Coverage for pygeodesy/ellipsoids.py: 96%
756 statements
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« prev ^ index » next coverage.py v7.2.2, created at 2024-03-08 13:06 -0500
2# -*- coding: utf-8 -*-
4u'''Ellipsoidal and spherical earth models.
6Classes L{a_f2Tuple}, L{Ellipsoid} and L{Ellipsoid2}, an L{Ellipsoids} registry and
72 dozen functions to convert I{equatorial} radius, I{polar} radius, I{eccentricities},
8I{flattenings} and I{inverse flattening}.
10See module L{datums} for L{Datum} and L{Transform} information and other details.
12Following is the list of predefined L{Ellipsoid}s, all instantiated lazily.
14@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, e21=0.99332946, 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)
15@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, e21=0.99332946, 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)
16@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, e21=0.99330562, 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)
17@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, e21=0.99330546, 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)
18@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, e21=0.99332563, 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)
19@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, e21=0.99332563, 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)
20@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, e21=0.99330562, 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)
21@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, e21=0.99323134, 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)
22@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, e21=0.99319649, 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)
23@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, e21=0.99319651, 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)
24@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, e21=0.99319652, e22=0.00685009, e32=0.00341335, A=6367386.69236201, L=10001867.62764496, R1=6371004.4187985, R2=6371002.77596616, R3=6370996.17427195, Rbiaxial=6367391.32901784, Rtriaxial=6372822.5494103)
25@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, e21=0.9940279, 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)
26@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, e21=0.99358976, 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)
27@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, e21=0.99330561, 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)
28@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, e21=0.99336215, 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)
29@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, e21=0.99336215, 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)
30@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, e21=0.99330658, 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)
31@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, e21=0.99330562, 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)
32@var Ellipsoids.GPES: Ellipsoid(name='GPES', a=6378135, b=6378135, f_=0, f=0, f2=0, n=0, e=0, e2=0, e21=1, e22=0, e32=0, A=6378135, L=10018751.02980197, R1=6378135, R2=6378135, R3=6378135, Rbiaxial=6378135, Rtriaxial=6378135)
33@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, e21=0.99330539, 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)
34@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, e21=0.99330562, 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)
35@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, e21=0.99330658, 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)
36@var Ellipsoids.IAU76: Ellipsoid(name='IAU76', a=6378140, b=6356755.28815753, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e21=0.99330562, e22=0.0067395, e32=0.00335843, A=6367452.13278844, L=10001970.4212264, R1=6371011.76271918, R2=6371010.17221946, R3=6371003.78129754, Rbiaxial=6367456.6214902, Rtriaxial=6372800.54945074)
37@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, e21=0.99330562, 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)
38@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, e21=0.99330562, e22=0.0067395, e32=0.00335843, A=6367448.44699641, L=10001964.63159783, R1=6371008.07219738, R2=6371006.48170097, R3=6371000.09079236, Rbiaxial=6367452.93568883, Rtriaxial=6372796.85654541)
39@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, e21=0.9933056, 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)
40@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, e21=0.99327733, 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)
41@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, e21=0.99330545, 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)
42@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, e21=0.99330658, 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)
43@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, e21=0.99330658, 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)
44@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, e21=0.98955621, 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)
45@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, e21=0.99330658, 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)
46@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, e21=0.99330658, 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)
47@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, e21=0.99330546, 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)
48@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, e21=0.99330562, 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)
49@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, e21=0.99330562, 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)
50@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, e21=0.99353046, 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)
51@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, e21=0.99330563, 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)
52@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, e21=0.99330562, 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)
53@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, e21=0.99330546, 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)
54@var Ellipsoids.Sphere: Ellipsoid(name='Sphere', a=6371008.771415, b=6371008.771415, f_=0, f=0, f2=0, n=0, e=0, e2=0, e21=1, 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)
55@var Ellipsoids.SphereAuthalic: Ellipsoid(name='SphereAuthalic', a=6371000, b=6371000, f_=0, f=0, f2=0, n=0, e=0, e2=0, e21=1, e22=0, e32=0, A=6371000, L=10007543.39801029, R1=6371000, R2=6371000, R3=6371000, Rbiaxial=6371000, Rtriaxial=6371000)
56@var Ellipsoids.SpherePopular: Ellipsoid(name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e21=1, e22=0, e32=0, A=6378137, L=10018754.17139462, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137)
57@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, e21=0.99322564, 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)
58@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, e21=0.99330658, e22=0.00673853, e32=0.00335795, A=6367478.63091189, L=10002012.04438139, R1=6371037.76231981, R2=6371036.17227197, R3=6371029.78316994, Rbiaxial=6367483.11833616, Rtriaxial=6372826.29723739)
59@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, e21=0.99330546, 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)
60@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, e21=0.99330568, 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)
61@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, e21=0.99330562, 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)
62'''
63# make sure int/int division yields float quotient, see .basics
64from __future__ import division as _; del _ # PYCHOK semicolon
66from pygeodesy.basics import copysign0, isbool, isint
67from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, INF, NINF, PI4, PI_2, PI_3, R_M, R_MA, R_FM, \
68 _EPSqrt, _EPStol as _TOL, _floatuple as _T, _isfinite, _SQRT2_2, \
69 _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \
70 _0_25, _3_0 # PYCHOK used!
71from pygeodesy.errors import _AssertionError, IntersectionError, _ValueError, _xattr, _xkwds_not
72from pygeodesy.fmath import cbrt, cbrt2, fdot, Fhorner, fpowers, Fsum, hypot, hypot_, \
73 hypot1, hypot2, sqrt3
74# from pygeodesy.fsums import Fsum # from .fmath
75from pygeodesy.interns import NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, _beta_, \
76 _Clarke1866_, _Clarke1880IGN_, _DOT_, _f_, _GRS80_, _height_, \
77 _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \
78 _Krassowsky1940_, _meridional_, _lat_, _negative_, _not_, \
79 _not_finite_, _prime_vertical_, _radius_, _Sphere_, _SPACE_, \
80 _vs_, _WGS72_, _WGS84_
81# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .named
82from pygeodesy.named import _lazyNamedEnumItem as _lazy, _NamedEnum, _NamedEnumItem, \
83 _NamedTuple, _Pass, _ALL_LAZY, _MODS
84from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple
85from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_, \
86 deprecated_property_RO, property_RO
87from pygeodesy.streprs import Fmt, fstr, instr, strs, unstr
88from pygeodesy.units import Bearing_, Distance, Float, Float_, Height, Lam_, Lat, Meter, \
89 Meter2, Meter3, Phi, Phi_, Radius, Radius_, Scalar
90from pygeodesy.utily import atan1, atan1d, atan2b, degrees90, m2radians, radians2m, sincos2d
92from math import asinh, atan, atanh, cos, degrees, exp, fabs, radians, sin, sinh, sqrt, tan
94__all__ = _ALL_LAZY.ellipsoids
95__version__ = '24.02.27'
97_f_0_0 = Float(f =_0_0) # zero flattening
98_f__0_0 = Float(f_=_0_0) # zero inverse flattening
99# see U{WGS84_f<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Constants.html>}
100_f__WGS84 = Float(f_=_1_0 / (1000000000 / 298257223563)) # 298.25722356299997 vs 298.257223563
103def _aux(lat, inverse, auxLat, clip=90):
104 '''Return a named auxiliary latitude in C{degrees}.
105 '''
106 return Lat(lat, clip=clip, name=_lat_ if inverse else auxLat.__name__)
109def _s2_c2(phi):
110 '''(INTERNAL) Return 2-tuple C{(sin(B{phi})**2, cos(B{phi})**2)}.
111 '''
112 if phi:
113 s2 = sin(phi)**2
114 if s2 > EPS:
115 c2 = _1_0 - s2
116 if c2 > EPS:
117 if c2 < EPS1:
118 return s2, c2
119 else:
120 return _1_0, _0_0 # phi == PI_2
121 return _0_0, _1_0 # phi == 0
124class a_f2Tuple(_NamedTuple):
125 '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial}
126 radius C{a} in C{meter} and scalar I{flattening} C{f}.
128 @see: Class L{Ellipsoid2}.
129 '''
130 _Names_ = (_a_, _f_) # name 'f' not 'f_'
131 _Units_ = (_Pass, _Pass)
133 def __new__(cls, a, f, **name):
134 '''New L{a_f2Tuple} ellipsoid specification.
136 @arg a: Equatorial radius (C{scalar} > 0).
137 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
138 @kwarg name: Optional name (C{str}).
140 @return: An L{a_f2Tuple}C{(a, f)} instance.
142 @raise UnitError: Invalid B{C{a}} or B{C{f}}.
144 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.
145 Negative C{B{f}} produces a I{prolate} ellipsoid.
146 '''
147 a = Radius_(a=a) # low=EPS, high=None
148 f = Float_( f=f, low=None, high=EPS1)
149 if fabs(f) < EPS: # force spherical
150 f = _f_0_0
151 return _NamedTuple.__new__(cls, a, f, **name)
153 @Property_RO
154 def b(self):
155 '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}.
156 '''
157 return a_f2b(self.a, self.f) # PYCHOK .a and .f
159 def ellipsoid(self, name=NN):
160 '''Return an L{Ellipsoid} for this 2-tuple C{(a, f)}.
162 @raise NameError: A registered C{ellipsoid} with the
163 same B{C{name}} already exists.
164 '''
165 return Ellipsoid(self.a, f=self.f, name=name or self.name) # PYCHOK .a and .f
167 @Property_RO
168 def f_(self):
169 '''Get the I{inverse} flattening (C{scalar}), M{1 / f} == M{a / (a - b)}.
170 '''
171 return f2f_(self.f) # PYCHOK .f
174class Circle4Tuple(_NamedTuple):
175 '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height},
176 both conventionally in C{meter} of a parallel I{circle of latitude} at
177 (geodetic) latitude C{lat} and the I{parametric (or reduced) auxiliary
178 latitude} C{beta}, both in C{degrees90}.
180 The C{height} is the (signed) distance along the z-axis between the
181 parallel and the equator. At near-polar C{lat}s, the C{radius} is C{0},
182 the C{height} is the ellipsoid's (signed) polar radius and C{beta}
183 equals C{lat}.
184 '''
185 _Names_ = (_radius_, _height_, _lat_, _beta_)
186 _Units_ = ( Radius, Height, Lat, Lat)
189class Curvature2Tuple(_NamedTuple):
190 '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in
191 C{meter}, conventionally.
192 '''
193 _Names_ = (_meridional_, _prime_vertical_)
194 _Units_ = ( Meter, Meter)
196 @property_RO
197 def transverse(self):
198 '''Get this I{prime_vertical}, aka I{transverse} radius of curvature.
