Property_RO 

def es(self): 

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

 

@see: Property L{e}. 

''' 

# note, self.e is always non-negative 

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

 

def es_atanh(self, x): 

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

for I{oblate} respectively I{prolate} ellipsoids 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>}. 

''' 

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

 

def _es_atanh(self, x): 

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

''' 

es = self.es 

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

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

_0_0)) # .isSpherical 

 

@Property_RO 

def es_c(self): 

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

''' 

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

 

def es_tauf(self, taup): 

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

<https://ArXiv.org/abs/1002.1417>}. 

 

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

SourceForge.io/html/classGeographicLib_1_1Math.html>} and 

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

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

''' 

t = Scalar(taup=taup) 

if self.f: # .isEllipsoidal 

a = abs(t) 

T = t * (self._exp_es_atanh_1 if a > 70 else self._1_e12) 

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

s = (a * _TOL) if a > _1_0 else _TOL 

for T, _, d in self._es_tauf3(t, T): 

if abs(d) < s: 

break 

t = Scalar(tauf=T) 

return t 

 

def _es_tauf3(self, taup, T): # in .utm.Utm._toLLEB 

'''(INTERNAL) Yield a 3-tuple C{(T, iteration, delta)} for up to 

9 Newton iterations, converging rapidly except when C{delta} 

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

''' 

F = Fsum(T) # τ0 

e = self._1_e12 

for i in range(1, 10): # max 2, mean 1.999, 1.963 or 1.954 

a, h = self._es_taupf2(T) 

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

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

T, d = F.fsum2_(d) # τi, (τi - τi-1) 

yield T, i, d 

 

def es_taupf(self, tau): 

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

<https://ArXiv.org/abs/1002.1417>}. 

 

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

SourceForge.io/html/classGeographicLib_1_1Math.html>}. 

''' 

t = Scalar(tau=tau) 

if self.f: # .isEllipsoidal 

t, _ = self._es_taupf2(t) 

t = Scalar(taupf=t) 

return t 

 

def _es_taupf2(self, tau): 

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

''' 

h = hypot1(tau) 

s = sinh(self._es_atanh(tau / h)) 

a = hypot1(s) * tau - h * s 

return a, h 

 

@Property_RO 

def _exp_es_atanh_1(self): 

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

''' 

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

 

@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 

 

@Property_RO 

def f_(self): 

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

''' 

return self._f_ 

 

@Property_RO 

def f1(self): 

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

 

@see: Property L{b_a}. 

''' 

return Float(f1=_1_0 - self.f) 

 

@Property_RO 

def f2(self): 

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

''' 

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

 

@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 not self.isEllipsoidal: 

# raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

return _MODS.karney._wrapped.Geodesic(self) 

 

@Property_RO 

def _geodesic_Math2(self): 

'''(INTERNAL) Get this ellipsoid's I{wrapped Karney} C{Geodesic} 

and I{Karney}'s C{Math} class, see L{geodesic}. 

''' 

return self.geodesic, _MODS.karney._wrapped.Math 

 

@Property_RO 

def geodesicx(self): 

'''Get this ellipsoid's L{GeodesicExact}. 

''' 

# if not self.isEllipsoidal: 

# raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

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

 

@Property_RO 

def geodsolve(self): 

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

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

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

variable C{PYGEODESY_GEODSOLVE}. 

''' 

# if not self.isEllipsoidal: 

# raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

return _MODS.geodsolve.GeodesicSolve(self, name=self.name) 

 

def height4(self, xyz, normal=True): 

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

on this ellipsoid's surface. 

 

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

L{Vector3Tuple} or L{Vector4Tuple}). 

@kwarg normal: If C{True} the projection is the nearest point on the 

ellipsoid's surface, otherwise the intersection of the 

radial line to the center and the ellipsoid's surface. 

 

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

C{x}, C{y} and C{z} of the projection on or the intersection 

with and with the height C{h} above the ellipsoid's surface 

in C{meter}, conventionally. 

 

@raise ValueError: Null B{C{xyz}}. 

