Coverage for pygeodesy/ellipsoids.py: 96%
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« prev ^ index » next coverage.py v7.2.2, created at 2024-05-25 12:04 -0400
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@var Ellipsoids.WGS84_NGS: Ellipsoid(name='WGS84_NGS', 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)
63'''
64# make sure int/int division yields float quotient, see .basics
65from __future__ import division as _; del _ # PYCHOK semicolon
67from pygeodesy.basics import copysign0, isbool, isint
68from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, INF, NINF, PI4, PI_2, PI_3, R_M, R_MA, R_FM, \
69 _EPSqrt, _EPStol as _TOL, _floatuple as _T, _isfinite, _SQRT2_2, \
70 _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \
71 _0_25, _3_0 # PYCHOK used!
72from pygeodesy.errors import _AssertionError, IntersectionError, _ValueError, _xattr, _xkwds_not
73from pygeodesy.fmath import cbrt, cbrt2, fdot, Fhorner, fpowers, hypot, hypot_, \
74 hypot1, hypot2, sqrt3, Fsum
75# from pygeodesy.fsums import Fsum # from .fmath
76from pygeodesy.interns import NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, _beta_, \
77 _Clarke1866_, _Clarke1880IGN_, _DOT_, _f_, _GRS80_, _height_, \
78 _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \
79 _Krassowsky1940_, _lat_, _meridional_, _negative_, _not_finite_, \
80 _prime_vertical_, _radius_, _Sphere_, _SPACE_, _vs_, _WGS72_, _WGS84_
81# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .named
82from pygeodesy.named import _lazyNamedEnumItem as _lazy, _name__, _name2__, _NamedEnum, \
83 _NamedEnumItem, _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__ = '24.05.21'
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 C{B{name}=NN} (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):
160 '''Return an L{Ellipsoid} for this 2-tuple C{(a, f)}.
162 @kwarg name: Optional C{B{name}=NN} (C{str}).
164 @raise NameError: A registered C{ellipsoid} with the
165 same B{C{name}} already exists.
166 '''
167 return Ellipsoid(self.a, f=self.f, name=self._name__(name)) # PYCHOK .a and .f
169 @Property_RO
170 def f_(self):
171 '''Get the I{inverse} flattening (C{scalar}), M{1 / f} == M{a / (a - b)}.
172 '''
173 return f2f_(self.f) # PYCHOK .f
176class Circle4Tuple(_NamedTuple):
177 '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height},
178 both conventionally in C{meter} of a parallel I{circle of latitude} at
179 (geodetic) latitude C{lat} and the I{parametric (or reduced) auxiliary
180 latitude} C{beta}, both in C{degrees90}.
182 The C{height} is the (signed) distance along the z-axis between the
183 parallel and the equator. At near-polar C{lat}s, the C{radius} is C{0},
184 the C{height} is the ellipsoid's (signed) polar radius and C{beta}
185 equals C{lat}.
186 '''
187 _Names_ = (_radius_, _height_, _lat_, _beta_)
188 _Units_ = ( Radius, Height, Lat, Lat)
191class Curvature2Tuple(_NamedTuple):
192 '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in
193 C{meter}, conventionally.
194 '''
195 _Names_ = (_meridional_, _prime_vertical_)
196 _Units_ = ( Meter, Meter)
198 @property_RO
199 def transverse(self):
200 '''Get this I{prime_vertical}, aka I{transverse} radius of curvature.
201 '''
202 return self.prime_vertical
205class Ellipsoid(_NamedEnumItem):
206 '''Ellipsoid with I{equatorial} and I{polar} radii, I{flattening}, I{inverse
207 flattening} and other, often used, I{cached} attributes, supporting
208 I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models.
209 '''
210 _a = 0 # equatorial radius, semi-axis (C{meter})
211 _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f
212 _f = 0 # (1st) flattening: (a - b) / a
213 _f_ = 0 # inverse flattening: 1 / f = a / (a - b)
215 _geodsolve = NN # means, use PYGEODESY_GEODSOLVE
216 _KsOrder = 8 # Krüger series order (4, 6 or 8)
217 _rhumbsolve = NN # means, use PYGEODESY_RHUMBSOLVE
219 def __init__(self, a, b=None, f_=None, f=None, **name):
220 '''New L{Ellipsoid} from the I{equatorial} radius I{and} either
221 the I{polar} radius or I{inverse flattening} or I{flattening}.
223 @arg a: Equatorial radius, semi-axis (C{meter}).
224 @arg b: Optional polar radius, semi-axis (C{meter}).
225 @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0).
226 @arg f: Flattening: M{(a - b) / a} (C{scalar}, near zero for
227 spherical).
228 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
230 @raise NameError: Ellipsoid with the same B{C{name}} already exists.
232 @raise ValueError: Invalid B{C{a}}, B{C{b}}, B{C{f_}} or B{C{f}} or
233 B{C{f_}} and B{C{f}} are incompatible.
235 @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced
236 to M{1 / f_ = 0}, spherical.
237 '''
238 ff_ = f, f_ # assertion below
239 n = _name__(**name) if name else NN
240 try:
241 a = Radius_(a=a) # low=EPS
242 if not _isfinite(a):
243 raise ValueError(_SPACE_(_a_, _not_finite_))
245 if b: # not in (_0_0, None)
246 b = Radius_(b=b) # low=EPS
247 f = a_b2f(a, b) if f is None else Float(f=f)
248 f_ = f2f_(f) if f_ is None else Float(f_=f_)
249 elif f is not None:
250 f = Float(f=f)
251 b = a_f2b(a, f)
252 f_ = f2f_(f) if f_ is None else Float(f_=f_)
253 elif f_:
254 f_ = Float(f_=f_)
255 b = a_f_2b(a, f_) # a * (f_ - 1) / f_
256 f = f_2f(f_)
257 else: # only a, spherical
258 f_ = f = 0
259 b = a # superfluous
261 if not f < _1_0: # sanity check, see .ecef.Ecef.__init__
262 raise ValueError(_SPACE_(_f_, _invalid_))
263 if not _isfinite(b):
264 raise ValueError(_SPACE_(_b_, _not_finite_))
266 if fabs(f) < EPS or a == b or not f_: # spherical
267 b = a
268 f = _f_0_0
269 f_ = _f__0_0
271 except (TypeError, ValueError) as x:
272 d = _xkwds_not(None, b=b, f_=f_, f=f)
273 t = instr(self, a=a, name=n, **d)
274 raise _ValueError(t, cause=x)
276 self._a = a
277 self._b = b
278 self._f = f
279 self._f_ = f_
281 self._register(Ellipsoids, n)
283 if f and f_: # see .test/testEllipsoidal.py
284 d = dict(eps=_TOL)
285 if None in ff_: # both f_ and f given
286 d.update(Error=_ValueError, txt=_incompatible_)
287 self._assert(_1_0 / f, f_=f_, **d)
288 self._assert(_1_0 / f_, f =f, **d)
289 self._assert(self.b2_a2, e21=self.e21, eps=EPS)
291 def __eq__(self, other):
292 '''Compare this and an other ellipsoid.
294 @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}).
296 @return: C{True} if equal, C{False} otherwise.
297 '''
298 return self is other or (isinstance(other, Ellipsoid) and
299 self.a == other.a and
300 (self.f == other.f or self.b == other.b))
302 def __hash__(self):
303 return self._hash # memoized
305 @Property_RO
306 def a(self):
307 '''Get the I{equatorial} radius, semi-axis (C{meter}).
308 '''
309 return self._a
311 equatoradius = a # = Requatorial
313 @Property_RO
314 def a2(self):
315 '''Get the I{equatorial} radius I{squared} (C{meter} I{squared}), M{a**2}.
316 '''
317 return Meter2(a2=self.a**2)
319 @Property_RO
320 def a2_(self):
321 '''Get the inverse of the I{equatorial} radius I{squared} (C{meter} I{squared}), M{1 / a**2}.
322 '''
323 return Float(a2_=_1_0 / self.a2)
325 @Property_RO
326 def a_b(self):
327 '''Get the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b} == M{1 / (1 - f)}.