199 '''
200 return self.prime_vertical
203class Ellipsoid(_NamedEnumItem):
204 '''Ellipsoid with I{equatorial} and I{polar} radii, I{flattening}, I{inverse
205 flattening} and other, often used, I{cached} attributes, supporting
206 I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models.
207 '''
208 _a = 0 # equatorial radius, semi-axis (C{meter})
209 _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f
210 _f = 0 # (1st) flattening: (a - b) / a
211 _f_ = 0 # inverse flattening: 1 / f = a / (a - b)
213 _geodsolve = NN # means, use PYGEODESY_GEODSOLVE
214 _KsOrder = 8 # Krüger series order (4, 6 or 8)
215 _rhumbsolve = NN # means, use PYGEODESY_RHUMBSOLVE
217 def __init__(self, a, b=None, f_=None, f=None, name=NN):
218 '''New L{Ellipsoid} from the I{equatorial} radius I{and} either
219 the I{polar} radius or I{inverse flattening} or I{flattening}.
221 @arg a: Equatorial radius, semi-axis (C{meter}).
222 @arg b: Optional polar radius, semi-axis (C{meter}).
223 @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0).
224 @arg f: Flattening: M{(a - b) / a} (C{scalar}, near zero for
225 spherical).
226 @kwarg name: Optional, unique name (C{str}).
228 @raise NameError: Ellipsoid with the same B{C{name}} already exists.
230 @raise ValueError: Invalid B{C{a}}, B{C{b}}, B{C{f_}} or B{C{f}} or
231 B{C{f_}} and B{C{f}} are incompatible.
233 @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced
234 to M{1 / f_ = 0}, spherical.
235 '''
236 ff_ = f, f_ # assertion below
237 try:
238 a = Radius_(a=a) # low=EPS
239 if not _isfinite(a):
240 raise ValueError(_SPACE_(_a_, _not_finite_))
242 if b: # not in (_0_0, None)
243 b = Radius_(b=b) # low=EPS
244 f = a_b2f(a, b) if f is None else Float(f=f)
245 f_ = f2f_(f) if f_ is None else Float(f_=f_)
246 elif f is not None:
247 f = Float(f=f)
248 b = a_f2b(a, f)
249 f_ = f2f_(f) if f_ is None else Float(f_=f_)
250 elif f_:
251 f_ = Float(f_=f_)
252 b = a_f_2b(a, f_) # a * (f_ - 1) / f_
253 f = f_2f(f_)
254 else: # only a, spherical
255 f_ = f = 0
256 b = a # superfluous
258 if not f < _1_0: # sanity check, see .ecef.Ecef.__init__
259 raise ValueError(_SPACE_(_f_, _invalid_))
260 if not _isfinite(b):
261 raise ValueError(_SPACE_(_b_, _not_finite_))
263 if fabs(f) < EPS or a == b or not f_: # spherical
264 b = a
265 f = _f_0_0
266 f_ = _f__0_0
268 except (TypeError, ValueError) as x:
269 d = _xkwds_not(None, b=b, f_=f_, f=f)
270 t = instr(self, a=a, name=name, **d)
271 raise _ValueError(t, cause=x)
273 self._a = a
274 self._b = b
275 self._f = f
276 self._f_ = f_
278 self._register(Ellipsoids, name)
280 if f and f_: # see .test/testEllipsoidal.py
281 d = dict(eps=_TOL)
282 if None in ff_: # both f_ and f given
283 d.update(Error=_ValueError, txt=_incompatible_)
284 self._assert(_1_0 / f, f_=f_, **d)
285 self._assert(_1_0 / f_, f =f, **d)
286 self._assert(self.b2_a2, e21=self.e21, eps=EPS)
288 def __eq__(self, other):
289 '''Compare this and an other ellipsoid.
291 @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}).
293 @return: C{True} if equal, C{False} otherwise.
294 '''
295 return self is other or (isinstance(other, Ellipsoid) and
296 self.a == other.a and
297 (self.f == other.f or self.b == other.b))
299 def __hash__(self):
300 return self._hash # memoized
302 @Property_RO
303 def a(self):
304 '''Get the I{equatorial} radius, semi-axis (C{meter}).
305 '''
306 return self._a
308 equatoradius = a # = Requatorial
310 @Property_RO
311 def a2(self):
312 '''Get the I{equatorial} radius I{squared} (C{meter} I{squared}), M{a**2}.
313 '''
314 return Meter2(a2=self.a**2)
316 @Property_RO
317 def a2_(self):
318 '''Get the inverse of the I{equatorial} radius I{squared} (C{meter} I{squared}), M{1 / a**2}.
319 '''
320 return Float(a2_=_1_0 / self.a2)
322 @Property_RO
323 def a_b(self):
324 '''Get the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b} == M{1 / (1 - f)}.
325 '''
326 return Float(a_b=self.a / self.b if self.f else _1_0)
328 @Property_RO
329 def a2_b(self):
330 '''Get the I{polar} meridional (or polar) radius of curvature (C{meter}), M{a**2 / b}.
332 @see: U{Radii of Curvature
333 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
334 and U{Moritz, H. (1980), Geodetic Reference System 1980
335 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}.
337 @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature
338 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{c2}
339 and L{R2} or L{Rauthalic}.
340 '''
341 return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar
343 @Property_RO
344 def a2_b2(self):
345 '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}),
346 M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e21}.
347 '''
348 return Float(a2_b2=self.a_b**2 if self.f else _1_0)
350 @Property_RO
351 def a_f(self):
352 '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}), see method C{toEllipsoid2}.
353 '''
354 return a_f2Tuple(self.a, self.f, name=self.name)
356 @Property_RO
357 def A(self):
358 '''Get the UTM I{meridional (or rectifying)} radius (C{meter}).
360 @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/>}.
361 '''
362 A, n = self.a, self.n
363 if n:
364 d = (n + _1_0) * 1048576 / A
365 if d: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8
366 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>
367 # <https://GeographicLib.SourceForge.io/C++/doc/transversemercator.html> and
368 # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> (3)
369 # A *= fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441) / 1048576) / (1 + n)
370 A = Radius(A=Fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441).fover(d))
371 return A
373 @Property_RO
374 def _albersCyl(self):
375 '''(INTERNAL) Helper for C{auxAuthalic}.
376 '''
377 return _MODS.albers.AlbersEqualAreaCylindrical(datum=self, name=self.name)
379 @Property_RO
380 def AlphaKs(self):
381 '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
382 '''
383 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon
384 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
385 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200),
386 _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned
387 _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned
388 _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned
389 _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned
390 _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned
391 _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned
392 _T(1424729850961/743921418240)) # PYCHOK unaligned
394 @Property_RO
395 def area(self):
396 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2}.
398 @see: Properties L{areax}, L{c2} and L{R2} and functions
399 L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
400 '''
401 return Meter2(area=self.c2 * PI4)
403 @Property_RO
404 def areax(self):
405 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more
406 accurate for very I{oblate} ellipsoids.
408 @see: Properties L{area}, L{c2x} and L{R2x}, class L{GeodesicExact} and
409 functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
410 '''
411 return Meter2(areax=self.c2x * PI4)
413 def _assert(self, val, eps=_TOL, f0=_0_0, Error=_AssertionError, txt=NN, **name_value):
414 '''(INTERNAL) Assert a C{name=value} vs C{val}.
415 '''
416 for n, v in name_value.items():
417 if fabs(v - val) > eps: # PYCHOK no cover
418 t = (v, _vs_, val)
419 t = _SPACE_.join(strs(t, prec=12, fmt=Fmt.g))
420 t = Fmt.EQUAL(self._DOT_(n), t)
421 raise Error(t, txt=txt or Fmt.exceeds_eps(eps))
422 return Float(v if self.f else f0, name=n)
423 raise Error(unstr(self._DOT_(self._assert.__name__), val,
424 eps=eps, f0=f0, **name_value))
426 def auxAuthalic(self, lat, inverse=False):
427 '''Compute the I{authalic} auxiliary latitude or the I{inverse} thereof.
429 @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}).
430 @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and
431 return the geodetic latitude (C{bool}).
433 @return: The I{authalic} (or geodetic) latitude in C{degrees90}.
435 @see: U{Inverse-/AuthalicLatitude<https://GeographicLib.SourceForge.io/
436 html/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude
437 <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and
438 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16.
439 '''
440 if self.f:
441 f = self._albersCyl._tanf if inverse else self._albersCyl._txif # PYCHOK attr
442 lat = atan1d(f(tan(Phi_(lat)))) # PYCHOK attr
443 return _aux(lat, inverse, Ellipsoid.auxAuthalic)
445 def auxConformal(self, lat, inverse=False):
446 '''Compute the I{conformal} auxiliary latitude or the I{inverse} thereof.
448 @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}).
449 @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and
450 return the geodetic latitude (C{bool}).
452 @return: The I{conformal} (or geodetic) latitude in C{degrees90}.
454 @see: U{Inverse-/ConformalLatitude<https://GeographicLib.SourceForge.io/
455 html/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude
456 <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and
457 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
458 '''
459 if self.f:
460 f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr
461 lat = atan1d(f(tan(Phi_(lat)))) # PYCHOK attr
462 return _aux(lat, inverse, Ellipsoid.auxConformal)
464 def auxGeocentric(self, lat, inverse=False):
465 '''Compute the I{geocentric} auxiliary latitude or the I{inverse} thereof.
467 @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}).
468 @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and
469 return the I{geocentric} latitude (C{bool}).
471 @return: The I{geocentric} (or geodetic) latitude in C{degrees90}.
473 @see: U{Inverse-/GeocentricLatitude<https://GeographicLib.SourceForge.io/
474 html/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude
475 <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and
476 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18.
477 '''
478 if self.f:
479 f = self.a2_b2 if inverse else self.b2_a2
480 lat = atan1d(f * tan(Phi_(lat)))
481 return _aux(lat, inverse, Ellipsoid.auxGeocentric)
483 def auxIsometric(self, lat, inverse=False):
484 '''Compute the I{isometric} auxiliary latitude or the I{inverse} thereof.
486 @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}).
487 @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and
488 return the geodetic latitude (C{bool}).
490 @return: The I{isometric} (or geodetic) latitude in C{degrees}.
492 @note: The I{isometric} latitude for geodetic C{+/-90} is far
493 outside the C{[-90..+90]} range but the inverse
494 thereof is the original geodetic latitude.
496 @see: U{Inverse-/IsometricLatitude<https://GeographicLib.SourceForge.io/
497 html/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude
498 <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and
499 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
500 '''
501 if self.f:
502 r = Phi_(lat, clip=0)
503 lat = degrees(atan1(self.es_tauf(sinh(r))) if inverse else
504 asinh(self.es_taupf(tan(r))))
505 # clip=0, since auxIsometric(+/-90) is far outside [-90..+90]
506 return _aux(lat, inverse, Ellipsoid.auxIsometric, clip=0)
508 def auxParametric(self, lat, inverse=False):
509 '''Compute the I{parametric} auxiliary latitude or the I{inverse} thereof.
511 @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}).
512 @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and
513 return the geodetic latitude (C{bool}).
515 @return: The I{parametric} (or geodetic) latitude in C{degrees90}.
517 @see: U{Inverse-/ParametricLatitude<https://GeographicLib.SourceForge.io/
518 html/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude
519 <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>},
520 and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18.
