# Coverage for pygeodesy/ellipsoids.py : 98%

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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.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.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.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.CGCS2000: Ellipsoid(name='CGCS2000', a=6378137, b=6356752.31414036, 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=6367449.14577105, L=10001965.72923046, R1=6371008.77138012, R2=6371007.18088352, R3=6371000.78997414, Rbiaxial=6367453.63446401, Rtriaxial=6372797.55593326) @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.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.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.GPES: Ellipsoid(name='GPES', a=6378135, b=6378135, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6378135, L=10018751.02980197, R1=6378135, R2=6378135, R3=6378135, Rbiaxial=6378135, Rtriaxial=6378135) @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.PZ90: Ellipsoid(name='PZ90', a=6378136, b=6356751.36174571, f_=298.2578393, f=0.0033528, f2=0.00336408, n=0.00167922, e=0.08181911, e2=0.00669437, e22=0.00673948, e32=0.00335842, A=6367448.16955443, L=10001964.19579298, R1=6371007.78724857, R2=6371006.1967588, R3=6370999.80587691, Rbiaxial=6367452.65822809, Rtriaxial=6372796.56780569) @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, isint from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, INF, NINF, PI4, PI_2, PI_3, R_M, R_MA, R_FM, \ _EPSqrt, _EPStol as _TOL, _floatuple as _T, _isfinite, _SQRT2_2, \ _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \ _0_25, _3_0 # PYCHOK used! 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 NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, \ _Clarke1866_, _Clarke1880IGN_, _DOT_, _f_, _GRS80_, _height_, \ _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \ _Krassowsky1940_, _meridional_, _lat_, _negative_, _not_finite_, \ _null_, _vs_, _prime_vertical_, _radius_, _Sphere_, _SPACE_, \ _WGS72_, _WGS84_ 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_, property_RO 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, m2radians, radians2m, sincos2d from math import asinh, atan, atanh, cos, degrees, exp, radians, sin, sinh, sqrt, tan __all__ = _ALL_LAZY.ellipsoids __version__ = '22.10.22' _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/C++/doc/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}). @kwarg name: Optional name (C{str}). @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) @property_RO def transverse(self): '''Get this I{prime_vertical}, aka I{transverse} radius of curvature. ''' return self.prime_vertical 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 f < _1_0: # sanity check, see .ecef.Ecef.__init__ raise ValueError(_SPACE_(_f_, _invalid_)) 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 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, cause=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, e21=self.e21, 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.f == other.f or self.b == other.b)) @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} I{squared}), 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} I{squared}), 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 (or polar) 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 / e21}. ''' 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}), see method C{toEllipsoid2}. ''' 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}). @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/ a-highly-accurate-world-wide-algorithm-for-the-transverse...>}. ''' A, n = self.a, self.n if n: d = (n + _1_0) * 1048576 / A if d: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html> # <https://GeographicLib.SourceForge.io/C++/doc/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(d)) 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/C++/doc/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} I{squared}), M{4 * PI * c2}. @see: Properties L{areax}, L{c2} and L{R2} and functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}. ''' return Meter2(area=self.c2 * PI4) @Property_RO def areax(self): '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more accurate for very I{oblate} ellipsoids. @see: Properties L{area}, L{c2x} and L{R2x}, class L{GeodesicExact} and functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}. ''' 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 the I{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 the I{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 the I{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 the I{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 the I{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 # synonymous def auxRectifying(self, lat, inverse=False): '''Compute the I{rectifying} auxiliary latitude or the I{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{e21}. ''' 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/C++/doc/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} I{squared}). @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/C++/doc/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} I{squared}), 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/C++/doc/GeodesicExact_8cpp_source.html>}. ''' return self._c2f(True) def _c2f(self, c2x): '''(INTERNAL) Helper for C{.c2} and C{.c2x}. ''' f, c2 = self.f, self.b2 if f: e = 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) 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>}, I{Karney's} C++ U{CircleRadius and CircleHeight <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>} and method C{Rlat}. ''' 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) @deprecated_Property_RO def e12(self): # see property ._e12 '''DEPRECATED, use property C{e21}.''' return self.e21 # @Property_RO # def _e12(self): # see property ._elliptic_e12 # # (INTERNAL) until e12 above can be replaced with e21. # return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2) @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 e21(self): '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2} == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}. ''' return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0) # _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2 _1_e21 = a2_b2 # == M{1 / e21} == 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_e12(self): # see I{Karney}'s Ellipsoid._e12 '''(INTERNAL) Elliptic helper for C{rhumbx.Rhumb}. ''' e12 = self.e2 / (_1_0 - self.e2) # NOT DEPRECATED .e12! return _MODS.elliptic.Elliptic(-e12) @Property_RO def _elliptic_e22(self): # aka ._elliptic_ep2 '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}. ''' return _MODS.elliptic.Elliptic(-self.e22abs) # complex equatoradius = a # Requatorial 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: Sine 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, cause=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 # signOf(es) == signOf(f) 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_e21) 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): # max 2 if abs(d) < s: break t = Scalar(tauf=T) return t def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most