328 '''
329 return Float(a_b=self.a / self.b if self.f else _1_0)
331 @Property_RO
332 def a2_b(self):
333 '''Get the I{polar} meridional (or polar) radius of curvature (C{meter}), M{a**2 / b}.
335 @see: U{Radii of Curvature
336 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
337 and U{Moritz, H. (1980), Geodetic Reference System 1980
338 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}.
340 @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature
341 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{c2}
342 and L{R2} or L{Rauthalic}.
343 '''
344 return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar
346 @Property_RO
347 def a2_b2(self):
348 '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}),
349 M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e21}.
350 '''
351 return Float(a2_b2=self.a_b**2 if self.f else _1_0)
353 @Property_RO
354 def a_f(self):
355 '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}), see method C{toEllipsoid2}.
356 '''
357 return a_f2Tuple(self.a, self.f, name=self.name)
359 @Property_RO
360 def A(self):
361 '''Get the UTM I{meridional (or rectifying)} radius (C{meter}).
363 @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/>}.
364 '''
365 A, n = self.a, self.n
366 if n:
367 d = (n + _1_0) * 1048576 / A
368 if d: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8
369 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>
370 # <https://GeographicLib.SourceForge.io/C++/doc/transversemercator.html> and
371 # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> (3)
372 # A *= fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441) / 1048576) / (1 + n)
373 A = Radius(A=Fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441).fover(d))
374 return A
376 @Property_RO
377 def _albersCyl(self):
378 '''(INTERNAL) Helper for C{auxAuthalic}.
379 '''
380 return _MODS.albers.AlbersEqualAreaCylindrical(datum=self, name=self.name)
382 @Property_RO
383 def AlphaKs(self):
384 '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
385 '''
386 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon
387 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
388 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200),
389 _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned
390 _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned
391 _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned
392 _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned
393 _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned
394 _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned
395 _T(1424729850961/743921418240)) # PYCHOK unaligned
397 @Property_RO
398 def area(self):
399 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2}.
401 @see: Properties L{areax}, L{c2} and L{R2} and functions
402 L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
403 '''
404 return Meter2(area=self.c2 * PI4)
406 @Property_RO
407 def areax(self):
408 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more
409 accurate for very I{oblate} ellipsoids.
411 @see: Properties L{area}, L{c2x} and L{R2x}, class L{GeodesicExact} and
412 functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}.
413 '''
414 return Meter2(areax=self.c2x * PI4)
416 def _assert(self, val, eps=_TOL, f0=_0_0, Error=_AssertionError, txt=NN, **name_value):
417 '''(INTERNAL) Assert a C{name=value} vs C{val}.
418 '''
419 for n, v in name_value.items():
420 if fabs(v - val) > eps: # PYCHOK no cover
421 t = (v, _vs_, val)
422 t = _SPACE_.join(strs(t, prec=12, fmt=Fmt.g))
423 t = Fmt.EQUAL(self._DOT_(n), t)
424 raise Error(t, txt=txt or Fmt.exceeds_eps(eps))
425 return Float(v if self.f else f0, name=n)
426 raise Error(unstr(self._DOT_(self._assert.__name__), val,
427 eps=eps, f0=f0, **name_value))
429 def auxAuthalic(self, lat, inverse=False):
430 '''Compute the I{authalic} auxiliary latitude or the I{inverse} thereof.
432 @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}).
433 @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and
434 return the geodetic latitude (C{bool}).
436 @return: The I{authalic} (or geodetic) latitude in C{degrees90}.
438 @see: U{Inverse-/AuthalicLatitude<https://GeographicLib.SourceForge.io/
439 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude
440 <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and
441 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16.
442 '''
443 if self.f:
444 f = self._albersCyl._tanf if inverse else self._albersCyl._txif # PYCHOK attr
445 lat = atan1d(f(tan(Phi_(lat)))) # PYCHOK attr
446 return _aux(lat, inverse, Ellipsoid.auxAuthalic)
448 def auxConformal(self, lat, inverse=False):
449 '''Compute the I{conformal} auxiliary latitude or the I{inverse} thereof.
451 @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}).
452 @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and
453 return the geodetic latitude (C{bool}).
455 @return: The I{conformal} (or geodetic) latitude in C{degrees90}.
457 @see: U{Inverse-/ConformalLatitude<https://GeographicLib.SourceForge.io/
458 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude
459 <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and
460 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
461 '''
462 if self.f:
463 f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr
464 lat = atan1d(f(tan(Phi_(lat)))) # PYCHOK attr
465 return _aux(lat, inverse, Ellipsoid.auxConformal)
467 def auxGeocentric(self, lat, inverse=False):
468 '''Compute the I{geocentric} auxiliary latitude or the I{inverse} thereof.
470 @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}).
471 @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and
472 return the I{geocentric} latitude (C{bool}).
474 @return: The I{geocentric} (or geodetic) latitude in C{degrees90}.
476 @see: U{Inverse-/GeocentricLatitude<https://GeographicLib.SourceForge.io/
477 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude
478 <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and
479 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18.
480 '''
481 if self.f:
482 f = self.a2_b2 if inverse else self.b2_a2
483 lat = atan1d(f * tan(Phi_(lat)))
484 return _aux(lat, inverse, Ellipsoid.auxGeocentric)
486 def auxIsometric(self, lat, inverse=False):
487 '''Compute the I{isometric} auxiliary latitude or the I{inverse} thereof.
489 @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}).
490 @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and
491 return the geodetic latitude (C{bool}).
493 @return: The I{isometric} (or geodetic) latitude in C{degrees}.
495 @note: The I{isometric} latitude for geodetic C{+/-90} is far
496 outside the C{[-90..+90]} range but the inverse
497 thereof is the original geodetic latitude.
499 @see: U{Inverse-/IsometricLatitude<https://GeographicLib.SourceForge.io/
500 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude
501 <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and
502 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16.
503 '''
504 if self.f:
505 r = Phi_(lat, clip=0)
506 lat = degrees(atan1(self.es_tauf(sinh(r))) if inverse else
507 asinh(self.es_taupf(tan(r))))
508 # clip=0, since auxIsometric(+/-90) is far outside [-90..+90]
509 return _aux(lat, inverse, Ellipsoid.auxIsometric, clip=0)
511 def auxParametric(self, lat, inverse=False):
512 '''Compute the I{parametric} auxiliary latitude or the I{inverse} thereof.
514 @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}).
515 @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and
516 return the geodetic latitude (C{bool}).
518 @return: The I{parametric} (or geodetic) latitude in C{degrees90}.
520 @see: U{Inverse-/ParametricLatitude<https://GeographicLib.SourceForge.io/
521 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude
522 <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>},
523 and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18.
524 '''
525 if self.f:
526 lat = self._beta(Lat(lat), inverse=inverse)
527 return _aux(lat, inverse, Ellipsoid.auxParametric)
529 auxReduced = auxParametric # synonymous
531 def auxRectifying(self, lat, inverse=False):
532 '''Compute the I{rectifying} auxiliary latitude or the I{inverse} thereof.
534 @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}).
535 @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and
536 return the geodetic latitude (C{bool}).
538 @return: The I{rectifying} (or geodetic) latitude in C{degrees90}.
540 @see: U{Inverse-/RectifyingLatitude<https://GeographicLib.SourceForge.io/
541 C++/doc/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude
542 <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and
543 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17.
544 '''
545 if self.f:
546 lat = Lat(lat)
547 if 0 < fabs(lat) < _90_0:
548 if inverse:
549 e = self._elliptic_e22
550 d = degrees90(e.fEinv(e.cE * lat / _90_0))
551 lat = self.auxParametric(d, inverse=True)
552 else:
553 lat = _90_0 * self.Llat(lat) / self.L
554 return _aux(lat, inverse, Ellipsoid.auxRectifying)
556 @Property_RO
557 def b(self):
558 '''Get the I{polar} radius, semi-axis (C{meter}).
559 '''
560 return self._b
562 polaradius = b # = Rpolar
564 @Property_RO
565 def b_a(self):
566 '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a == f1 == 1 - f}.
568 @see: Property L{f1}.