521 '''
522 if self.f:
523 lat = self._beta(Lat(lat), inverse=inverse)
524 return _aux(lat, inverse, Ellipsoid.auxParametric)
526 auxReduced = auxParametric # synonymous
528 def auxRectifying(self, lat, inverse=False):
529 '''Compute the I{rectifying} auxiliary latitude or the I{inverse} thereof.
531 @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}).
532 @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and
533 return the geodetic latitude (C{bool}).
535 @return: The I{rectifying} (or geodetic) latitude in C{degrees90}.
537 @see: U{Inverse-/RectifyingLatitude<https://GeographicLib.SourceForge.io/
538 html/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude
539 <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and
540 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17.
541 '''
542 if self.f:
543 lat = Lat(lat)
544 if 0 < fabs(lat) < _90_0:
545 if inverse:
546 e = self._elliptic_e22
547 d = degrees90(e.fEinv(e.cE * lat / _90_0))
548 lat = self.auxParametric(d, inverse=True)
549 else:
550 lat = _90_0 * self.Llat(lat) / self.L
551 return _aux(lat, inverse, Ellipsoid.auxRectifying)
553 @Property_RO
554 def b(self):
555 '''Get the I{polar} radius, semi-axis (C{meter}).
556 '''
557 return self._b
559 polaradius = b # = Rpolar
561 @Property_RO
562 def b_a(self):
563 '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a == f1 == 1 - f}.
565 @see: Property L{f1}.
566 '''
567 return self._assert(self.b / self.a, b_a=self.f1, f0=_1_0)
569 @Property_RO
570 def b2(self):
571 '''Get the I{polar} radius I{squared} (C{float}), M{b**2}.
572 '''
573 return Meter2(b2=self.b**2)
575 @Property_RO
576 def b2_a(self):
577 '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see C{rocMeridional}C{(0)}.
579 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
580 '''
581 return Radius(b2_a=self.b2 / self.a if self.f else self.b)
583 @Property_RO
584 def b2_a2(self):
585 '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2}
586 == M{(1 - f)**2} == M{1 - e**2} == C{e21}.
587 '''
588 return Float(b2_a2=self.b_a**2 if self.f else _1_0)
590 def _beta(self, lat, inverse=False):
591 '''(INTERNAL) Get the I{parametric (or reduced) auxiliary latitude} or inverse thereof.
592 '''
593 s, c = sincos2d(lat) # like Karney's tand(lat)
594 s *= self.a_b if inverse else self.b_a
595 return atan1d(s, c)
597 @Property_RO
598 def BetaKs(self):
599 '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
600 '''
601 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon
602 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
603 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600),
604 _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned
605 _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned
606 _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned
607 _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned
608 _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne
609 _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned
610 _T(191773887257/3719607091200)) # PYCHOK unaligned
612 @deprecated_Property_RO
613 def c(self): # PYCHOK no cover
614 '''DEPRECATED, use property C{R2} or C{Rauthalic}.'''
615 return self.R2
617 @Property_RO
618 def c2(self):
619 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}).
621 @see: Properties L{c2x}, L{area}, L{R2}, L{Rauthalic}, I{Karney's} U{equation (60)
622 <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area
623 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>},
624 U{Authalic radius<https://WikiPedia.org/wiki/Earth_radius#Authalic_radius>}, U{Surface area
625 <https://WikiPedia.org/wiki/Ellipsoid>} and U{surface area
626 <https://www.Numericana.com/answer/geometry.htm#oblate>}.
627 '''
628 return self._c2f(False)
630 @Property_RO
631 def c2x(self):
632 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}), more accurate for very I{oblate}
633 ellipsoids.
635 @see: Properties L{c2}, L{areax}, L{R2x}, L{Rauthalicx}, class L{GeodesicExact} and I{Karney}'s comments at C++
636 attribute U{GeodesicExact._c2<https://GeographicLib.SourceForge.io/C++/doc/GeodesicExact_8cpp_source.html>}.
637 '''
638 return self._c2f(True)
640 def _c2f(self, c2x):
641 '''(INTERNAL) Helper for C{.c2} and C{.c2x}.
642 '''
643 f, c2 = self.f, self.b2
644 if f:
645 e = self.e
646 if e > EPS0:
647 if f > 0: # .isOblate
648 c2 *= (asinh(sqrt(self.e22abs)) if c2x else atanh(e)) / e
649 elif f < 0: # .isProlate
650 c2 *= atan1(e) / e # XXX asin?
651 c2 = Meter2(c2=(self.a2 + c2) * _0_5)
652 return c2
654 def circle4(self, lat):
655 '''Get the equatorial or a parallel I{circle of latitude}.
657 @arg lat: Geodetic latitude (C{degrees90}, C{str}).
659 @return: A L{Circle4Tuple}C{(radius, height, lat, beta)}
660 instance.
662 @raise RangeError: Latitude B{C{lat}} outside valid range and
663 L{pygeodesy.rangerrors} set to C{True}.
665 @raise TypeError: Invalid B{C{lat}}.
667 @raise ValueError: Invalid B{C{lat}}.
669 @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude}
670 <https://WikiPedia.org/wiki/Latitude>}, I{Karney's} C++ U{CircleRadius and CircleHeight
671 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>}
672 and method C{Rlat}.
673 '''
674 lat = Lat(lat)
675 if lat:
676 b = lat
677 if fabs(lat) < _90_0:
678 if self.f:
679 b = self._beta(lat)
680 z, r = sincos2d(b)
681 r *= self.a
682 z *= self.b
683 else: # near-polar
684 r, z = _0_0, copysign0(self.b, lat)
685 else: # equator
686 r = self.a
687 z = lat = b = _0_0
688 return Circle4Tuple(r, z, lat, b)
690 def degrees2m(self, deg, lat=0):
691 '''Convert an angle to the distance along the equator or
692 along a parallel of (geodetic) latitude.
694 @arg deg: The angle (C{degrees}).
695 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
697 @return: Distance (C{meter}, same units as the equatorial
698 and polar radii) or C{0} for near-polar B{C{lat}}.
700 @raise RangeError: Latitude B{C{lat}} outside valid range and
701 L{pygeodesy.rangerrors} set to C{True}.
703 @raise ValueError: Invalid B{C{deg}} or B{C{lat}}.
704 '''
705 return self.radians2m(radians(deg), lat=lat)
707 def distance2(self, lat0, lon0, lat1, lon1):
708 '''I{Approximate} the distance and (initial) bearing between
709 two points based on the U{local, flat earth approximation
710 <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny
711 <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
713 I{Suitable only for distances of several hundred Km or Miles
714 and only between points not near-polar}.
716 @arg lat0: From latitude (C{degrees}).
717 @arg lon0: From longitude (C{degrees}).
718 @arg lat1: To latitude (C{degrees}).
719 @arg lon1: To longitude (C{degrees}).
721 @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance}
722 in same units as this ellipsoid's axes.
724 @note: The meridional and prime_vertical radii of curvature are
725 taken and scaled I{at the initial latitude}, see C{roc2}.
727 @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}.
728 '''
729 phi0 = Phi_(lat0=lat0)
730 m, n = self.roc2_(phi0, scaled=True)
731 m *= Phi_(lat1=lat1) - phi0
732 n *= Lam_(lon1=lon1) - Lam_(lon0=lon0)
733 return Distance2Tuple(hypot(m, n), atan2b(n, m))
735 @Property_RO
736 def e(self):
737 '''Get the I{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}.
739 @see: Property L{es}.
740 '''
741 return Float(e=sqrt(self.e2abs) if self.e2 else _0_0)
743 @deprecated_Property_RO
744 def e12(self): # see property ._e12
745 '''DEPRECATED, use property C{e21}.'''
746 return self.e21
748# @Property_RO
749# def _e12(self): # see property ._elliptic_e12
750# # (INTERNAL) until e12 above can be replaced with e21.
751# return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2)
753 @Property_RO
754 def e2(self):
755 '''Get the I{signed, (1st) eccentricity squared} (C{float}), M{f * (2 - f)
756 == 1 - (b / a)**2}, see C{a_b2e2}.
757 '''
758 return self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f))
760 @Property_RO
761 def e2abs(self):
762 '''Get the I{unsigned, (1st) eccentricity squared} (C{float}).
763 '''
764 return fabs(self.e2)
766 @Property_RO
767 def e21(self):
768 '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2}
769 == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}.
770 '''
771 return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0)
773# _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2
774 _1_e21 = a2_b2 # == M{1 / e21} == M{1 / (1 - e**2)}
776 @Property_RO
777 def e22(self):
778 '''Get the I{signed, 2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)
779 == e2 / (1 - f)**2 == (a / b)**2 - 1}, see C{a_b2e22}.
780 '''
781 return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f))
783 @Property_RO
784 def e22abs(self):
785 '''Get the I{unsigned, 2nd eccentricity squared} (C{float}).
786 '''
787 return fabs(self.e22)
789 @Property_RO
790 def e32(self):
791 '''Get the I{signed, 3rd eccentricity squared} (C{float}), M{e2 / (2 - e2)
792 == (a**2 - b**2) / (a**2 + b**2)}, see C{a_b2e32}.
793 '''
794 return self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f))
796 @Property_RO
797 def e32abs(self):
798 '''Get the I{unsigned, 3rd eccentricity squared} (C{float}).
799 '''
800 return fabs(self.e32)
802 @Property_RO
803 def e4(self):
804 '''Get the I{unsignd, (1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}.
805 '''
806 return Float(e4=self.e2**2 if self.e2 else _0_0)
808 eccentricity = e # eccentricity
809# eccentricity2 = e2 # eccentricity squared
810 eccentricity1st2 = e2 # first eccentricity squared
811 eccentricity2nd2 = e22 # second eccentricity squared
812 eccentricity3rd2 = e32 # third eccentricity squared
814 def ecef(self, Ecef=None):
815 '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter.
817 @kwarg Ecef: ECEF class to use, default L{EcefKarney}.
819 @return: An ECEF converter for this C{ellipsoid}.
821 @raise TypeError: Invalid B{C{Ecef}}.
823 @see: Module L{pygeodesy.ecef}.
824 '''
825 return _MODS.ecef._4Ecef(self, Ecef)
827 @Property_RO
828 def _elliptic_e12(self): # see I{Karney}'s Ellipsoid._e12
829 '''(INTERNAL) Elliptic helper for C{Rhumb}.
830 '''
831 e12 = self.e2 / (self.e2 - _1_0) # NOT DEPRECATED .e12!
832 return _MODS.elliptic.Elliptic(e12)
834 @Property_RO
835 def _elliptic_e22(self): # aka ._elliptic_ep2
836 '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}.
837 '''
838 return _MODS.elliptic.Elliptic(-self.e22abs) # complex
840 equatoradius = a # Requatorial
842 def e2s(self, s):
843 '''Compute norm M{sqrt(1 - e2 * s**2)}.
845 @arg s: Sine value (C{scalar}).
847 @return: Norm (C{float}).
849 @raise ValueError: Invalid B{C{s}}.
850 '''
851 return sqrt(self.e2s2(s)) if self.e2 else _1_0
853 def e2s2(self, s):
854 '''Compute M{1 - e2 * s**2}.