 

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

 

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

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

''' 

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

r = v.length 

if r < EPS0: # EPS 

raise _ValueError(xyz=xyz, txt=_null_) 

 

i = None 

a, b = self.a, self.b 

if self.isSpherical: 

v = v.times(a / r) 

h = r - a 

 

elif normal: # perpendicular to ellipsoid 

x, y = hypot(v.x, v.y), abs(v.z) 

if x < EPS0: # PYCHOK no cover 

z = copysign0(b, v.z) 

v = Vector3Tuple(v.x, v.y, z) 

h = y - b # polar 

elif y < EPS0: # PYCHOK no cover 

t = a / r 

v = v.times_(t, t, 0) # force z=0.0 

h = x - a # equatorial 

else: # normal in 1st quadrant 

x, y, i = _normal3(x, y, self) 

t, v = v, v.times_(x, x, y) 

h = t.minus(v).length 

 

else: # radial to ellipsoid's center 

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

if t < EPS0: # EPS 

raise _ValueError(xyz=xyz, txt=_null_) 

t = a * b / t 

v = v.times(t) 

h = r * (_1_0 - t) 

 

return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, 

name=self.height4.__name__) 

 

def _hubeny_2(self, phi2, phi1, lam21): 

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

 

def _Kseries(self, *AB8Ks): 

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

or Beta series coefficients per I{Karney}'s U{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 (L{KsOrder}C{-tuple}). 

 

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

<https://GeographicLib.SourceForge.io/html/tmseries30.html>}. 

''' 

k = self.KsOrder 

if self.n: 

ns = fpowers(self.n, k) 

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

else: 

ks = (_0_0,) * k 

return ks 

 

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

def KsOrder(self): 

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

''' 

return self._KsOrder 

 

@KsOrder.setter # PYCHOK setter! 

def KsOrder(self, order): 

'''Set the I{Krüger} series' order. 

 

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

 

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

''' 

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

raise _ValueError(order=order) 

if order != self._KsOrder: 

Ellipsoid.AlphaKs._update(self) 

Ellipsoid.BetaKs._update(self) 

self._KsOrder = order 

 

@Property_RO 

def L(self): 

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

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

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

''' 

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

return Distance(L=self.b * r) 

 

def Llat(self, lat): 

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

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

 

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

 

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

hemisphere (C{meter}). 

''' 

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

return Distance(Llat=self.b * r) 

 

Lmeridian = Llat # meridional distance 

 

@Property_RO 

def Mabcd(self): 

'''Get the OSGR meridional coefficients (C{4-Tuple}), C{Airy130} only. 

''' 

if self.n: 

n1, n2, n3 = fpowers(self.n, 3) # PYCHOK false! 

Mabcd = (Fsum(4, n1 * 4, n2 * 5, n3 * 5).fover(_4_0), 

Fsum( n1 * 24, n2 * 24, n3 * 21).fover(_8_0), 

Fsum( n2 * 15, n3 * 15).fover(_8_0), 

(n3 * 35 / _24_0)) 

else: 

Mabcd = _1_0, _0_0, _0_0, _0_0 

return Mabcd 

 

@deprecated_Property_RO 

def majoradius(self): # PYCHOK no cover 

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

return self.a 

 

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

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

along a parallel of (geodetic) latitude. 

 

@arg distance: Distance (C{meter}). 

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

 

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

 

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

L{pygeodesy.rangerrors} set to C{True}. 

 

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

''' 

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

 

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

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

along a parallel of (geodetic) latitude. 

 

@arg distance: Distance (C{meter}). 

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

 

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

 

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

L{pygeodesy.rangerrors} set to C{True}. 

 

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

''' 

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

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

 

@deprecated_Property_RO 

def minoradius(self): # PYCHOK no cover 

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

return self.b 

 

@Property_RO 

def n(self): 

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

''' 

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

 

flattening = f 

flattening1st = f 

flattening2nd = f2 

flattening3rd = n 

 

@deprecated_Property_RO 

def quarteradius(self): # PYCHOK no cover 

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

return self.L 

 

@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}. 

''' 

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

return Radius(R1=r) 

 

Rmean = R1 

 

@Property_RO 

def R2(self): 

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

 

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

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

''' 

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

 

Rauthalic = R2 

 

@Property_RO 

def R2x(self): 

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

 

@see: C{R2}, C{c2x} and C{areax}. 

''' 

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

 

Rauthalicx = R2x 

 

@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>} and C{volume}. 

''' 

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

return Radius(R3=r) 

 

Rvolumetric = R3 

 

def radians2m(self, rad, lat=0): 

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

along a parallel of (geodetic) latitude. 

 

@arg rad: The angle (C{radians}). 

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

 

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

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

 

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

L{pygeodesy.rangerrors} set to C{True}. 