569 '''
570 return self._assert(self.b / self.a, b_a=self.f1, f0=_1_0)
572 @Property_RO
573 def b2(self):
574 '''Get the I{polar} radius I{squared} (C{float}), M{b**2}.
575 '''
576 return Meter2(b2=self.b**2)
578 @Property_RO
579 def b2_a(self):
580 '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see C{rocMeridional}C{(0)}.
582 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
583 '''
584 return Radius(b2_a=self.b2 / self.a if self.f else self.b)
586 @Property_RO
587 def b2_a2(self):
588 '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2}
589 == M{(1 - f)**2} == M{1 - e**2} == C{e21}.
590 '''
591 return Float(b2_a2=self.b_a**2 if self.f else _1_0)
593 def _beta(self, lat, inverse=False):
594 '''(INTERNAL) Get the I{parametric (or reduced) auxiliary latitude} or inverse thereof.
595 '''
596 s, c = sincos2d(lat) # like Karney's tand(lat)
597 s *= self.a_b if inverse else self.b_a
598 return atan1d(s, c)
600 @Property_RO
601 def BetaKs(self):
602 '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>} (C{KsOrder}C{-tuple}).
603 '''
604 return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon
605 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8
606 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600),
607 _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned
608 _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned
609 _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned
610 _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned
611 _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne
612 _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned
613 _T(191773887257/3719607091200)) # PYCHOK unaligned
615 @deprecated_Property_RO
616 def c(self): # PYCHOK no cover
617 '''DEPRECATED, use property C{R2} or C{Rauthalic}.'''
618 return self.R2
620 @Property_RO
621 def c2(self):
622 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}).
624 @see: Properties L{c2x}, L{area}, L{R2}, L{Rauthalic}, I{Karney's} U{equation (60)
625 <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area
626 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>},
627 U{Authalic radius<https://WikiPedia.org/wiki/Earth_radius#Authalic_radius>}, U{Surface area
628 <https://WikiPedia.org/wiki/Ellipsoid>} and U{surface area
629 <https://www.Numericana.com/answer/geometry.htm#oblate>}.
630 '''
631 return self._c2f(False)
633 @Property_RO
634 def c2x(self):
635 '''Get the I{authalic} earth radius I{squared} (C{meter} I{squared}), more accurate for very I{oblate}
636 ellipsoids.
638 @see: Properties L{c2}, L{areax}, L{R2x}, L{Rauthalicx}, class L{GeodesicExact} and I{Karney}'s comments at C++
639 attribute U{GeodesicExact._c2<https://GeographicLib.SourceForge.io/C++/doc/GeodesicExact_8cpp_source.html>}.
640 '''
641 return self._c2f(True)
643 def _c2f(self, c2x):
644 '''(INTERNAL) Helper for C{.c2} and C{.c2x}.
645 '''
646 f, c2 = self.f, self.b2
647 if f:
648 e = self.e
649 if e > EPS0:
650 if f > 0: # .isOblate
651 c2 *= (asinh(sqrt(self.e22abs)) if c2x else atanh(e)) / e
652 elif f < 0: # .isProlate
653 c2 *= atan1(e) / e # XXX asin?
654 c2 = Meter2(c2=(self.a2 + c2) * _0_5)
655 return c2
657 def circle4(self, lat):
658 '''Get the equatorial or a parallel I{circle of latitude}.
660 @arg lat: Geodetic latitude (C{degrees90}, C{str}).
662 @return: A L{Circle4Tuple}C{(radius, height, lat, beta)}
663 instance.
665 @raise RangeError: Latitude B{C{lat}} outside valid range and
666 L{pygeodesy.rangerrors} set to C{True}.
668 @raise TypeError: Invalid B{C{lat}}.
670 @raise ValueError: Invalid B{C{lat}}.
672 @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude}
673 <https://WikiPedia.org/wiki/Latitude>}, I{Karney's} C++ U{CircleRadius and CircleHeight
674 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>}
675 and method C{Rlat}.
676 '''
677 lat = Lat(lat)
678 if lat:
679 b = lat
680 if fabs(lat) < _90_0:
681 if self.f:
682 b = self._beta(lat)
683 z, r = sincos2d(b)
684 r *= self.a
685 z *= self.b
686 else: # near-polar
687 r, z = _0_0, copysign0(self.b, lat)
688 else: # equator
689 r = self.a
690 z = lat = b = _0_0
691 return Circle4Tuple(r, z, lat, b)
693 def degrees2m(self, deg, lat=0):
694 '''Convert an angle to the distance along the equator or
695 along a parallel of (geodetic) latitude.
697 @arg deg: The angle (C{degrees}).
698 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
700 @return: Distance (C{meter}, same units as the equatorial
701 and polar radii) or C{0} for near-polar B{C{lat}}.
703 @raise RangeError: Latitude B{C{lat}} outside valid range and
704 L{pygeodesy.rangerrors} set to C{True}.
706 @raise ValueError: Invalid B{C{deg}} or B{C{lat}}.
707 '''
708 return self.radians2m(radians(deg), lat=lat)
710 def distance2(self, lat0, lon0, lat1, lon1):
711 '''I{Approximate} the distance and (initial) bearing between
712 two points based on the U{local, flat earth approximation
713 <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny
714 <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula.
716 I{Suitable only for distances of several hundred Km or Miles
717 and only between points not near-polar}.
719 @arg lat0: From latitude (C{degrees}).
720 @arg lon0: From longitude (C{degrees}).
721 @arg lat1: To latitude (C{degrees}).
722 @arg lon1: To longitude (C{degrees}).
724 @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance}
725 in same units as this ellipsoid's axes.
727 @note: The meridional and prime_vertical radii of curvature are
728 taken and scaled I{at the initial latitude}, see C{roc2}.
730 @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}.
731 '''
732 phi0 = Phi_(lat0=lat0)
733 m, n = self.roc2_(phi0, scaled=True)
734 m *= Phi_(lat1=lat1) - phi0
735 n *= Lam_(lon1=lon1) - Lam_(lon0=lon0)
736 return Distance2Tuple(hypot(m, n), atan2b(n, m))
738 @Property_RO
739 def e(self):
740 '''Get the I{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}.
742 @see: Property L{es}.
743 '''
744 return Float(e=sqrt(self.e2abs) if self.e2 else _0_0)
746 @deprecated_Property_RO
747 def e12(self): # see property ._e12
748 '''DEPRECATED, use property C{e21}.'''
749 return self.e21
751# @Property_RO
752# def _e12(self): # see property ._elliptic_e12
753# # (INTERNAL) until e12 above can be replaced with e21.
754# return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2)
756 @Property_RO
757 def e2(self):
758 '''Get the I{signed, (1st) eccentricity squared} (C{float}), M{f * (2 - f)
759 == 1 - (b / a)**2}, see C{a_b2e2}.
760 '''
761 return self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f))
763 @Property_RO
764 def e2abs(self):
765 '''Get the I{unsigned, (1st) eccentricity squared} (C{float}).
766 '''
767 return fabs(self.e2)
769 @Property_RO
770 def e21(self):
771 '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2}
772 == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}.
773 '''
774 return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0)
776# _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2
777 _1_e21 = a2_b2 # == M{1 / e21} == M{1 / (1 - e**2)}
779 @Property_RO
780 def e22(self):
781 '''Get the I{signed, 2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)
782 == e2 / (1 - f)**2 == (a / b)**2 - 1}, see C{a_b2e22}.
783 '''
784 return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f))
786 @Property_RO
787 def e22abs(self):
788 '''Get the I{unsigned, 2nd eccentricity squared} (C{float}).
789 '''
790 return fabs(self.e22)
792 @Property_RO
793 def e32(self):
794 '''Get the I{signed, 3rd eccentricity squared} (C{float}), M{e2 / (2 - e2)
795 == (a**2 - b**2) / (a**2 + b**2)}, see C{a_b2e32}.
796 '''
797 return self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f))
799 @Property_RO
800 def e32abs(self):
801 '''Get the I{unsigned, 3rd eccentricity squared} (C{float}).
802 '''
803 return fabs(self.e32)
805 @Property_RO
806 def e4(self):
807 '''Get the I{unsignd, (1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}.