856 @arg s: Sine value (C{scalar}).
858 @return: Result (C{float}).
860 @raise ValueError: Invalid B{C{s}}.
861 '''
862 r = _1_0
863 if self.e2:
864 try:
865 r -= self.e2 * Scalar(s=s)**2
866 if r < 0:
867 raise ValueError(_negative_)
868 except (TypeError, ValueError) as x:
869 t = self._DOT_(Ellipsoid.e2s2.__name__)
870 raise _ValueError(t, s, cause=x)
871 return r
873 @Property_RO
874 def es(self):
875 '''Get the I{signed (1st) eccentricity} (C{float}).
877 @see: Property L{e}.
878 '''
879 # note, self.e is always non-negative
880 return Float(es=copysign0(self.e, self.f)) # see .ups
882 def es_atanh(self, x):
883 '''Compute M{es * atanh(es * x)} or M{-es * atan(es * x)}
884 for I{oblate} respectively I{prolate} ellipsoids where
885 I{es} is the I{signed} (1st) eccentricity.
887 @raise ValueError: Invalid B{C{x}}.
889 @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/
890 html/classGeographicLib_1_1Math.html>}.
891 '''
892 return self._es_atanh(Scalar(x=x)) if self.f else _0_0
894 def _es_atanh(self, x): # see .albers._atanhee, .AuxLat._atanhee
895 '''(INTERNAL) Helper for .es_atanh, ._es_taupf2 and ._exp_es_atanh.
896 '''
897 es = self.es # signOf(es) == signOf(f)
898 return es * (atanh(es * x) if es > 0 else # .isOblate
899 (-atan(es * x) if es < 0 else # .isProlate
900 _0_0)) # .isSpherical
902 @Property_RO
903 def es_c(self):
904 '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}.
905 '''
906 return Float(es_c=(self._exp_es_atanh_1 * self.b_a) if self.f else _1_0)
908 def es_tauf(self, taup):
909 '''Compute I{Karney}'s U{equations (19), (20) and (21)
910 <https://ArXiv.org/abs/1002.1417>}.
912 @see: I{Karney}'s C++ method U{Math::tauf<https://GeographicLib.
913 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>} and
914 and I{Veness}' JavaScript method U{toLatLon<https://www.
915 Movable-Type.co.UK/scripts/latlong-utm-mgrs.html>}.
916 '''
917 t = Scalar(taup=taup)
918 if self.f: # .isEllipsoidal
919 a = fabs(t)
920 T = t * (self._exp_es_atanh_1 if a > 70 else self._1_e21)
921 if fabs(T * _EPSqrt) < _2_0: # handles +/- INF and NAN
922 s = (a * _TOL) if a > _1_0 else _TOL
923 for T, _, d in self._es_tauf3(t, T): # max 2
924 if fabs(d) < s:
925 break
926 t = Scalar(tauf=T)
927 return t
929 def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB
930 '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most
931 B{C{N}} Newton iterations, converging rapidly except when C{delta}
932 toggles on +/-1.12e-16 or +/-4.47e-16, see C{.utm.Utm._toLLEB}.
933 '''
934 e = self._1_e21
935 _F2_ = Fsum(T).fsum2_ # τ0
936 _tf2 = self._es_taupf2
937 for i in range(1, N + 1):
938 a, h = _tf2(T)
939 d = (taup - a) * (e + T**2) / (hypot1(a) * h)
940 # = (taup - a) / hypot1(a) / ((e + T**2) / h)
941 T, d = _F2_(d) # τi, (τi - τi-1)
942 yield T, i, d
944 def es_taupf(self, tau):
945 '''Compute I{Karney}'s U{equations (7), (8) and (9)
946 <https://ArXiv.org/abs/1002.1417>}.
948 @see: I{Karney}'s C++ method U{Math::taupf<https://GeographicLib.
949 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}.
950 '''
951 t = Scalar(tau=tau)
952 if self.f: # .isEllipsoidal
953 t, _ = self._es_taupf2(t)
954 t = Scalar(taupf=t)
955 return t
957 def _es_taupf2(self, tau):
958 '''(INTERNAL) Return 2-tuple C{(es_taupf(tau), hypot1(tau))}.
959 '''
960 if _isfinite(tau):
961 h = hypot1(tau)
962 s = sinh(self._es_atanh(tau / h))
963 a = hypot1(s) * tau - h * s
964 else:
965 a, h = tau, INF
966 return a, h
968 @Property_RO
969 def _exp_es_atanh_1(self):
970 '''(INTERNAL) Helper for .es_c and .es_tauf.
971 '''
972 return exp(self._es_atanh(_1_0)) if self.es else _1_0
974 @Property_RO
975 def f(self):
976 '''Get the I{flattening} (C{scalar}), M{(a - b) / a}, C{0} for spherical, negative for prolate.
977 '''
978 return self._f
980 @Property_RO
981 def f_(self):
982 '''Get the I{inverse flattening} (C{scalar}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}.
983 '''
984 return self._f_
986 @Property_RO
987 def f1(self):
988 '''Get the I{1 - flattening} (C{float}), M{f1 == 1 - f == b / a}.
990 @see: Property L{b_a}.
991 '''
992 return Float(f1=_1_0 - self.f)
994 @Property_RO
995 def f2(self):
996 '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}.
997 '''
998 return self._assert(self.a_b - _1_0, f2=f2f2(self.f))
1000 @deprecated_Property_RO
1001 def geodesic(self):
1002 '''DEPRECATED, use property C{geodesicw}.'''
1003 return self.geodesicw
1005 def geodesic_(self, exact=True):
1006 '''Get the an I{exact} C{Geodesic...} instance for this ellipsoid.
1008 @kwarg exact: If C{bool} return L{GeodesicExact}C{(exact=B{exact}, ...)},
1009 otherwise a L{Geodesic}, L{GeodesicExact} or L{GeodesicSolve}
1010 instance for I{this} ellipsoid.
1012 @return: The C{exact} geodesic (C{Geodesic...}).
1014 @raise TypeError: Invalid B{C{exact}}.
1016 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1017 '''
1018 if isbool(exact): # for consistenccy with C{.rhumb_}
1019 g = _MODS.geodesicx.GeodesicExact(self, C4order=30 if exact else 24,
1020 name=self.name)
1021 else:
1022 g = exact
1023 E = _xattr(g, ellipsoid=None)
1024 if not (E is self and isinstance(g, self._Geodesics)):
1025 raise _ValueError(exact=g, ellipsoid=E, txt=_not_(self.name))
1026 return g
1028 @property_RO
1029 def _Geodesics(self):
1030 '''(INTERNAL) Get all C{Geodesic...} classes, I{once}.
1031 '''
1032 Ellipsoid._Geodesics = t = (_MODS.geodesicw._wrapped.Geodesic, # overwrite property_RO
1033 _MODS.geodesicx.GeodesicExact,
1034 _MODS.geodsolve.GeodesicSolve)
1035 return t
1037 @property_RO
1038 def geodesicw(self):
1039 '''Get this ellipsoid's I{wrapped} U{geodesicw.Geodesic
1040 <https://GeographicLib.SourceForge.io/Python/doc/code.html>}, provided
1041 I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1042 package is installed.
1043 '''
1044 # if not self.isEllipsoidal:
1045 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1046 return _MODS.geodesicw.Geodesic(self)
1048 @property_RO
1049 def geodesicx(self):
1050 '''Get this ellipsoid's I{exact} L{GeodesicExact}.
1051 '''
1052 # if not self.isEllipsoidal:
1053 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1054 return _MODS.geodesicx.GeodesicExact(self, name=self.name)
1056 @property
1057 def geodsolve(self):
1058 '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility
1059 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1060 provided the path to the C{GeodSolve} executable is specified with env
1061 variable C{PYGEODESY_GEODSOLVE} or re-/set with this property..
1062 '''
1063 # if not self.isEllipsoidal:
1064 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1065 return _MODS.geodsolve.GeodesicSolve(self, path=self._geodsolve, name=self.name)
1067 @geodsolve.setter # PYCHOK setter!
1068 def geodsolve(self, path):
1069 '''Re-/set the (fully qualified) path to the U{GeodSolve
1070 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1071 overriding env variable C{PYGEODESY_GEODSOLVE} (C{str}).
1072 '''
1073 self._geodsolve = path
1075 def hartzell4(self, pov, los=None):
1076 '''Compute the intersection of this ellipsoid's surface and a Line-Of-Sight
1077 from a Point-Of-View in space.
1079 @arg pov: Point-Of-View outside this ellipsoid (C{Cartesian}, L{Ecef9Tuple}
1080 or L{Vector3d}).
1081 @kwarg los: Line-Of-Sight, I{direction} to this ellipsoid (L{Vector3d}) or
1082 C{None} to point to this ellipsoid's center.
1084 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1085 C{y} and C{z} of the projection on or the intersection with this
1086 ellipsoid and the I{distance} C{h} from B{C{pov}} to C{(x, y, z)}
1087 along B{C{los}}, all in C{meter}, conventionally.
1089 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, or B{C{pov}}
1090 is inside this ellipsoid or B{C{los}} points
1091 outside this ellipsoid or points in an opposite
1092 direction.
1094 @raise TypeError: Invalid B{C{pov}} or B{C{los}}.
1096 @see: U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.
1097 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and
1098 methods L{Ellipsoid.height4} and L{Triaxial.hartzell4}.
1099 '''
1100 try:
1101 v, d = _MODS.triaxials._hartzell2(pov, los, self._triaxial)
1102 except Exception as x:
1103 raise IntersectionError(pov=pov, los=los, cause=x)
1104 return Vector4Tuple(v.x, v.y, v.z, d, name=self.hartzell4.__name__)
1106 @Property_RO
1107 def _hash(self):
1108 return hash((self.a, self.f))
1110 def height4(self, xyz, normal=True):
1111 '''Compute the projection on and the height of a cartesian above or below
1112 this ellipsoid's surface.
1114 @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d},
1115 L{Vector3Tuple} or L{Vector4Tuple}).
1116 @kwarg normal: If C{True}, the projection is perpendicular to (the nearest
1117 point on) this ellipsoid's surface, otherwise the C{radial}
1118 line to this ellipsoid's center (C{bool}).
1120 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1121 C{y} and C{z} of the projection on and the height C{h} above or
1122 below this ellipsoid's surface, all in C{meter}, conventionally.
1124 @raise ValueError: Null B{C{xyz}}.
1126 @raise TypeError: Non-cartesian B{C{xyz}}.
1128 @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>}
1129 and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse and
1130 methods L{Ellipsoid.hartzell4} and L{Triaxial.height4}.