 

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

''' 

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

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

 

@Property_RO 

def Rbiaxial(self): 

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

 

@see: C{Rtriaxial} 

''' 

b = (sqrt((_1_0 + self.b2_a2) * _0_5) * self.a) if self.f else self.a 

return Radius(Rbiaxial=b) 

 

Requatorial = a # for consistent naming 

 

def Rgeocentric(self, lat): 

'''Compute the I{geocentric} earth radius of (geodetic) 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>} 

''' 

r, a = self.a, Phi_(lat) 

if a and self.f: 

if abs(a) < PI_2: 

s2, c2 = _s2_c2(a) 

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)) 

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

else: 

r = self.b 

return Radius(Rgeocentric=r) 

 

@Property_RO 

def Rgeometric(self): 

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

 

@see: C{R1}. 

''' 

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

return Radius(Rgeometric=g) 

 

def Rlat(self, lat): 

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

 

@arg lat: Latitude (C{degrees90}). 

 

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

 

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

L{pygeodesy.rangerrors} set to C{True}. 

 

@raise TypeError: Invalid B{C{lat}}. 

 

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

 

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

 

@see: Method C{Rparallel}. 

''' 

# r = a - (a - b) * |lat| / 90 

r = self.a 

if self.f and lat: # .isEllipsoidal 

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

r = Radius(Rlat=r) 

return r 

 

Rpolar = b # for consistent naming 

 

@deprecated_Property_RO 

def Rquadratic(self): # PYCHOK no cover 

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

return self.Rbiaxial 

 

@deprecated_Property_RO 

def Rr(self): # PYCHOK no cover 

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

return self.Rrectifying 

 

@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>}. 

''' 

r = (cbrt2((_1_0 + sqrt3(self.b_a)) * _0_5) * self.a) if self.f else self.a 

return Radius(Rrectifying=r) 

 

def roc1_(self, sa, ca=None): 

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

of (geodetic) 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: # .isSpherical 

n = self.a 

elif ca is None: 

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

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

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 of (geodetic) latitude. 

 

@arg lat: Latitude (C{degrees90}). 

@kwarg scaled: Scale prime_vertical by C{cos(radians(B{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 of (geodetic) latitude. 

 

@arg phi: Latitude (C{radians}). 

@kwarg scaled: Scale prime_vertical by C{cos(B{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 the meridional and prime vertical U{Radii of Curvature 

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

''' 

a = abs(Phi(phi)) 

if self.f: 

r = self.e2s2(sin(a)) 

if r > EPS02: 

n = self.a / sqrt(r) 

m = n * self.e12 / r # PYCHOK attr 

else: 

m = n = _0_0 # PYCHOK attr 

else: 

m = n = self.a 

if scaled and a: 

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 

of (geodetic) 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{pygeodesy.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>} 

''' 

if self.f: 

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 

b = n / (c2 + s2) # == 1 / (c2 / m + s2 / n) 

else: 

b = self.b # == self.a 

return Radius(rocBearing=b) 

 

def rocGauss(self, lat): 

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

g = self.b 

if self.f: 

s2, c2 = _s2_c2(Phi_(lat)) 

g = g / (c2 + self.b2_a2 * s2) 

return Radius(rocGauss=g) 

 

def rocMean(self, lat): 

'''Compute the I{mean} radius of curvature of (geodetic) 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>} 

''' 

if self.f: 

m, n = self.roc2_(Phi_(lat)) 

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

else: 

m = self.a 

return Radius(rocMean=m) 

 

def rocMeridional(self, lat): 

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

 

rocPolar = a2_b # synonymous 

 

def rocPrimeVertical(self, lat): 

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

of (geodetic) 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 

 

@deprecated_Property_RO 

def Rs(self): # PYCHOK no cover 

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

return self.Rgeometric 

 

@Property_RO 

def Rtriaxial(self): 

'''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}. 

 

@see: C{Rbiaxial} 

''' 

t = (sqrt((_3_0 + self.b2_a2) * _0_25) * self.a) if self.f else self.a 

return Radius(Rtriaxial=t) 

 

def toStr(self, prec=8, name=NN, **unused): # PYCHOK expected 

'''Return this ellipsoid as a text string. 

 

@kwarg prec: Number of decimal digits, unstripped (C{int}). 

@kwarg name: Override name (C{str}) or C{None} to exclude 

this ellipsoid's name. 