808 '''
809 return Float(e4=self.e2**2 if self.e2 else _0_0)
811 eccentricity = e # eccentricity
812# eccentricity2 = e2 # eccentricity squared
813 eccentricity1st2 = e2 # first eccentricity squared
814 eccentricity2nd2 = e22 # second eccentricity squared
815 eccentricity3rd2 = e32 # third eccentricity squared
817 def ecef(self, Ecef=None):
818 '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter.
820 @kwarg Ecef: ECEF class to use, default L{EcefKarney}.
822 @return: An ECEF converter for this C{ellipsoid}.
824 @raise TypeError: Invalid B{C{Ecef}}.
826 @see: Module L{pygeodesy.ecef}.
827 '''
828 return _MODS.ecef._4Ecef(self, Ecef)
830 @Property_RO
831 def _elliptic_e12(self): # see I{Karney}'s Ellipsoid._e12
832 '''(INTERNAL) Elliptic helper for C{Rhumb}.
833 '''
834 e12 = self.e2 / (self.e2 - _1_0) # NOT DEPRECATED .e12!
835 return _MODS.elliptic.Elliptic(e12)
837 @Property_RO
838 def _elliptic_e22(self): # aka ._elliptic_ep2
839 '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}.
840 '''
841 return _MODS.elliptic.Elliptic(-self.e22abs) # complex
843 equatoradius = a # Requatorial
845 def e2s(self, s):
846 '''Compute norm M{sqrt(1 - e2 * s**2)}.
848 @arg s: Sine value (C{scalar}).
850 @return: Norm (C{float}).
852 @raise ValueError: Invalid B{C{s}}.
853 '''
854 return sqrt(self.e2s2(s)) if self.e2 else _1_0
856 def e2s2(self, s):
857 '''Compute M{1 - e2 * s**2}.
859 @arg s: Sine value (C{scalar}).
861 @return: Result (C{float}).
863 @raise ValueError: Invalid B{C{s}}.
864 '''
865 r = _1_0
866 if self.e2:
867 try:
868 r -= self.e2 * Scalar(s=s)**2
869 if r < 0:
870 raise ValueError(_negative_)
871 except (TypeError, ValueError) as x:
872 t = self._DOT_(Ellipsoid.e2s2.__name__)
873 raise _ValueError(t, s, cause=x)
874 return r
876 @Property_RO
877 def es(self):
878 '''Get the I{signed (1st) eccentricity} (C{float}).
880 @see: Property L{e}.
881 '''
882 # note, self.e is always non-negative
883 return Float(es=copysign0(self.e, self.f)) # see .ups
885 def es_atanh(self, x):
886 '''Compute M{es * atanh(es * x)} or M{-es * atan(es * x)}
887 for I{oblate} respectively I{prolate} ellipsoids where
888 I{es} is the I{signed} (1st) eccentricity.
890 @raise ValueError: Invalid B{C{x}}.
892 @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/
893 C++/doc/classGeographicLib_1_1Math.html>}.
894 '''
895 return self._es_atanh(Scalar(x=x)) if self.f else _0_0
897 def _es_atanh(self, x): # see .albers._atanhee, .AuxLat._atanhee
898 '''(INTERNAL) Helper for .es_atanh, ._es_taupf2 and ._exp_es_atanh.
899 '''
900 es = self.es # signOf(es) == signOf(f)
901 return es * (atanh(es * x) if es > 0 else # .isOblate
902 (-atan(es * x) if es < 0 else # .isProlate
903 _0_0)) # .isSpherical
905 @Property_RO
906 def es_c(self):
907 '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}.
908 '''
909 return Float(es_c=(self._exp_es_atanh_1 * self.b_a) if self.f else _1_0)
911 def es_tauf(self, taup):
912 '''Compute I{Karney}'s U{equations (19), (20) and (21)
913 <https://ArXiv.org/abs/1002.1417>}.
915 @see: I{Karney}'s C++ method U{Math::tauf<https://GeographicLib.
916 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>} and
917 and I{Veness}' JavaScript method U{toLatLon<https://www.
918 Movable-Type.co.UK/scripts/latlong-utm-mgrs.html>}.
919 '''
920 t = Scalar(taup=taup)
921 if self.f: # .isEllipsoidal
922 a = fabs(t)
923 T = t * (self._exp_es_atanh_1 if a > 70 else self._1_e21)
924 if fabs(T * _EPSqrt) < _2_0: # handles +/- INF and NAN
925 s = (a * _TOL) if a > _1_0 else _TOL
926 for T, _, d in self._es_tauf3(t, T): # max 2
927 if fabs(d) < s:
928 break
929 t = Scalar(tauf=T)
930 return t
932 def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB
933 '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most
934 B{C{N}} Newton iterations, converging rapidly except when C{delta}
935 toggles on +/-1.12e-16 or +/-4.47e-16, see C{.utm.Utm._toLLEB}.
936 '''
937 e = self._1_e21
938 _F2_ = Fsum(T).fsum2f_ # τ0
939 _tf2 = self._es_taupf2
940 for i in range(1, N + 1):
941 a, h = _tf2(T)
942 d = (taup - a) * (e + T**2) / (hypot1(a) * h)
943 # = (taup - a) / hypot1(a) / ((e + T**2) / h)
944 T, d = _F2_(d) # τi, (τi - τi-1)
945 yield T, i, d
947 def es_taupf(self, tau):
948 '''Compute I{Karney}'s U{equations (7), (8) and (9)
949 <https://ArXiv.org/abs/1002.1417>}.
951 @see: I{Karney}'s C++ method U{Math::taupf<https://GeographicLib.
952 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}.
953 '''
954 t = Scalar(tau=tau)
955 if self.f: # .isEllipsoidal
956 t, _ = self._es_taupf2(t)
957 t = Scalar(taupf=t)
958 return t
960 def _es_taupf2(self, tau):
961 '''(INTERNAL) Return 2-tuple C{(es_taupf(tau), hypot1(tau))}.
962 '''
963 if _isfinite(tau):
964 h = hypot1(tau)
965 s = sinh(self._es_atanh(tau / h))
966 a = hypot1(s) * tau - h * s
967 else:
968 a, h = tau, INF
969 return a, h
971 @Property_RO
972 def _exp_es_atanh_1(self):
973 '''(INTERNAL) Helper for .es_c and .es_tauf.
974 '''
975 return exp(self._es_atanh(_1_0)) if self.es else _1_0
977 @Property_RO
978 def f(self):
979 '''Get the I{flattening} (C{scalar}), M{(a - b) / a}, C{0} for spherical, negative for prolate.
980 '''
981 return self._f
983 @Property_RO
984 def f_(self):
985 '''Get the I{inverse flattening} (C{scalar}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see C{a_b2f_}.
986 '''
987 return self._f_
989 @Property_RO
990 def f1(self):
991 '''Get the I{1 - flattening} (C{float}), M{f1 == 1 - f == b / a}.
993 @see: Property L{b_a}.
994 '''
995 return Float(f1=_1_0 - self.f)
997 @Property_RO
998 def f2(self):
999 '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see C{a_b2f2}.
1000 '''
1001 return self._assert(self.a_b - _1_0, f2=f2f2(self.f))
1003 @deprecated_Property_RO
1004 def geodesic(self):
1005 '''DEPRECATED, use property C{geodesicw}.'''
1006 return self.geodesicw
1008 def geodesic_(self, exact=True):
1009 '''Get the an I{exact} C{Geodesic...} instance for this ellipsoid.
1011 @kwarg exact: If C{bool} return L{GeodesicExact}C{(exact=B{exact}, ...)},
1012 otherwise a L{Geodesic}, L{GeodesicExact} or L{GeodesicSolve}
1013 instance for I{this} ellipsoid.
1015 @return: The C{exact} geodesic (C{Geodesic...}).
1017 @raise TypeError: Invalid B{C{exact}}.
1019 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1020 '''
1021 if isbool(exact): # for consistenccy with C{.rhumb_}
1022 g = _MODS.geodesicx.GeodesicExact(self, C4order=30 if exact else 24,
1023 name=self.name)
1024 else:
1025 g = exact
1026 E = _xattr(g, ellipsoid=None)
1027 if not (E is self and isinstance(g, self._Geodesics)):
1028 raise _ValueError(exact=g, ellipsoid=E, txt_not_=self.name)
1029 return g
1031 @property_RO
1032 def _Geodesics(self):
1033 '''(INTERNAL) Get all C{Geodesic...} classes, I{once}.