1131 '''
1132 v = _MODS.vector3d._otherV3d(xyz=xyz)
1133 r = v.length
1135 a, b, i = self.a, self.b, None
1136 if r < EPS0: # EPS
1137 v = v.times(_0_0)
1138 h = -a
1140 elif self.isSpherical:
1141 v = v.times(a / r)
1142 h = r - a
1144 elif normal: # perpendicular to ellipsoid
1145 x, y = hypot(v.x, v.y), fabs(v.z)
1146 if x < EPS0: # PYCHOK no cover
1147 z = copysign0(b, v.z)
1148 v = Vector3Tuple(v.x, v.y, z)
1149 h = y - b # polar
1150 elif y < EPS0: # PYCHOK no cover
1151 t = a / r
1152 v = v.times_(t, t, 0) # force z=0.0
1153 h = x - a # equatorial
1154 else: # normal in 1st quadrant
1155 x, y, i = _normalTo3(x, y, self)
1156 t, v = v, v.times_(x, x, y)
1157 h = t.minus(v).length
1159 else: # radial to ellipsoid's center
1160 h = hypot_(a * v.z, b * v.x, b * v.y)
1161 t = (a * b / h) if h > EPS0 else _0_0 # EPS
1162 v = v.times(t)
1163 h = r * (_1_0 - t)
1165 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i,
1166 name=self.height4.__name__)
1168 def _hubeny_2(self, phi2, phi1, lam21, scaled=True, squared=True):
1169 '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_},
1170 returning the I{angular} distance in C{radians squared} or C{radians}
1171 '''
1172 m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=scaled)
1173 h, r = (hypot2, self.a2_) if squared else (hypot, _1_0 / self.a)
1174 return h(m * (phi2 - phi1), n * lam21) * r
1176 @Property_RO
1177 def isEllipsoidal(self):
1178 '''Is this model I{ellipsoidal} (C{bool})?
1179 '''
1180 return self.f != 0
1182 @Property_RO
1183 def isOblate(self):
1184 '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or
1185 spherical otherwise.
1186 '''
1187 return self.f > 0
1189 @Property_RO
1190 def isProlate(self):
1191 '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or
1192 spherical otherwise.
1193 '''
1194 return self.f < 0
1196 @Property_RO
1197 def isSpherical(self):
1198 '''Is this ellipsoid I{spherical} (C{bool})?
1199 '''
1200 return self.f == 0
1202 def _Kseries(self, *AB8Ks):
1203 '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha
1204 or Beta series coefficients per I{Karney}'s U{equations (35)
1205 and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}.
1207 @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series
1208 coefficient tuples.
1210 @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}).
1212 @see: I{Karney}'s 30-th order U{TMseries30
1213 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}.
1214 '''
1215 k = self.KsOrder
1216 if self.n:
1217 ns = fpowers(self.n, k)
1218 ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k))
1219 else:
1220 ks = _0_0s(k)
1221 return ks
1223 @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''')
1224 def KsOrder(self):
1225 '''Get the I{Krüger} series' order (C{int} 4, 6 or 8).
1226 '''
1227 return self._KsOrder
1229 @KsOrder.setter # PYCHOK setter!
1230 def KsOrder(self, order):
1231 '''Set the I{Krüger} series' order (C{int} 4, 6 or 8).
1233 @raise ValueError: Invalid B{C{order}}.
1234 '''
1235 if not (isint(order) and order in (4, 6, 8)):
1236 raise _ValueError(order=order)
1237 if self._KsOrder != order:
1238 Ellipsoid.AlphaKs._update(self)
1239 Ellipsoid.BetaKs._update(self)
1240 self._KsOrder = order
1242 @Property_RO
1243 def L(self):
1244 '''Get the I{quarter meridian} C{L}, aka the C{polar distance}
1245 along a meridian between the equator and a pole (C{meter}),
1246 M{b * Elliptic(-e2 / (1 - e2)).cE} or M{b * PI / 2}.
1247 '''
1248 r = self._elliptic_e22.cE if self.f else PI_2
1249 return Distance(L=self.b * r)
1251 def Llat(self, lat):
1252 '''Return the I{meridional length}, the distance along a meridian
1253 between the equator and a (geodetic) latitude, see C{L}.
1255 @arg lat: Geodetic latitude (C{degrees90}).
1257 @return: The meridional length at B{C{lat}}, negative on southern
1258 hemisphere (C{meter}).
1259 '''
1260 r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phi_(lat)
1261 return Distance(Llat=self.b * r)
1263 Lmeridian = Llat # meridional distance
1265 @property_RO
1266 def _Lpd(self):
1267 '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}.
1268 '''
1269 return Meter(_Lpd=self.L / _90_0)
1271 @property_RO
1272 def _Lpr(self):
1273 '''Get the I{quarter meridian} per radian (C{meter}), M{self.L / PI_2}.
1274 '''
1275 return Meter(_Lpr=self.L / PI_2)
1277 @deprecated_Property_RO
1278 def majoradius(self): # PYCHOK no cover
1279 '''DEPRECATED, use property C{a} or C{Requatorial}.'''
1280 return self.a
1282 def m2degrees(self, distance, lat=0):
1283 '''Convert a distance to an angle along the equator or
1284 along a parallel of (geodetic) latitude.
1286 @arg distance: Distance (C{meter}).
1287 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1289 @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}.
1291 @raise RangeError: Latitude B{C{lat}} outside valid range and
1292 L{pygeodesy.rangerrors} set to C{True}.
1294 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1295 '''
1296 return degrees(self.m2radians(distance, lat=lat))
1298 def m2radians(self, distance, lat=0):
1299 '''Convert a distance to an angle along the equator or
1300 along a parallel of (geodetic) latitude.
1302 @arg distance: Distance (C{meter}).
1303 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1305 @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}.
1307 @raise RangeError: Latitude B{C{lat}} outside valid range and
1308 L{pygeodesy.rangerrors} set to C{True}.
1310 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1311 '''
1312 r = self.circle4(lat).radius if lat else self.a
1313 return m2radians(distance, radius=r, lat=0)
1315 @deprecated_Property_RO
1316 def minoradius(self): # PYCHOK no cover
1317 '''DEPRECATED, use property C{b}, C{polaradius} or C{Rpolar}.'''
1318 return self.b
1320 @Property_RO
1321 def n(self):
1322 '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}.
1323 '''
1324 return self._assert(a_b2n(self.a, self.b), n=f2n(self.f))
1326 flattening = f
1327 flattening1st = f
1328 flattening2nd = f2
1329 flattening3rd = n
1331 polaradius = b # Rpolar
1333# @Property_RO
1334# def Q(self):
1335# '''Get the I{meridian arc unit} C{Q}, the mean, meridional length I{per radian} C({float}).
1336#
1337# @note: C{Q * PI / 2} ≈ C{L}, the I{quarter meridian}.
1338#
1339# @see: Property C{A} and U{Engsager, K., Poder, K.<https://StudyLib.net/doc/7443565/
1340# a-highly-accurate-world-wide-algorithm-for-the-transverse...>}.
1341# '''
1342# n = self.n
1343# d = (n + _1_0) / self.a
1344# return Float(Q=Fhorner(n**2, _1_0, _0_25, _1_16th, _0_25).fover(d) if d else self.b)
1346# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1347# # Q = (1 - 3/4 * e'2 + 45/64 * e'4 - 175/256 * e'6 + 11025/16384 * e'8) * rocPolar
1348# # = (4 + e'2 * (-3 + e'2 * (45/16 + e'2 * (-175/64 + e'2 * 11025/4096)))) * rocPolar / 4
1349# return Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(4 / self.rocPolar)
1351 @deprecated_Property_RO
1352 def quarteradius(self): # PYCHOK no cover
1353 '''DEPRECATED, use property C{L} or method C{Llat}.'''
1354 return self.L
1356 @Property_RO
1357 def R1(self):
1358 '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}.
1360 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}
1361 and method C{Rgeometric}.
1362 '''
1363 r = Fsum(self.a, self.a, self.b).fover(_3_0) if self.f else self.a
1364 return Radius(R1=r)
1366 Rmean = R1
1368 @Property_RO
1369 def R2(self):
1370 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}.
1372 @see: C{R2x}, C{c2}, C{area} and U{Earth radius
1373 <https://WikiPedia.org/wiki/Earth_radius>}.
1374 '''
1375 return Radius(R2=sqrt(self.c2) if self.f else self.a)
1377 Rauthalic = R2
1379# @Property_RO
1380# def R2(self):
1381# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1382# # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar
1383# # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3
1384# return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar)
1386 @Property_RO
1387 def R2x(self):
1388 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}.
1390 @see: C{R2}, C{c2x} and C{areax}.
1391 '''
1392 return Radius(R2x=sqrt(self.c2x) if self.f else self.a)
1394 Rauthalicx = R2x
1396 @Property_RO
1397 def R3(self):
1398 '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}.
1400 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}.
1401 '''
1402 r = (cbrt(self.b_a) * self.a) if self.f else self.a
1403 return Radius(R3=r)
1405 Rvolumetric = R3
1407 def radians2m(self, rad, lat=0):
1408 '''Convert an angle to the distance along the equator or
1409 along a parallel of (geodetic) latitude.
1411 @arg rad: The angle (C{radians}).
1412 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1414 @return: Distance (C{meter}, same units as the equatorial
1415 and polar radii) or C{0} for near-polar B{C{lat}}.
1417 @raise RangeError: Latitude B{C{lat}} outside valid range and
1418 L{pygeodesy.rangerrors} set to C{True}.
1420 @raise ValueError: Invalid B{C{rad}} or B{C{lat}}.
1421 '''
1422 r = self.circle4(lat).radius if lat else self.a
1423 return radians2m(rad, radius=r, lat=0)
1425 @Property_RO
1426 def Rbiaxial(self):
1427 '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}.
1429 @see: C{Rtriaxial}
1430 '''
1431 a, b = self.a, self.b
1432 if b < a:
1433 b = sqrt(_0_5 + self.b2_a2 * _0_5) * a
1434 elif b > a:
1435 b *= sqrt(_0_5 + self.a2_b2 * _0_5)
1436 return Radius(Rbiaxial=b)
1438 Requatorial = a # for consistent naming
1440 def Rgeocentric(self, lat):
1441 '''Compute the I{geocentric} earth radius of (geodetic) latitude.
1443 @arg lat: Latitude (C{degrees90}).
1445 @return: Geocentric earth radius (C{meter}).
1447 @raise ValueError: Invalid B{C{lat}}.
1449 @see: U{Geocentric Radius
1450 <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>}
1451 '''
1452 r, a = self.a, Phi_(lat)
1453 if a and self.f:
1454 if fabs(a) < PI_2:
1455 s2, c2 = _s2_c2(a)
1456 b2_a2_s2 = self.b2_a2 * s2
1457 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2))
1458 # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2)))
1459 # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2))
1460 # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2))
1461 # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2))
1462 r *= sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2))
1463 else:
1464 r = self.b
1465 return Radius(Rgeocentric=r)
1467 @Property_RO
1468 def Rgeometric(self):
1469 '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}.
1471 @see: C{R1}.
1472 '''
1473 g = sqrt(self.a * self.b) if self.f else self.a
1474 return Radius(Rgeometric=g)
1476 def rhumb_(self, exact=True):
1477 '''Get the an I{exact} C{Rhumb...} instance for this ellipsoid.
1479 @kwarg exact: If C{bool} or C{None} return L{Rhumb}C{(exact=B{exact}, ...)},
1480 otherwise a L{Rhumb}, L{RhumbAux} or L{RhumbSolve} instance
1481 for I{this} ellipsoid.