B{C{N}} Newton iterations, converging rapidly except when C{delta} toggles on +/-1.12e-16 or +/-4.47e-16, see C{.utm.Utm._toLLEB}. ''' e = self._1_e21 _F2_ = Fsum(T).fsum2_ # τ0 _tf2 = self._es_taupf2 for i in range(1, N + 1): a, h = _tf2(T) d = (taup - a) * (e + T**2) / (hypot1(a) * h) # = (taup - a) / hypot1(a) / ((e + T**2) / h) T, d = _F2_(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))}. ''' if _isfinite(tau): h = hypot1(tau) s = sinh(self._es_atanh(tau / h)) a = hypot1(s) * tau - h * s else: a, h = tau, INF 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/C++/doc/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 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/C++/doc/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 projection on and the height of a cartesian above or below 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 = _normalTo3(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/C++/doc/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_0s(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 (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{b * PI / 2}. ''' r = self._elliptic_e22.cE if self.f else PI_2 return Distance(L=self.b * r) @Property_RO def _L_90(self): '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}. ''' return self.L / _90_0 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 @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 polaradius = b # Rpolar # @Property_RO # def Q(self): # '''Get the I{meridian arc unit} C{Q}, the mean, meridional length I{per radian} C({float}). # # @note: C{Q * PI / 2} ≈ C{L}, the I{quarter meridian}. # # @see: Property C{A} and U{Engsager, K., Poder, K.<https://StudyLib.net/doc/7443565/ # a-highly-accurate-world-wide-algorithm-for-the-transverse...>}. # ''' # n = self.n # d = (n + _1_0) / self.a # return Float(Q=Fhorner(n**2, _1_0, _0_25, _1_16th, _0_25).fover(d) if d else self.b) # # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf> # # Q = (1 - 3/4 * e'2 + 45/64 * e'4 - 175/256 * e'6 + 11025/16384 * e'8) * rocPolar # # = (4 + e'2 * (-3 + e'2 * (45/16 + e'2 * (-175/64 + e'2 * 11025/4096)))) * rocPolar / 4 # return Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(4 / self.rocPolar) @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 == a * (1 - f / 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 R2(self): # # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf> # # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar # # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3 # return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar) @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} ''' a, b = self.a, self.b q = (sqrt((_1_0 + self.b2_a2) * _0_5) * a) if a > b else ( (sqrt((_1_0 + self.a2_b2) * _0_5) * b) if a < b else a) return Radius(Rbiaxial=q) 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) @Property_RO def rhumbsolve(self): '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}, provided the path to the C{RhumbSolve} executable is specified with env variable C{PYGEODESY_RHUMBSOLVE}. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.rhumbsolve.RhumbSolve(self, name=self.name) @Property_RO def rhumbx(self): '''Get this ellipsoid's L{Rhumb}. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.rhumbx.Rhumb(self, name=self.name) 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{circle4}. ''' # 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_}, 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_}, property L{rocEquatorial2}, 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.e21 / 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) @Property_RO def rocEquatorial2(self): '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}. @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a}, L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return Curvature2Tuple(Radius(rocMeridional0=self.b2_a if self.f else self.a), Radius(rocPrimeVertical0=self.a)) 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: Non-directional 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: Non-directional 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_}, 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 if lat else \ self.rocEquatorial2.meridional rocPolar = a2_b # synonymous def rocPrimeVertical(self, lat): '''Compute the I{prime-vertical}, I{normal} radius of curvature of (geodetic) latitude, aka the I{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_}, 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 if lat else \ self.rocEquatorial2.prime_vertical rocTransverse = rocPrimeVertical # synonymous @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} ''' a, b = self.a, self.b q = (sqrt((_3_0 + self.b2_a2) * _0_25) * a) if a > b else ( (sqrt((_3_0 * self.a2_b2 + _1_0) * _0_25) * b) if a < b else a) return Radius(Rtriaxial=q) def toEllipsoid2(self, name=NN): '''Get a copy of this ellipsoid as an L{Ellipsoid2}. @kwarg name: Optional, unique name (C{str}). @see: Property C{a_f}. ''' return Ellipsoid2(self, None, name=name) 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: This C{Ellipsoid}'s attributes (C{str}). ''' E = self.__class__ 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 * PI_3 * _4_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. ''' if f is None and isinstance(a, Ellipsoid): Ellipsoid.__init__(self, a.a, f =a.f, b=a.b, f_=a.f_, name=name) else: 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 = (n + _1_0)**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 _normalTo3(px, py, E): # in .height4 above '''(INTERNAL) Nearest point on a 2-D ellipse in 1st quadrant. @see: Functions C{.triaxial._normalTo4} and C{-To5}. ''' 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://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS> # <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)), CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC) # 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, None)), # 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 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11 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_, _NLATvar_ from pygeodesy.lazily import printf 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 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10), nl=1) printf('# 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)) printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20)) printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20)) printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property # __doc__ of this file, force all into registry t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_) printf(_NLATvar_.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 # Ellipsoids.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 # Ellipsoids.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 # Ellipsoids.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 # Ellipsoids.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 # Ellipsoids._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=6367453.6345163295, Rtriaxial=6362105.2243882557 # 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