1034 '''
1035 Ellipsoid._Geodesics = t = (_MODS.geodesicw._wrapped.Geodesic, # overwrite property_RO
1036 _MODS.geodesicx.GeodesicExact,
1037 _MODS.geodsolve.GeodesicSolve)
1038 return t
1040 @property_RO
1041 def geodesicw(self):
1042 '''Get this ellipsoid's I{wrapped} U{geodesicw.Geodesic
1043 <https://GeographicLib.SourceForge.io/Python/doc/code.html>}, provided
1044 I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>}
1045 package is installed.
1046 '''
1047 # if not self.isEllipsoidal:
1048 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1049 return _MODS.geodesicw.Geodesic(self)
1051 @property_RO
1052 def geodesicx(self):
1053 '''Get this ellipsoid's I{exact} L{GeodesicExact}.
1054 '''
1055 # if not self.isEllipsoidal:
1056 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1057 return _MODS.geodesicx.GeodesicExact(self, name=self.name)
1059 @property
1060 def geodsolve(self):
1061 '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility
1062 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1063 provided the path to the C{GeodSolve} executable is specified with env
1064 variable C{PYGEODESY_GEODSOLVE} or re-/set with this property..
1065 '''
1066 # if not self.isEllipsoidal:
1067 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1068 return _MODS.geodsolve.GeodesicSolve(self, path=self._geodsolve, name=self.name)
1070 @geodsolve.setter # PYCHOK setter!
1071 def geodsolve(self, path):
1072 '''Re-/set the (fully qualified) path to the U{GeodSolve
1073 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1074 overriding env variable C{PYGEODESY_GEODSOLVE} (C{str}).
1075 '''
1076 self._geodsolve = path
1078 def hartzell4(self, pov, los=None):
1079 '''Compute the intersection of this ellipsoid's surface and a Line-Of-Sight
1080 from a Point-Of-View in space.
1082 @arg pov: Point-Of-View outside this ellipsoid (C{Cartesian}, L{Ecef9Tuple}
1083 or L{Vector3d}).
1084 @kwarg los: Line-Of-Sight, I{direction} to this ellipsoid (L{Vector3d}) or
1085 C{None} to point to this ellipsoid's center.
1087 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1088 C{y} and C{z} of the projection on or the intersection with this
1089 ellipsoid and the I{distance} C{h} from B{C{pov}} to C{(x, y, z)}
1090 along B{C{los}}, all in C{meter}, conventionally.
1092 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, or B{C{pov}}
1093 is inside this ellipsoid or B{C{los}} points
1094 outside this ellipsoid or points in an opposite
1095 direction.
1097 @raise TypeError: Invalid B{C{pov}} or B{C{los}}.
1099 @see: U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.
1100 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and
1101 methods L{Ellipsoid.height4} and L{Triaxial.hartzell4}.
1102 '''
1103 try:
1104 v, d = _MODS.triaxials._hartzell2(pov, los, self._triaxial)
1105 except Exception as x:
1106 raise IntersectionError(pov=pov, los=los, cause=x)
1107 return Vector4Tuple(v.x, v.y, v.z, d, name__=self.hartzell4)
1109 @Property_RO
1110 def _hash(self):
1111 return hash((self.a, self.f))
1113 def height4(self, xyz, normal=True):
1114 '''Compute the projection on and the height of a cartesian above or below
1115 this ellipsoid's surface.
1117 @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d},
1118 L{Vector3Tuple} or L{Vector4Tuple}).
1119 @kwarg normal: If C{True}, the projection is perpendicular to (the nearest
1120 point on) this ellipsoid's surface, otherwise the C{radial}
1121 line to this ellipsoid's center (C{bool}).
1123 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x},
1124 C{y} and C{z} of the projection on and the height C{h} above or
1125 below this ellipsoid's surface, all in C{meter}, conventionally.
1127 @raise ValueError: Null B{C{xyz}}.
1129 @raise TypeError: Non-cartesian B{C{xyz}}.
1131 @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>}
1132 and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse and
1133 methods L{Ellipsoid.hartzell4} and L{Triaxial.height4}.
1134 '''
1135 v = _MODS.vector3d._otherV3d(xyz=xyz)
1136 r = v.length
1138 a, b, i = self.a, self.b, None
1139 if r < EPS0: # EPS
1140 v = v.times(_0_0)
1141 h = -a
1143 elif self.isSpherical:
1144 v = v.times(a / r)
1145 h = r - a
1147 elif normal: # perpendicular to ellipsoid
1148 x, y = hypot(v.x, v.y), fabs(v.z)
1149 if x < EPS0: # PYCHOK no cover
1150 z = copysign0(b, v.z)
1151 v = Vector3Tuple(v.x, v.y, z)
1152 h = y - b # polar
1153 elif y < EPS0: # PYCHOK no cover
1154 t = a / r
1155 v = v.times_(t, t, 0) # force z=0.0
1156 h = x - a # equatorial
1157 else: # normal in 1st quadrant
1158 x, y, i = _normalTo3(x, y, self)
1159 t, v = v, v.times_(x, x, y)
1160 h = t.minus(v).length
1162 else: # radial to ellipsoid's center
1163 h = hypot_(a * v.z, b * v.x, b * v.y)
1164 t = (a * b / h) if h > EPS0 else _0_0 # EPS
1165 v = v.times(t)
1166 h = r * (_1_0 - t)
1168 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, name__=self.height4)
1170 def _hubeny_2(self, phi2, phi1, lam21, scaled=True, squared=True):
1171 '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_},
1172 returning the I{angular} distance in C{radians squared} or C{radians}
1173 '''
1174 m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=scaled)
1175 h, r = (hypot2, self.a2_) if squared else (hypot, _1_0 / self.a)
1176 return h(m * (phi2 - phi1), n * lam21) * r
1178 @Property_RO
1179 def isEllipsoidal(self):
1180 '''Is this model I{ellipsoidal} (C{bool})?
1181 '''
1182 return self.f != 0
1184 @Property_RO
1185 def isOblate(self):
1186 '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or
1187 spherical otherwise.
1188 '''
1189 return self.f > 0
1191 @Property_RO
1192 def isProlate(self):
1193 '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or
1194 spherical otherwise.
1195 '''
1196 return self.f < 0
1198 @Property_RO
1199 def isSpherical(self):
1200 '''Is this ellipsoid I{spherical} (C{bool})?
1201 '''
1202 return self.f == 0
1204 def _Kseries(self, *AB8Ks):
1205 '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha
1206 or Beta series coefficients per I{Karney}'s U{equations (35)
1207 and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}.
1209 @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series
1210 coefficient tuples.
1212 @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}).
1214 @see: I{Karney}'s 30-th order U{TMseries30
1215 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}.
1216 '''
1217 k = self.KsOrder
1218 if self.n:
1219 ns = fpowers(self.n, k)
1220 ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k))
1221 else:
1222 ks = _0_0s(k)
1223 return ks
1225 @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''')
1226 def KsOrder(self):
1227 '''Get the I{Krüger} series' order (C{int} 4, 6 or 8).
1228 '''
1229 return self._KsOrder
1231 @KsOrder.setter # PYCHOK setter!
1232 def KsOrder(self, order):
1233 '''Set the I{Krüger} series' order (C{int} 4, 6 or 8).
1235 @raise ValueError: Invalid B{C{order}}.
1236 '''
1237 if not (isint(order) and order in (4, 6, 8)):
1238 raise _ValueError(order=order)
1239 if self._KsOrder != order:
1240 Ellipsoid.AlphaKs._update(self)
1241 Ellipsoid.BetaKs._update(self)
1242 self._KsOrder = order
1244 @Property_RO
1245 def L(self):
1246 '''Get the I{quarter meridian} C{L}, aka the C{polar distance}
1247 along a meridian between the equator and a pole (C{meter}),
1248 M{b * Elliptic(-e2 / (1 - e2)).cE} or M{b * PI / 2}.
1249 '''
1250 r = self._elliptic_e22.cE if self.f else PI_2
1251 return Distance(L=self.b * r)
1253 def Llat(self, lat):
1254 '''Return the I{meridional length}, the distance along a meridian
1255 between the equator and a (geodetic) latitude, see C{L}.