1483 @return: The C{exact} rhumb (C{Rhumb...}).
1485 @raise TypeError: Invalid B{C{exact}}.
1487 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1488 '''
1489 if isbool(exact): # use Rhumb for backward compatibility
1490 r = _MODS.rhumb.ekx.Rhumb(self, exact=exact, name=self.name)
1491 else:
1492 r = exact
1493 E = _xattr(r, ellipsoid=None)
1494 if not (E is self and isinstance(r, self._Rhumbs)):
1495 raise _ValueError(exact=r, ellipsosid=E, txt=_not_(self.name))
1496 return r
1498 @property_RO
1499 def rhumbaux(self):
1500 '''Get this ellipsoid's I{Auxiliary} C{rhumb.RhumbAux}.
1501 '''
1502 # if not self.isEllipsoidal:
1503 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1504 return _MODS.rhumb.aux_.RhumbAux(self, name=self.name)
1506 @property_RO
1507 def rhumbekx(self):
1508 '''Get this ellipsoid's I{Elliptic, Krüger} C{rhumb.Rhumb}.
1509 '''
1510 # if not self.isEllipsoidal:
1511 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1512 return _MODS.rhumb.ekx.Rhumb(self, name=self.name)
1514 @property_RO
1515 def _Rhumbs(self):
1516 '''(INTERNAL) Get all C{Rhumb...} classes, I{once}.
1517 '''
1518 p = _MODS.rhumb
1519 Ellipsoid._Rhumbs = t = (p.aux_.RhumbAux, # overwrite property_RO
1520 p.ekx.Rhumb, p.solve.RhumbSolve)
1521 return t
1523 @property
1524 def rhumbsolve(self):
1525 '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility
1526 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1527 provided the path to the C{RhumbSolve} executable is specified with env
1528 variable C{PYGEODESY_RHUMBSOLVE} or re-/set with this property.
1529 '''
1530 # if not self.isEllipsoidal:
1531 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1532 return _MODS.rhumb.solve.RhumbSolve(self, path=self._rhumbsolve, name=self.name)
1534 @rhumbsolve.setter # PYCHOK setter!
1535 def rhumbsolve(self, path):
1536 '''Re-/set the (fully qualified) path to the U{RhumbSolve
1537 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1538 overriding env variable C{PYGEODESY_RHUMBSOLVE} (C{str}).
1539 '''
1540 self._rhumbsolve = path
1542 @deprecated_property_RO
1543 def rhumbx(self):
1544 '''DEPRECATED on 2023.11.28, use property C{rhumbekx}. '''
1545 return self.rhumbekx
1547 def Rlat(self, lat):
1548 '''I{Approximate} the earth radius of (geodetic) latitude.
1550 @arg lat: Latitude (C{degrees90}).
1552 @return: Approximate earth radius (C{meter}).
1554 @raise RangeError: Latitude B{C{lat}} outside valid range and
1555 L{pygeodesy.rangerrors} set to C{True}.
1557 @raise TypeError: Invalid B{C{lat}}.
1559 @raise ValueError: Invalid B{C{lat}}.
1561 @note: C{Rlat(B{90})} equals C{Rpolar}.
1563 @see: Method C{circle4}.
1564 '''
1565 # r = a - (a - b) * |lat| / 90
1566 r = self.a
1567 if self.f and lat: # .isEllipsoidal
1568 r -= (r - self.b) * fabs(Lat(lat)) / _90_0
1569 r = Radius(Rlat=r)
1570 return r
1572 Rpolar = b # for consistent naming
1574 def roc1_(self, sa, ca=None):
1575 '''Compute the I{prime-vertical}, I{normal} radius of curvature
1576 of (geodetic) latitude, I{unscaled}.
1578 @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]).
1579 @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0])
1580 to use an alternate formula.
1582 @return: The prime-vertical radius of curvature (C{float}).
1584 @note: The delta between both formulae with C{Ellipsoids.WGS84}
1585 is less than 2 nanometer over the entire latitude range.
1587 @see: Method L{roc2_} and class L{EcefYou}.
1588 '''
1589 if not self.f: # .isSpherical
1590 n = self.a
1591 elif ca is None:
1592 r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward
1593 n = sqrt(self.a2 / r) if r > EPS02 else _0_0
1594 elif ca: # derived from EcefYou.forward
1595 h = hypot(ca, self.b_a * sa) if sa else fabs(ca)
1596 n = self.a / h
1597 elif sa:
1598 n = self.a2_b / fabs(sa)
1599 else:
1600 n = self.a
1601 return n
1603 def roc2(self, lat, scaled=False):
1604 '''Compute the I{meridional} and I{prime-vertical}, I{normal}
1605 radii of curvature of (geodetic) latitude.
1607 @arg lat: Latitude (C{degrees90}).
1608 @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}).
1610 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with
1611 the radii of curvature.
1613 @raise ValueError: Invalid B{C{lat}}.
1615 @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation
1616 <https://www.EdWilliams.org/avform.htm#flat>} and meridional and
1617 prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/
1618 Earth_radius#Radii_of_curvature>}.
1619 '''
1620 return self.roc2_(Phi_(lat), scaled=scaled)
1622 def roc2_(self, phi, scaled=False):
1623 '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of
1624 curvature of (geodetic) latitude.
1626 @arg phi: Latitude (C{radians}).
1627 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}).
1629 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the
1630 radii of curvature.
1632 @raise ValueError: Invalid B{C{phi}}.
1634 @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local,
1635 flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>}
1636 and the meridional and prime vertical U{Radii of Curvature
1637 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1638 '''
1639 a = fabs(Phi(phi))
1640 if self.f:
1641 r = self.e2s2(sin(a))
1642 if r > EPS02:
1643 n = self.a / sqrt(r)
1644 m = n * self.e21 / r # PYCHOK attr
1645 else:
1646 m = n = _0_0 # PYCHOK attr
1647 else:
1648 m = n = self.a
1649 if scaled and a:
1650 n *= cos(a) if a < PI_2 else _0_0
1651 return Curvature2Tuple(Radius(rocMeridional=m),
1652 Radius(rocPrimeVertical=n))
1654 def rocBearing(self, lat, bearing):
1655 '''Compute the I{directional} radius of curvature of (geodetic)
1656 latitude and compass direction.
1658 @arg lat: Latitude (C{degrees90}).
1659 @arg bearing: Direction (compass C{degrees360}).
1661 @return: Directional radius of curvature (C{meter}).
1663 @raise RangeError: Latitude B{C{lat}} outside valid range and
1664 L{pygeodesy.rangerrors} set to C{True}.
1666 @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}.
1668 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
1669 '''
1670 if self.f:
1671 s2, c2 = _s2_c2(Bearing_(bearing))
1672 m, n = self.roc2_(Phi_(lat))
1673 if n < m: # == n / (c2 * n / m + s2)
1674 c2 *= n / m
1675 elif m < n: # == m / (c2 + s2 * m / n)
1676 s2 *= m / n
1677 n = m
1678 b = n / (c2 + s2) # == 1 / (c2 / m + s2 / n)
1679 else:
1680 b = self.b # == self.a
1681 return Radius(rocBearing=b)
1683 @Property_RO
1684 def rocEquatorial2(self):
1685 '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature
1686 at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}.
1688 @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a},
1689 L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature
1690 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1691 '''
1692 return Curvature2Tuple(Radius(rocMeridional0=self.b2_a if self.f else self.a),
1693 Radius(rocPrimeVertical0=self.a))
1695 def rocGauss(self, lat):
1696 '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude.
1698 @arg lat: Latitude (C{degrees90}).
1700 @return: Gaussian radius of curvature (C{meter}).
1702 @raise ValueError: Invalid B{C{lat}}.
1704 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1705 Earth_radius#Radii_of_curvature>}
1706 '''
1707 # using ...
1708 # m, n = self.roc2_(Phi_(lat))
1709 # return sqrt(m * n)
1710 # ... requires 1 or 2 sqrt
1711 g = self.b
1712 if self.f:
1713 s2, c2 = _s2_c2(Phi_(lat))
1714 g = g / (c2 + self.b2_a2 * s2)
1715 return Radius(rocGauss=g)
1717 def rocMean(self, lat):
1718 '''Compute the I{mean} radius of curvature of (geodetic) latitude.
1720 @arg lat: Latitude (C{degrees90}).
1722 @return: Mean radius of curvature (C{meter}).
1724 @raise ValueError: Invalid B{C{lat}}.
1726 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1727 Earth_radius#Radii_of_curvature>}
1728 '''
1729 if self.f:
1730 m, n = self.roc2_(Phi_(lat))
1731 m *= n * _2_0 / (m + n) # == 2 / (1 / m + 1 / n)
1732 else:
1733 m = self.a
1734 return Radius(rocMean=m)
1736 def rocMeridional(self, lat):
1737 '''Compute the I{meridional} radius of curvature of (geodetic) latitude.
1739 @arg lat: Latitude (C{degrees90}).
1741 @return: Meridional radius of curvature (C{meter}).
1743 @raise ValueError: Invalid B{C{lat}}.
1745 @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation
1746 <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of
1747 Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1748 '''
1749 return self.roc2_(Phi_(lat)).meridional if lat else \
1750 self.rocEquatorial2.meridional
1752 rocPolar = a2_b # synonymous
1754 def rocPrimeVertical(self, lat):
1755 '''Compute the I{prime-vertical}, I{normal} radius of curvature of
1756 (geodetic) latitude, aka the I{transverse} radius of curvature.
1758 @arg lat: Latitude (C{degrees90}).
1760 @return: Prime-vertical radius of curvature (C{meter}).
1762 @raise ValueError: Invalid B{C{lat}}.
1764 @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth
1765 approximation<https://www.EdWilliams.org/avform.htm#flat>} and
1766 U{Radii of Curvature<https://WikiPedia.org/wiki/
1767 Earth_radius#Radii_of_curvature>}.
1768 '''
1769 return self.roc2_(Phi_(lat)).prime_vertical if lat else \
1770 self.rocEquatorial2.prime_vertical
1772 rocTransverse = rocPrimeVertical # synonymous
1774 @deprecated_Property_RO
1775 def Rquadratic(self): # PYCHOK no cover
1776 '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.'''
1777 return self.Rbiaxial
1779 @deprecated_Property_RO
1780 def Rr(self): # PYCHOK no cover
1781 '''DEPRECATED, use property C{Rrectifying}.'''
1782 return self.Rrectifying
1784 @Property_RO
1785 def Rrectifying(self):
1786 '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}.
1788 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}.
1789 '''
1790 r = (cbrt2((_1_0 + sqrt3(self.b_a)) * _0_5) * self.a) if self.f else self.a
1791 return Radius(Rrectifying=r)
1793 @deprecated_Property_RO
1794 def Rs(self): # PYCHOK no cover
1795 '''DEPRECATED, use property C{Rgeometric}.'''
1796 return self.Rgeometric
1798 @Property_RO
1799 def Rtriaxial(self):
1800 '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}.