 

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

''' 

E = Ellipsoid 

return self._instr(name, prec, _a_, E.b.name, E.f_.name, _f_, E.f2.name, E.n.name, 

E.e.name, E.e2.name, E.e22.name, E.e32.name, 

E.A.name, E.L.name, E.R1.name, E.R2.name, E.R3.name, 

E.Rbiaxial.name, E.Rtriaxial.name) 

 

@Property_RO 

def volume(self): 

'''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}. 

 

@see: C{R3}. 

''' 

return Meter3(volume=self.a2 * self.b * (PI4 / _3_0)) 

 

 

class Ellipsoid2(Ellipsoid): 

'''An L{Ellipsoid} 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 I{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. 

''' 

Ellipsoid.__init__(self, a, f=f, name=name) 

 

 

def _spherical_a_b(a, b): 

'''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}. 

''' 

return a < EPS or b < EPS or abs(a - b) < EPS 

 

 

def _spherical_f(f): 

'''(INTERNAL) C{True} for spherical or invalid C{f}. 

''' 

return abs(f) < EPS or f > EPS1 

 

 

def _spherical_f_(f_): 

'''(INTERNAL) C{True} for spherical or invalid C{f_}. 

''' 

return abs(f_) < EPS or abs(f_) > _1_EPS 

 

 

def a_b2e(a, b): 

'''Return C{e}, the I{1st 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 I{unsigned}, (1st) eccentricity (C{float} or C{0}), 

M{sqrt(1 - (b / a)**2)}. 

 

@note: The result is always I{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{1st 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 I{signed}, (1st) 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 I{signed}, 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 I{signed}, 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 ((a - b) / a) 

return _f_0_0 if _spherical_f(f) else Float(f=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 _f__0_0 if _spherical_f_(f_) else Float(f_=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 I{signed}, 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 I{signed}, 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)}. 

''' 

b = a if _spherical_f(f) else (a * (_1_0 - f)) 

return Radius_(b=a if _spherical_a_b(a, b) else b) 

 

 

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_}. 

''' 

b = a if _spherical_f_(f_) else (a * (f_ - _1_0) / f_) 

return Radius_(b=a if _spherical_a_b(a, b) else b) 

 

 

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 

a = b if abs(t < EPS) else (b / t) 

return Radius_(a=b if _spherical_a_b(a, b) else a) 

 

 

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 

a = b if _spherical_f_(f_) or abs(t - f_) < EPS \ 

or abs(t) < EPS else (b * f_ / t) 

return Radius_(a=b if _spherical_a_b(a, b) else a) 

 

 

def f2e2(f): 

'''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}. 

 

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

 

@return: The I{signed}, (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(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 I{signed}, 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 I{signed}, 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_(f_) else _1_0 / f_ 

return _f_0_0 if _spherical_f(f) else Float(f=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(f) else _1_0 / f 

return _f__0_0 if _spherical_f_(f_) else Float(f_=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 I{signed}, 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(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 I{signed}, 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(f) else (f / float(_2_0 - f))) 

 

 

def n2e2(n): 

'''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}. 

 

@arg n: The 3rd flattening (-1 <= C{scalar} < 1). 

 

@return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF), 

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 

(NINF 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 NINF), 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 (NINF if t < EPS else (_2_0 * n / t)) 

return _f_0_0 if _spherical_f(f) else Float(f=f) 

 

 

def n2f_(n): 

'''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}. 

 

@arg n: The 3rd flattening (-1 <= C{scalar} < 1). 

 

@return: The inverse flattening (C{float} or C{0}), M{1 / f}. 

 

@see: L{n2f} and L{f2f_}. 

''' 

return f2f_(n2f(n)) 

 

 

def _normal3(px, py, E): # in .height4 above 

'''(INTERNAL) Nearest point on a 2-D ellipse in 1st quadrant. 

''' 

a, b = E.a, E.b 

if min(px, py, a, b) < EPS0: 

raise _AssertionError(px=px, py=py, a=a, b=b, E=E) 

 

a2 = a - b * E.b_a 

b2 = b - a * E.a_b 

tx = ty = _SQRT2_2 

for i in range(10): # max 5 

ex = a2 * tx**3 

ey = b2 * ty**3 

 

qx = px - ex 

qy = py - ey 

q = hypot(qx, qy) 

if q < EPS0: # PYCHOK no cover 

break 

r = hypot(ex - tx * a, ey - ty * b) / q 

 

sx, tx = tx, min(_1_0, max(0, (ex + qx * r) / a)) 

sy, ty = ty, min(_1_0, max(0, (ey + qy * r) / b)) 

t = hypot(ty, tx) 

if t < EPS0: # PYCHOK no cover 

break 

tx = tx / t # /= chokes PyChecker 

ty = ty / t 

if max(abs(sx - tx), abs(sy - ty)) < EPS: 

break 

 

tx *= a / px 

ty *= b / py 

return tx, ty, i # x and y as fractions 

 

 

class Ellipsoids(_NamedEnum): 

'''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class 

to accommodate the L{_LazyNamedEnumItem} properties. 