1257 @arg lat: Geodetic latitude (C{degrees90}).
1259 @return: The meridional length at B{C{lat}}, negative on southern
1260 hemisphere (C{meter}).
1261 '''
1262 r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phi_(lat)
1263 return Distance(Llat=self.b * r)
1265 Lmeridian = Llat # meridional distance
1267 @property_RO
1268 def _Lpd(self):
1269 '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}.
1270 '''
1271 return Meter(_Lpd=self.L / _90_0)
1273 @property_RO
1274 def _Lpr(self):
1275 '''Get the I{quarter meridian} per radian (C{meter}), M{self.L / PI_2}.
1276 '''
1277 return Meter(_Lpr=self.L / PI_2)
1279 @deprecated_Property_RO
1280 def majoradius(self): # PYCHOK no cover
1281 '''DEPRECATED, use property C{a} or C{Requatorial}.'''
1282 return self.a
1284 def m2degrees(self, distance, lat=0):
1285 '''Convert a distance to an angle along the equator or
1286 along a parallel of (geodetic) latitude.
1288 @arg distance: Distance (C{meter}).
1289 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1291 @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}.
1293 @raise RangeError: Latitude B{C{lat}} outside valid range and
1294 L{pygeodesy.rangerrors} set to C{True}.
1296 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1297 '''
1298 return degrees(self.m2radians(distance, lat=lat))
1300 def m2radians(self, distance, lat=0):
1301 '''Convert a distance to an angle along the equator or
1302 along a parallel of (geodetic) latitude.
1304 @arg distance: Distance (C{meter}).
1305 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1307 @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}.
1309 @raise RangeError: Latitude B{C{lat}} outside valid range and
1310 L{pygeodesy.rangerrors} set to C{True}.
1312 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}.
1313 '''
1314 r = self.circle4(lat).radius if lat else self.a
1315 return m2radians(distance, radius=r, lat=0)
1317 @deprecated_Property_RO
1318 def minoradius(self): # PYCHOK no cover
1319 '''DEPRECATED, use property C{b}, C{polaradius} or C{Rpolar}.'''
1320 return self.b
1322 @Property_RO
1323 def n(self):
1324 '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}.
1325 '''
1326 return self._assert(a_b2n(self.a, self.b), n=f2n(self.f))
1328 flattening = f
1329 flattening1st = f
1330 flattening2nd = f2
1331 flattening3rd = n
1333 polaradius = b # Rpolar
1335# @Property_RO
1336# def Q(self):
1337# '''Get the I{meridian arc unit} C{Q}, the mean, meridional length I{per radian} C({float}).
1338#
1339# @note: C{Q * PI / 2} ≈ C{L}, the I{quarter meridian}.
1340#
1341# @see: Property C{A} and U{Engsager, K., Poder, K.<https://StudyLib.net/doc/7443565/
1342# a-highly-accurate-world-wide-algorithm-for-the-transverse...>}.
1343# '''
1344# n = self.n
1345# d = (n + _1_0) / self.a
1346# return Float(Q=Fhorner(n**2, _1_0, _0_25, _1_16th, _0_25).fover(d) if d else self.b)
1348# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1349# # Q = (1 - 3/4 * e'2 + 45/64 * e'4 - 175/256 * e'6 + 11025/16384 * e'8) * rocPolar
1350# # = (4 + e'2 * (-3 + e'2 * (45/16 + e'2 * (-175/64 + e'2 * 11025/4096)))) * rocPolar / 4
1351# return Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(4 / self.rocPolar)
1353 @deprecated_Property_RO
1354 def quarteradius(self): # PYCHOK no cover
1355 '''DEPRECATED, use property C{L} or method C{Llat}.'''
1356 return self.L
1358 @Property_RO
1359 def R1(self):
1360 '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}.
1362 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}
1363 and method C{Rgeometric}.
1364 '''
1365 r = Fsum(self.a, self.a, self.b).fover(_3_0) if self.f else self.a
1366 return Radius(R1=r)
1368 Rmean = R1
1370 @Property_RO
1371 def R2(self):
1372 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}.
1374 @see: C{R2x}, C{c2}, C{area} and U{Earth radius
1375 <https://WikiPedia.org/wiki/Earth_radius>}.
1376 '''
1377 return Radius(R2=sqrt(self.c2) if self.f else self.a)
1379 Rauthalic = R2
1381# @Property_RO
1382# def R2(self):
1383# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf>
1384# # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar
1385# # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3
1386# return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar)
1388 @Property_RO
1389 def R2x(self):
1390 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}.
1392 @see: C{R2}, C{c2x} and C{areax}.
1393 '''
1394 return Radius(R2x=sqrt(self.c2x) if self.f else self.a)
1396 Rauthalicx = R2x
1398 @Property_RO
1399 def R3(self):
1400 '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}.
1402 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}.
1403 '''
1404 r = (cbrt(self.b_a) * self.a) if self.f else self.a
1405 return Radius(R3=r)
1407 Rvolumetric = R3
1409 def radians2m(self, rad, lat=0):
1410 '''Convert an angle to the distance along the equator or
1411 along a parallel of (geodetic) latitude.
1413 @arg rad: The angle (C{radians}).
1414 @kwarg lat: Parallel latitude (C{degrees90}, C{str}).
1416 @return: Distance (C{meter}, same units as the equatorial
1417 and polar radii) or C{0} for near-polar B{C{lat}}.
1419 @raise RangeError: Latitude B{C{lat}} outside valid range and
1420 L{pygeodesy.rangerrors} set to C{True}.
1422 @raise ValueError: Invalid B{C{rad}} or B{C{lat}}.
1423 '''
1424 r = self.circle4(lat).radius if lat else self.a
1425 return radians2m(rad, radius=r, lat=0)
1427 @Property_RO
1428 def Rbiaxial(self):
1429 '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}.
1431 @see: C{Rtriaxial}
1432 '''
1433 a, b = self.a, self.b
1434 if b < a:
1435 b = sqrt(_0_5 + self.b2_a2 * _0_5) * a
1436 elif b > a:
1437 b *= sqrt(_0_5 + self.a2_b2 * _0_5)
1438 return Radius(Rbiaxial=b)
1440 Requatorial = a # for consistent naming
1442 def Rgeocentric(self, lat):
1443 '''Compute the I{geocentric} earth radius of (geodetic) latitude.
1445 @arg lat: Latitude (C{degrees90}).
1447 @return: Geocentric earth radius (C{meter}).
1449 @raise ValueError: Invalid B{C{lat}}.
1451 @see: U{Geocentric Radius
1452 <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>}
1453 '''
1454 r, a = self.a, Phi_(lat)
1455 if a and self.f:
1456 if fabs(a) < PI_2:
1457 s2, c2 = _s2_c2(a)
1458 b2_a2_s2 = self.b2_a2 * s2
1459 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2))
1460 # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2)))
1461 # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2))
1462 # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2))
1463 # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2))
1464 r *= sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2))
1465 else:
1466 r = self.b
1467 return Radius(Rgeocentric=r)
1469 @Property_RO
1470 def Rgeometric(self):
1471 '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}.
1473 @see: C{R1}.
1474 '''
1475 g = sqrt(self.a * self.b) if self.f else self.a
1476 return Radius(Rgeometric=g)
1478 def rhumb_(self, exact=True):
1479 '''Get the an I{exact} C{Rhumb...} instance for this ellipsoid.
1481 @kwarg exact: If C{bool} or C{None} return L{Rhumb}C{(exact=B{exact}, ...)},
1482 otherwise a L{Rhumb}, L{RhumbAux} or L{RhumbSolve} instance
1483 for I{this} ellipsoid.
1485 @return: The C{exact} rhumb (C{Rhumb...}).
1487 @raise TypeError: Invalid B{C{exact}}.
1489 @raise ValueError: Incompatible B{C{exact}} ellipsoid.