1802 @see: C{Rbiaxial}
1803 '''
1804 a, b = self.a, self.b
1805 q = (sqrt((_3_0 + self.b2_a2) * _0_25) * a) if a > b else (
1806 (sqrt((_3_0 * self.a2_b2 + _1_0) * _0_25) * b) if a < b else a)
1807 return Radius(Rtriaxial=q)
1809 def toEllipsoid2(self, name=NN):
1810 '''Get a copy of this ellipsoid as an L{Ellipsoid2}.
1812 @kwarg name: Optional, unique name (C{str}).
1814 @see: Property C{a_f}.
1815 '''
1816 return Ellipsoid2(self, None, name=name)
1818 def toStr(self, prec=8, name=NN, **unused): # PYCHOK expected
1819 '''Return this ellipsoid as a text string.
1821 @kwarg prec: Number of decimal digits, unstripped (C{int}).
1822 @kwarg name: Override name (C{str}) or C{None} to exclude
1823 this ellipsoid's name.
1825 @return: This C{Ellipsoid}'s attributes (C{str}).
1826 '''
1827 E = Ellipsoid
1828 t = E.a, E.b, E.f_, E.f, E.f2, E.n, E.e, E.e2, E.e21, E.e22, E.e32, \
1829 E.A, E.L, E.R1, E.R2, E.R3, E.Rbiaxial, E.Rtriaxial
1830 if Fmt.form: # terse form[at]
1831 t = t[:4]
1832 return self._instr(name, prec, props=t)
1834 def toTriaxial(self, name=NN):
1835 '''Convert this ellipsoid to a L{Triaxial_}.
1837 @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis
1838 pointing east and C{Z} pointing north.
1840 @see: Method L{Triaxial_.toEllipsoid}.
1841 '''
1842 T = self._triaxial
1843 return T.copy(name=name) if name else T
1845 @property_RO
1846 def _triaxial(self):
1847 '''(INTERNAL) Get this ellipsoid's un-/ordered C{Triaxial/_}.
1848 '''
1849 a, b, m = self.a, self.b, _MODS.triaxials
1850 T = m.Triaxial if a > b else m.Triaxial_
1851 return T(a, a, b, name=self.name)
1853 @Property_RO
1854 def volume(self):
1855 '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}.
1857 @see: C{R3}.
1858 '''
1859 return Meter3(volume=self.a2 * self.b * PI_3 * _4_0)
1862class Ellipsoid2(Ellipsoid):
1863 '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}.
1864 '''
1865 def __init__(self, a, f, name=NN):
1866 '''New L{Ellipsoid2}.
1868 @arg a: Equatorial radius, semi-axis (C{meter}).
1869 @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}).
1870 @kwarg name: Optional, unique name (C{str}).
1872 @raise NameError: Ellipsoid with that B{C{name}} already exists.
1874 @raise ValueError: Invalid B{C{a}} or B{C{f}}.
1876 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.
1877 Negative C{B{f}} produces a I{prolate} ellipsoid.
1878 '''
1879 if f is None and isinstance(a, Ellipsoid):
1880 Ellipsoid.__init__(self, a.a, f =a.f,
1881 b=a.b, f_=a.f_, name=name)
1882 else:
1883 Ellipsoid.__init__(self, a, f=f, name=name)
1886def _spherical_a_b(a, b):
1887 '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}.
1888 '''
1889 return a < EPS0 or b < EPS0 or fabs(a - b) < EPS0
1892def _spherical_f(f):
1893 '''(INTERNAL) C{True} for spherical or invalid C{f}.
1894 '''
1895 return fabs(f) < EPS or f > EPS1
1898def _spherical_f_(f_):
1899 '''(INTERNAL) C{True} for spherical or invalid C{f_}.
1900 '''
1901 return fabs(f_) < EPS or fabs(f_) > _1_EPS
1904def a_b2e(a, b):
1905 '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius.
1907 @arg a: Equatorial radius (C{scalar} > 0).
1908 @arg b: Polar radius (C{scalar} > 0).
1910 @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}),
1911 M{sqrt(1 - (b / a)**2)}.
1913 @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids.
1914 '''
1915 return Float(e=sqrt(fabs(a_b2e2(a, b)))) # == sqrt(fabs(a - b) * (a + b)) / a)
1918def a_b2e2(a, b):
1919 '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius.
1921 @arg a: Equatorial radius (C{scalar} > 0).
1922 @arg b: Polar radius (C{scalar} > 0).
1924 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}),
1925 M{1 - (b / a)**2}.
1927 @note: The result is positive for I{oblate}, negative for I{prolate}
1928 or C{0} for I{near-spherical} ellipsoids.
1929 '''
1930 return Float(e2=_0_0 if _spherical_a_b(a, b) else ((a - b) * (a + b) / a**2))
1933def a_b2e22(a, b):
1934 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1936 @arg a: Equatorial radius (C{scalar} > 0).
1937 @arg b: Polar radius (C{scalar} > 0).
1939 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}),
1940 M{(a / b)**2 - 1}.
1942 @note: The result is positive for I{oblate}, negative for I{prolate}
1943 or C{0} for I{near-spherical} ellipsoids.
1944 '''
1945 return Float(e22=_0_0 if _spherical_a_b(a, b) else ((a - b) * (a + b) / b**2))
1948def a_b2e32(a, b):
1949 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1951 @arg a: Equatorial radius (C{scalar} > 0).
1952 @arg b: Polar radius (C{scalar} > 0).
1954 @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}),
1955 M{(a**2 - b**2) / (a**2 + b**2)}.
1957 @note: The result is positive for I{oblate}, negative for I{prolate}
1958 or C{0} for I{near-spherical} ellipsoids.
1959 '''
1960 a2, b2 = a**2, b**2
1961 return Float(e32=_0_0 if _spherical_a_b(a2, b2) else ((a2 - b2) / (a2 + b2)))
1964def a_b2f(a, b):
1965 '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius.
1967 @arg a: Equatorial radius (C{scalar} > 0).
1968 @arg b: Polar radius (C{scalar} > 0).
1970 @return: The flattening (C{scalar} or C{0}), M{(a - b) / a}.
1972 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1973 for I{near-spherical} ellipsoids.
1974 '''
1975 f = 0 if _spherical_a_b(a, b) else ((a - b) / a)
1976 return _f_0_0 if _spherical_f(f) else Float(f=f)
1979def a_b2f_(a, b):
1980 '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius.
1982 @arg a: Equatorial radius (C{scalar} > 0).
1983 @arg b: Polar radius (C{scalar} > 0).
1985 @return: The inverse flattening (C{scalar} or C{0}), M{a / (a - b)}.
1987 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1988 for I{near-spherical} ellipsoids.
1989 '''
1990 f_ = 0 if _spherical_a_b(a, b) else (a / float(a - b))
1991 return _f__0_0 if _spherical_f_(f_) else Float(f_=f_)
1994def a_b2f2(a, b):
1995 '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius.
1997 @arg a: Equatorial radius (C{scalar} > 0).
1998 @arg b: Polar radius (C{scalar} > 0).
2000 @return: The I{signed}, 2nd flattening (C{scalar} or C{0}), M{(a - b) / b}.
2002 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2003 for I{near-spherical} ellipsoids.
2004 '''
2005 t = 0 if _spherical_a_b(a, b) else float(a - b)
2006 return Float(f2=_0_0 if fabs(t) < EPS0 else (t / b))
2009def a_b2n(a, b):
2010 '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius.
2012 @arg a: Equatorial radius (C{scalar} > 0).
2013 @arg b: Polar radius (C{scalar} > 0).
2015 @return: The I{signed}, 3rd flattening (C{scalar} or C{0}), M{(a - b) / (a + b)}.
2017 @note: The result is positive for I{oblate}, negative for I{prolate}
2018 or C{0} for I{near-spherical} ellipsoids.
2019 '''
2020 t = 0 if _spherical_a_b(a, b) else float(a - b)
2021 return Float(n=_0_0 if fabs(t) < EPS0 else (t / (a + b)))
2024def a_f2b(a, f):
2025 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}.
2027 @arg a: Equatorial radius (C{scalar} > 0).
2028 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2030 @return: The polar radius (C{float}), M{a * (1 - f)}.
2031 '''
2032 b = a if _spherical_f(f) else (a * (_1_0 - f))
2033 return Radius_(b=a if _spherical_a_b(a, b) else b)
2036def a_f_2b(a, f_):
2037 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}.
2039 @arg a: Equatorial radius (C{scalar} > 0).
2040 @arg f_: Inverse flattening (C{scalar} >>> 1).
2042 @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}.
2043 '''
2044 b = a if _spherical_f_(f_) else (a * (f_ - _1_0) / f_)
2045 return Radius_(b=a if _spherical_a_b(a, b) else b)
2048def b_f2a(b, f):
2049 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}.
2051 @arg b: Polar radius (C{scalar} > 0).
2052 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2054 @return: The equatorial radius (C{float}), M{b / (1 - f)}.
2055 '''
2056 t = _1_0 - f
2057 a = b if fabs(t) < EPS0 else (b / t)
2058 return Radius_(a=b if _spherical_a_b(a, b) else a)
2061def b_f_2a(b, f_):
2062 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}.
2064 @arg b: Polar radius (C{scalar} > 0).
2065 @arg f_: Inverse flattening (C{scalar} >>> 1).
2067 @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}.
2068 '''
2069 t = f_ - _1_0
2070 a = b if _spherical_f_(f_) or fabs(t - f_) < EPS0 \
2071 or fabs(t) < EPS0 else (b * f_ / t)
2072 return Radius_(a=b if _spherical_a_b(a, b) else a)
2075def e2f(e):
2076 '''Return C{f}, the I{flattening} for a given I{1st eccentricity}.
2078 @arg e: The (1st) eccentricity (0 <= C{float} < 1)
2080 @return: The flattening (C{scalar} or C{0}).
2082 @see: Function L{e22f}.
2083 '''
2084 return e22f(e**2)
2087def e22f(e2):
2088 '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}.
2090 @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1)
2092 @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(e2 - 1) + 1)}.
2093 '''
2094 return Float(f=e2 / (sqrt(_1_0 - e2) + _1_0)) if e2 else _f_0_0
2097def f2e2(f):
2098 '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}.
2100 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2102 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1),
2103 M{f * (2 - f)}.
2105 @note: The result is positive for I{oblate}, negative for I{prolate}
2106 or C{0} for I{near-spherical} ellipsoids.
2108 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2109 html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2110 <https://WikiPedia.org/wiki/Flattening>}.
2111 '''
2112 return Float(e2=_0_0 if _spherical_f(f) else (f * (_2_0 - f)))
2115def f2e22(f):
2116 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}.
2118 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2120 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or
2121 C{INF}), M{f * (2 - f) / (1 - f)**2}.
2123 @note: The result is positive for I{oblate}, negative for I{prolate}
2124 or C{0} for near-spherical ellipsoids.
2126 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2127 html/classGeographicLib_1_1Ellipsoid.html>}.