''' 

def _Lazy(self, a, b, f_, **kwds): 

'''(INTERNAL) Instantiate the L{Ellipsoid}. 

''' 

return Ellipsoid(a, b=b, f_=f_, **kwds) 

 

Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton 

'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.''' 

# <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> 

# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids> 

Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid> 

Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909 

AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448 

# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid 

Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192 

ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System" 

Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)), 

BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)), 

# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)), 

# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)), 

Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)), 

Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)), 

Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)), 

Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc 

CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures 

Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium 

Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)), 

Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)), 

Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)), # Everest 1830 1975 Def'n 

Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)), 

# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)), 

GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)), 

# GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid" 

GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516 

# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)), 

GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.257222101)), # ITRS, ETRS89 

# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, _0_0)), # aka Intl1924 f_=297 

Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)), 

# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)), 

IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302 

IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy 

IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)), 

Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795 

Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l 

Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling 

Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling 

Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France 

Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)), 

Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)), 

# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab. 

NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab. 

# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965 

OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)), 

OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)), 

# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect? 

Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972 

SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System 

SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American 

Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)), 

# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)), 

# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)), 

WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)), 

WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)), 

WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52 

WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # GPS b=6356752.3142451793 

# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)), 

Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo 

SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo 

SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)) # EPSG:3857 Spheroid 

) 

 

 

if __name__ == '__main__': 

 

from pygeodesy.interns import _COMMA_, _NL_, _NL_hash_, _NL_var_ 

from pygeodesy.named import nameof 

 

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 = NN(_COMMA_, _NL_hash_, _SPACE_)(E.toStr(prec=10), # re-callable 

'e=%s, f_=%s, f=%s, n=%s (%s)' % (fstr(E.e, prec=13, fmt=Fmt.e), 

fstr(E.f_, prec=13, fmt=Fmt.e), 

fstr(E.f, prec=13, fmt=Fmt.e), 

fstr(E.n, prec=13, fmt=Fmt.e), 

fstr(e, prec=9, fmt=Fmt.e),), 

'%s=(%s)' % (Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20),), 

'%s= (%s)' % (Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20),), 

'%s= %s' % (nameof(Ellipsoid.KsOrder), E.KsOrder), # property 

'%s= (%s)' % (Ellipsoid.Mabcd.name, fstr(E.Mabcd, prec=20),)) 

print('%s%s: %s' % (_NL_hash_, _DOT_(E.classname, E.name), t)) 

 

# __doc__ of this file, force all into registry 

t = [NN] + Ellipsoids.toRepr(all=True).split(_NL_) 

print(_NL_var_.join(i.strip(_COMMA_) for i in t)) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2022 -- 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. 

 

# % python3 -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, A=6367449.1458234144, L=10001965.7293127216, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.7900091587, Rbiaxial=6367453.6345163295, Rtriaxial=6372797.5559594007, 

# e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00), 

# AlphaKs=(0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0), 

# BetaKs= (0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0), 

# KsOrder= 8, 

# Mabcd= (1.00168275103155868244, 0.00504613293193329013, 0.00000529596776243447, 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, A=6367449.1457710434, L=10001965.7292304579, R1=6371008.7713801153, R2=6371007.1808835147, R3=6371000.7899741363, Rbiaxial=6367453.6344640013, Rtriaxial=6372797.5559332585, 

# e=8.1819191042833e-02, f_=2.98257222101e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (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, A=6371008.7714149999, L=10007557.1761167478, R1=6371008.7714149999, R2=6371008.7714149999, R3=6371008.7714149999, Rbiaxial=6371008.7714149999, Rtriaxial=6371008.7714149999, 

# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (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, A=6378137, L=10018754.171394622, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137, 

# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (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, A=6367449.1458234144, L=10035500.5204500332, R1=6363880.5428301189, R2=6363878.9413582645, R3=6363872.5644020075, Rbiaxial=6367453.6345163304, Rtriaxial=6362105.2243882548, 

# e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (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)