1490 '''
1491 if isbool(exact): # use Rhumb for backward compatibility
1492 r = _MODS.rhumb.ekx.Rhumb(self, exact=exact, name=self.name)
1493 else:
1494 r = exact
1495 E = _xattr(r, ellipsoid=None)
1496 if not (E is self and isinstance(r, self._Rhumbs)):
1497 raise _ValueError(exact=r, ellipsosid=E, txt_not_=self.name)
1498 return r
1500 @property_RO
1501 def rhumbaux(self):
1502 '''Get this ellipsoid's I{Auxiliary} C{rhumb.RhumbAux}.
1503 '''
1504 # if not self.isEllipsoidal:
1505 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1506 return _MODS.rhumb.aux_.RhumbAux(self, name=self.name)
1508 @property_RO
1509 def rhumbekx(self):
1510 '''Get this ellipsoid's I{Elliptic, Krüger} C{rhumb.Rhumb}.
1511 '''
1512 # if not self.isEllipsoidal:
1513 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1514 return _MODS.rhumb.ekx.Rhumb(self, name=self.name)
1516 @property_RO
1517 def _Rhumbs(self):
1518 '''(INTERNAL) Get all C{Rhumb...} classes, I{once}.
1519 '''
1520 p = _MODS.rhumb
1521 Ellipsoid._Rhumbs = t = (p.aux_.RhumbAux, # overwrite property_RO
1522 p.ekx.Rhumb, p.solve.RhumbSolve)
1523 return t
1525 @property
1526 def rhumbsolve(self):
1527 '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility
1528 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>},
1529 provided the path to the C{RhumbSolve} executable is specified with env
1530 variable C{PYGEODESY_RHUMBSOLVE} or re-/set with this property.
1531 '''
1532 # if not self.isEllipsoidal:
1533 # raise _IsnotError(_ellipsoidal_, ellipsoid=self)
1534 return _MODS.rhumb.solve.RhumbSolve(self, path=self._rhumbsolve, name=self.name)
1536 @rhumbsolve.setter # PYCHOK setter!
1537 def rhumbsolve(self, path):
1538 '''Re-/set the (fully qualified) path to the U{RhumbSolve
1539 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable,
1540 overriding env variable C{PYGEODESY_RHUMBSOLVE} (C{str}).
1541 '''
1542 self._rhumbsolve = path
1544 @deprecated_property_RO
1545 def rhumbx(self):
1546 '''DEPRECATED on 2023.11.28, use property C{rhumbekx}. '''
1547 return self.rhumbekx
1549 def Rlat(self, lat):
1550 '''I{Approximate} the earth radius of (geodetic) latitude.
1552 @arg lat: Latitude (C{degrees90}).
1554 @return: Approximate earth radius (C{meter}).
1556 @raise RangeError: Latitude B{C{lat}} outside valid range and
1557 L{pygeodesy.rangerrors} set to C{True}.
1559 @raise TypeError: Invalid B{C{lat}}.
1561 @raise ValueError: Invalid B{C{lat}}.
1563 @note: C{Rlat(B{90})} equals C{Rpolar}.
1565 @see: Method C{circle4}.
1566 '''
1567 # r = a - (a - b) * |lat| / 90
1568 r = self.a
1569 if self.f and lat: # .isEllipsoidal
1570 r -= (r - self.b) * fabs(Lat(lat)) / _90_0
1571 r = Radius(Rlat=r)
1572 return r
1574 Rpolar = b # for consistent naming
1576 def roc1_(self, sa, ca=None):
1577 '''Compute the I{prime-vertical}, I{normal} radius of curvature
1578 of (geodetic) latitude, I{unscaled}.
1580 @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]).
1581 @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0])
1582 to use an alternate formula.
1584 @return: The prime-vertical radius of curvature (C{float}).
1586 @note: The delta between both formulae with C{Ellipsoids.WGS84}
1587 is less than 2 nanometer over the entire latitude range.
1589 @see: Method L{roc2_} and class L{EcefYou}.
1590 '''
1591 if not self.f: # .isSpherical
1592 n = self.a
1593 elif ca is None:
1594 r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward
1595 n = sqrt(self.a2 / r) if r > EPS02 else _0_0
1596 elif ca: # derived from EcefYou.forward
1597 h = hypot(ca, self.b_a * sa) if sa else fabs(ca)
1598 n = self.a / h
1599 elif sa:
1600 n = self.a2_b / fabs(sa)
1601 else:
1602 n = self.a
1603 return n
1605 def roc2(self, lat, scaled=False):
1606 '''Compute the I{meridional} and I{prime-vertical}, I{normal}
1607 radii of curvature of (geodetic) latitude.
1609 @arg lat: Latitude (C{degrees90}).
1610 @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}).
1612 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with
1613 the radii of curvature.
1615 @raise ValueError: Invalid B{C{lat}}.
1617 @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation
1618 <https://www.EdWilliams.org/avform.htm#flat>} and meridional and
1619 prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/
1620 Earth_radius#Radii_of_curvature>}.
1621 '''
1622 return self.roc2_(Phi_(lat), scaled=scaled)
1624 def roc2_(self, phi, scaled=False):
1625 '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of
1626 curvature of (geodetic) latitude.
1628 @arg phi: Latitude (C{radians}).
1629 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}).
1631 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the
1632 radii of curvature.
1634 @raise ValueError: Invalid B{C{phi}}.
1636 @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local,
1637 flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>}
1638 and the meridional and prime vertical U{Radii of Curvature
1639 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1640 '''
1641 a = fabs(Phi(phi))
1642 if self.f:
1643 r = self.e2s2(sin(a))
1644 if r > EPS02:
1645 n = self.a / sqrt(r)
1646 m = n * self.e21 / r # PYCHOK attr
1647 else:
1648 m = n = _0_0 # PYCHOK attr
1649 else:
1650 m = n = self.a
1651 if scaled and a:
1652 n *= cos(a) if a < PI_2 else _0_0
1653 return Curvature2Tuple(Radius(rocMeridional=m),
1654 Radius(rocPrimeVertical=n))
1656 def rocBearing(self, lat, bearing):
1657 '''Compute the I{directional} radius of curvature of (geodetic)
1658 latitude and compass direction.
1660 @arg lat: Latitude (C{degrees90}).
1661 @arg bearing: Direction (compass C{degrees360}).
1663 @return: Directional radius of curvature (C{meter}).
1665 @raise RangeError: Latitude B{C{lat}} outside valid range and
1666 L{pygeodesy.rangerrors} set to C{True}.
1668 @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}.
1670 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}
1671 '''
1672 if self.f:
1673 s2, c2 = _s2_c2(Bearing_(bearing))
1674 m, n = self.roc2_(Phi_(lat))
1675 if n < m: # == n / (c2 * n / m + s2)
1676 c2 *= n / m
1677 elif m < n: # == m / (c2 + s2 * m / n)
1678 s2 *= m / n
1679 n = m
1680 b = n / (c2 + s2) # == 1 / (c2 / m + s2 / n)
1681 else:
1682 b = self.b # == self.a
1683 return Radius(rocBearing=b)
1685 @Property_RO
1686 def rocEquatorial2(self):
1687 '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature
1688 at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}.
1690 @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a},
1691 L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature
1692 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1693 '''
1694 return Curvature2Tuple(Radius(rocMeridional0=self.b2_a if self.f else self.a),
1695 Radius(rocPrimeVertical0=self.a))
1697 def rocGauss(self, lat):
1698 '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude.
1700 @arg lat: Latitude (C{degrees90}).
1702 @return: Gaussian radius of curvature (C{meter}).
1704 @raise ValueError: Invalid B{C{lat}}.
1706 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1707 Earth_radius#Radii_of_curvature>}
1708 '''
1709 # using ...
1710 # m, n = self.roc2_(Phi_(lat))
1711 # return sqrt(m * n)
1712 # ... requires 1 or 2 sqrt
1713 g = self.b
1714 if self.f:
1715 s2, c2 = _s2_c2(Phi_(lat))
1716 g = g / (c2 + self.b2_a2 * s2)
1717 return Radius(rocGauss=g)
1719 def rocMean(self, lat):
1720 '''Compute the I{mean} radius of curvature of (geodetic) latitude.