2128 '''
2129 # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2
2130 t = (_1_0 - f)**2
2131 return Float(e22=INF if t < EPS0 else (f2e2(f) / t)) # PYCHOK type
2134def f2e32(f):
2135 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}.
2137 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2139 @return: The I{signed}, 3rd eccentricity I{squared} (C{float}),
2140 M{f * (2 - f) / (1 + (1 - f)**2)}.
2142 @note: The result is positive for I{oblate}, negative for I{prolate}
2143 or C{0} for I{near-spherical} ellipsoids.
2145 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2146 html/classGeographicLib_1_1Ellipsoid.html>}.
2147 '''
2148 # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2)
2149 e2 = f2e2(f)
2150 return Float(e32=e2 / (_2_0 - e2))
2153def f_2f(f_):
2154 '''Return C{f}, the I{flattening} for a given I{inverse flattening}.
2156 @arg f_: Inverse flattening (C{scalar} >>> 1).
2158 @return: The flattening (C{scalar} or C{0}), M{1 / f_}.
2160 @note: The result is positive for I{oblate}, negative for I{prolate}
2161 or C{0} for I{near-spherical} ellipsoids.
2162 '''
2163 f = 0 if _spherical_f_(f_) else _1_0 / f_
2164 return _f_0_0 if _spherical_f(f) else Float(f=f) # PYCHOK type
2167def f2f_(f):
2168 '''Return C{f_}, the I{inverse flattening} for a given I{flattening}.
2170 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2172 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2174 @note: The result is positive for I{oblate}, negative for I{prolate}
2175 or C{0} for I{near-spherical} ellipsoids.
2176 '''
2177 f_ = 0 if _spherical_f(f) else _1_0 / f
2178 return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) # PYCHOK type
2181def f2f2(f):
2182 '''Return C{f2}, the I{2nd flattening} for a given I{flattening}.
2184 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2186 @return: The I{signed}, 2nd flattening (C{scalar} or C{INF}), M{f / (1 - f)}.
2188 @note: The result is positive for I{oblate}, negative for I{prolate}
2189 or C{0} for I{near-spherical} ellipsoids.
2191 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2192 html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2193 <https://WikiPedia.org/wiki/Flattening>}.
2194 '''
2195 t = _1_0 - f
2196 return Float(f2=_0_0 if _spherical_f(f) else (INF if fabs(t) < EPS
2197 else (f / t))) # PYCHOK type
2200def f2n(f):
2201 '''Return C{n}, the I{3rd flattening} for a given I{flattening}.
2203 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2205 @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1),
2206 M{f / (2 - f)}.
2208 @note: The result is positive for I{oblate}, negative for I{prolate}
2209 or C{0} for I{near-spherical} ellipsoids.
2211 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2212 html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2213 <https://WikiPedia.org/wiki/Flattening>}.
2214 '''
2215 return Float(n=_0_0 if _spherical_f(f) else (f / float(_2_0 - f)))
2218def n2e2(n):
2219 '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}.
2221 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2223 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF),
2224 M{4 * n / (1 + n)**2}.
2226 @note: The result is positive for I{oblate}, negative for I{prolate}
2227 or C{0} for I{near-spherical} ellipsoids.
2229 @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}.
2230 '''
2231 t = (n + _1_0)**2
2232 return Float(e2=_0_0 if fabs(n) < EPS0 else
2233 (NINF if t < EPS0 else (_4_0 * n / t)))
2236def n2f(n):
2237 '''Return C{f}, the I{flattening} for a given I{3rd flattening}.
2239 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2241 @return: The flattening (C{scalar} or NINF), M{2 * n / (1 + n)}.
2243 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2244 html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2245 <https://WikiPedia.org/wiki/Flattening>}.
2246 '''
2247 t = n + _1_0
2248 f = 0 if fabs(n) < EPS0 else (NINF if t < EPS0 else (_2_0 * n / t))
2249 return _f_0_0 if _spherical_f(f) else Float(f=f)
2252def n2f_(n):
2253 '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}.
2255 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2257 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2259 @see: L{n2f} and L{f2f_}.
2260 '''
2261 return f2f_(n2f(n))
2264def _normalTo3(px, py, E, eps=EPS): # in .height4 above
2265 '''(INTERNAL) Nearest point on a 2-D ellipse in 1st quadrant.
2267 @see: Functions C{pygeodesy.triaxial._normalTo4} and C{-To5}.
2268 '''
2269 a, b, e0 = E.a, E.b, EPS0
2270 if min(px, py, a, b) < e0:
2271 raise _AssertionError(px=px, py=py, a=a, b=b, E=E)
2273 a2 = a - b * E.b_a
2274 b2 = b - a * E.a_b
2275 tx = ty = _SQRT2_2
2276 _a, _h = fabs, hypot
2277 for i in range(16): # max 5
2278 ex = a2 * tx**3
2279 ey = b2 * ty**3
2281 qx = px - ex
2282 qy = py - ey
2283 q = _h(qx, qy)
2284 if q < e0: # PYCHOK no cover
2285 break
2286 r = _h(ex - tx * a,
2287 ey - ty * b) / q
2289 sx, tx = tx, min(_1_0, max(0, (ex + qx * r) / a))
2290 sy, ty = ty, min(_1_0, max(0, (ey + qy * r) / b))
2291 t = _h(ty, tx)
2292 if t < e0: # PYCHOK no cover
2293 break
2294 tx = tx / t # /= chokes PyChecker
2295 ty = ty / t
2296 if _a(sx - tx) < eps and \
2297 _a(sy - ty) < eps:
2298 break
2300 tx *= a / px
2301 ty *= b / py
2302 return tx, ty, i # x and y as fractions
2305class Ellipsoids(_NamedEnum):
2306 '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class
2307 to accommodate the L{_LazyNamedEnumItem} properties.
2308 '''
2309 def _Lazy(self, a, b, f_, **kwds):
2310 '''(INTERNAL) Instantiate the L{Ellipsoid}.
2311 '''
2312 return Ellipsoid(a, b=b, f_=f_, **kwds)
2314Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton
2315'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.'''
2316# <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html>
2317# <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS>
2318# <https://kb.OSU.edu/dspace/handle/1811/77986>
2319# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids>
2320# <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html>
2321# <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java>
2322Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid>
2323 Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909
2324 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448
2325# APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965
2326# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid
2327# AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America
2328 Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192
2329 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System"
2330 Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)),
2331 BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)),
2332# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)),
2333 CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC)
2334# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)),
2335 Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)),
2336 Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)),
2337 Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)),
2338 Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc
2339 CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures
2340 Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium
2341 Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)),
2342# Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)),
2343# Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)),
2344# Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)),
2345 Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)),
2346 Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)),
2347 Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)),
2348# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)),
2349 GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)),
2350 GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid"
2351 GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516
2352# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)),
2353 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.25722210088)), # IUGG, ITRS, ETRS89
2354# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297
2355 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)),
2356# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)),
2357 IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union
2358 IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302
2359 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy
2360 IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)),
2361 Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795
2362 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l
2363 Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2364 Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2365# Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961
2366# Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979
2367 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France
2368 Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)),
2369 Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)),
2370# MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983
2371# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab.
2372 NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab.
2373# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965
2374 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)),
2375 OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)),
2376# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect?
2377 Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972
2378# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)),
2379 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11
2380# SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia
2381 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System
2382 SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American
2383 Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo
2384 SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo
2385 SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)), # EPSG:3857 Spheroid
2386 Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)),
2387# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)),
2388# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)),
2389 WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)),
2390 WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)),
2391 WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52
2392 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)) # GPS b=6356752.3142451793
2393)
2395_EWGS84 = Ellipsoids.WGS84 # (INTERNAL) shared
2397if __name__ == '__main__':
2399 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_
2400 from pygeodesy import nameof, printf
2402 for E in (_EWGS84, Ellipsoids.GRS80, # NAD83,
2403 Ellipsoids.Sphere, Ellipsoids.SpherePopular,
2404 Ellipsoid(_EWGS84.b, _EWGS84.a, name='_Prolate')):
2405 e = f2n(E.f) - E.n
2406 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10), nl=1)
2407 printf('# e=%s, f_=%s, f=%s, n=%s (%s)', fstr(E.e, prec=13, fmt=Fmt.e),
2408 fstr(E.f_, prec=13, fmt=Fmt.e),
2409 fstr(E.f, prec=13, fmt=Fmt.e),
2410 fstr(E.n, prec=13, fmt=Fmt.e),
2411 fstr(e, prec=9, fmt=Fmt.e))
2412 printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20))
2413 printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20))
2414 printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property
2416 # __doc__ of this file, force all into registry
2417 t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_)
2418 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))
2420# % python3 -m pygeodesy.ellipsoids
2422# 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, e21=0.99330562, e22=0.0067394967, e32=0.0033584313, A=6367449.1458234144, L=10001965.7293127235, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.7900091587, Rbiaxial=6367453.6345163295, Rtriaxial=6372797.5559594007
2423# e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00)
2424# AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2425# BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2426# KsOrder 8
2428# Ellipsoids.GRS80: name='GRS80', a=6378137, b=6356752.3141403468, f_=298.2572221009, f=0.0033528107, f2=0.0033640898, n=0.0016792204, e=0.081819191, e2=0.00669438, e21=0.99330562, e22=0.0067394968, e32=0.0033584313, A=6367449.1457710434, L=10001965.7292304561, R1=6371008.7713801153, R2=6371007.1808835147, R3=6371000.7899741363, Rbiaxial=6367453.6344640013, Rtriaxial=6372797.5559332585
2429# e=8.1819191042833e-02, f_=2.9825722210088e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00)
2430# AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2431# BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2432# KsOrder 8
2434# Ellipsoids.Sphere: name='Sphere', a=6371008.7714149999, b=6371008.7714149999, f_=0, f=0, f2=0, n=0, e=0, e2=0, e21=1, 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
2435# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2436# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2437# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2438# KsOrder 8
2440# Ellipsoids.SpherePopular: name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e21=1, e22=0, e32=0, A=6378137, L=10018754.171394622, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137
2441# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2442# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2443# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2444# KsOrder 8
2446# 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, e21=1.0067394967, e22=-0.00669438, e32=-0.0033584313, A=6367449.1458234144, L=10035500.5204500314, R1=6363880.5428301189, R2=6363878.9413582645, R3=6363872.5644020075, Rbiaxial=6367453.6345163295, Rtriaxial=6362105.2243882557
2447# e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00)
2448# AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0
2449# BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0
2450# KsOrder 8
2452# **) MIT License
2453#
2454# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
2455#
2456# Permission is hereby granted, free of charge, to any person obtaining a
2457# copy of this software and associated documentation files (the "Software"),
2458# to deal in the Software without restriction, including without limitation
2459# the rights to use, copy, modify, merge, publish, distribute, sublicense,
2460# and/or sell copies of the Software, and to permit persons to whom the
2461# Software is furnished to do so, subject to the following conditions:
2462#
2463# The above copyright notice and this permission notice shall be included
2464# in all copies or substantial portions of the Software.
2465#
2466# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
2467# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
2468# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
2469# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
2470# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
2471# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
2472# OTHER DEALINGS IN THE SOFTWARE.