1722 @arg lat: Latitude (C{degrees90}).
1724 @return: Mean radius of curvature (C{meter}).
1726 @raise ValueError: Invalid B{C{lat}}.
1728 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/
1729 Earth_radius#Radii_of_curvature>}
1730 '''
1731 if self.f:
1732 m, n = self.roc2_(Phi_(lat))
1733 m *= n * _2_0 / (m + n) # == 2 / (1 / m + 1 / n)
1734 else:
1735 m = self.a
1736 return Radius(rocMean=m)
1738 def rocMeridional(self, lat):
1739 '''Compute the I{meridional} radius of curvature of (geodetic) latitude.
1741 @arg lat: Latitude (C{degrees90}).
1743 @return: Meridional radius of curvature (C{meter}).
1745 @raise ValueError: Invalid B{C{lat}}.
1747 @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation
1748 <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of
1749 Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}.
1750 '''
1751 return self.roc2_(Phi_(lat)).meridional if lat else \
1752 self.rocEquatorial2.meridional
1754 rocPolar = a2_b # synonymous
1756 def rocPrimeVertical(self, lat):
1757 '''Compute the I{prime-vertical}, I{normal} radius of curvature of
1758 (geodetic) latitude, aka the I{transverse} radius of curvature.
1760 @arg lat: Latitude (C{degrees90}).
1762 @return: Prime-vertical radius of curvature (C{meter}).
1764 @raise ValueError: Invalid B{C{lat}}.
1766 @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth
1767 approximation<https://www.EdWilliams.org/avform.htm#flat>} and
1768 U{Radii of Curvature<https://WikiPedia.org/wiki/
1769 Earth_radius#Radii_of_curvature>}.
1770 '''
1771 return self.roc2_(Phi_(lat)).prime_vertical if lat else \
1772 self.rocEquatorial2.prime_vertical
1774 rocTransverse = rocPrimeVertical # synonymous
1776 @deprecated_Property_RO
1777 def Rquadratic(self): # PYCHOK no cover
1778 '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.'''
1779 return self.Rbiaxial
1781 @deprecated_Property_RO
1782 def Rr(self): # PYCHOK no cover
1783 '''DEPRECATED, use property C{Rrectifying}.'''
1784 return self.Rrectifying
1786 @Property_RO
1787 def Rrectifying(self):
1788 '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}.
1790 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}.
1791 '''
1792 r = (cbrt2((_1_0 + sqrt3(self.b_a)) * _0_5) * self.a) if self.f else self.a
1793 return Radius(Rrectifying=r)
1795 @deprecated_Property_RO
1796 def Rs(self): # PYCHOK no cover
1797 '''DEPRECATED, use property C{Rgeometric}.'''
1798 return self.Rgeometric
1800 @Property_RO
1801 def Rtriaxial(self):
1802 '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}.
1804 @see: C{Rbiaxial}
1805 '''
1806 a, b = self.a, self.b
1807 q = (sqrt((_3_0 + self.b2_a2) * _0_25) * a) if a > b else (
1808 (sqrt((_3_0 * self.a2_b2 + _1_0) * _0_25) * b) if a < b else a)
1809 return Radius(Rtriaxial=q)
1811 def toEllipsoid2(self, **name):
1812 '''Get a copy of this ellipsoid as an L{Ellipsoid2}.
1814 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
1816 @see: Property C{a_f}.
1817 '''
1818 return Ellipsoid2(self, None, **name)
1820 def toStr(self, prec=8, terse=0, **name): # PYCHOK expected
1821 '''Return this ellipsoid as a text string.
1823 @kwarg prec: Number of decimal digits, unstripped (C{int}).
1824 @kwarg terse: Limit the number of items (C{int}, 0...18).
1825 @kwarg name: Optional C{B{name}=NN} (C{str}) or C{None} to
1826 exclude this ellipsoid's name.
1828 @return: This C{Ellipsoid}'s attributes (C{str}).
1829 '''
1830 E = Ellipsoid
1831 t = E.a, E.b, E.f_, E.f, E.f2, E.n, E.e, E.e2, E.e21, E.e22, E.e32, \
1832 E.A, E.L, E.R1, E.R2, E.R3, E.Rbiaxial, E.Rtriaxial
1833 if terse:
1834 t = t[:terse]
1835 n, _ = _name2__(**name) # name=None
1836 return self._instr(n, prec, props=t)
1838 def toTriaxial(self, **name):
1839 '''Convert this ellipsoid to a L{Triaxial_}.
1841 @kwarg name: Optional C{B{name}=NN} (C{str}).
1843 @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis
1844 pointing east and C{Z} pointing north.
1846 @see: Method L{Triaxial_.toEllipsoid}.
1847 '''
1848 T = self._triaxial
1849 return T.copy(**name) if name else T
1851 @property_RO
1852 def _triaxial(self):
1853 '''(INTERNAL) Get this ellipsoid's un-/ordered C{Triaxial/_}.
1854 '''
1855 a, b, m = self.a, self.b, _MODS.triaxials
1856 T = m.Triaxial if a > b else m.Triaxial_
1857 return T(a, a, b, name=self.name)
1859 @Property_RO
1860 def volume(self):
1861 '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}.
1863 @see: C{R3}.
1864 '''
1865 return Meter3(volume=self.a2 * self.b * PI_3 * _4_0)
1868class Ellipsoid2(Ellipsoid):
1869 '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}.
1870 '''
1871 def __init__(self, a, f=None, **name):
1872 '''New L{Ellipsoid2}.
1874 @arg a: Equatorial radius, semi-axis (C{meter}) or a previous
1875 L{Ellipsoid} instance.
1876 @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}),
1877 if B{C{a}} is in C{meter}.
1878 @kwarg name: Optional, unique C{B{name}=NN} (C{str}).
1880 @raise NameError: Ellipsoid with that B{C{name}} already exists.
1882 @raise ValueError: Invalid B{C{a}} or B{C{f}}.
1884 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}.
1885 Negative C{B{f}} produces a I{prolate} ellipsoid.
1886 '''
1887 if f is None and isinstance(a, Ellipsoid):
1888 Ellipsoid.__init__(self, a.a, f =a.f,
1889 b=a.b, f_=a.f_, **name)
1890 else:
1891 Ellipsoid.__init__(self, a, f=f, **name)
1894def _spherical_a_b(a, b):
1895 '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}.
1896 '''
1897 return a < EPS0 or b < EPS0 or fabs(a - b) < EPS0
1900def _spherical_f(f):
1901 '''(INTERNAL) C{True} for spherical or invalid C{f}.
1902 '''
1903 return fabs(f) < EPS or f > EPS1
1906def _spherical_f_(f_):
1907 '''(INTERNAL) C{True} for spherical or invalid C{f_}.
1908 '''
1909 return fabs(f_) < EPS or fabs(f_) > _1_EPS
1912def a_b2e(a, b):
1913 '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius.
1915 @arg a: Equatorial radius (C{scalar} > 0).
1916 @arg b: Polar radius (C{scalar} > 0).
1918 @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}),
1919 M{sqrt(1 - (b / a)**2)}.
1921 @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids.
1922 '''
1923 return Float(e=sqrt(fabs(a_b2e2(a, b)))) # == sqrt(fabs(a - b) * (a + b)) / a)
1926def a_b2e2(a, b):
1927 '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius.
1929 @arg a: Equatorial radius (C{scalar} > 0).
1930 @arg b: Polar radius (C{scalar} > 0).
1932 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}),
1933 M{1 - (b / a)**2}.
1935 @note: The result is positive for I{oblate}, negative for I{prolate}
1936 or C{0} for I{near-spherical} ellipsoids.
1937 '''
1938 return Float(e2=_0_0 if _spherical_a_b(a, b) else ((a - b) * (a + b) / a**2))
1941def a_b2e22(a, b):
1942 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1944 @arg a: Equatorial radius (C{scalar} > 0).
1945 @arg b: Polar radius (C{scalar} > 0).
1947 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}),
1948 M{(a / b)**2 - 1}.
1950 @note: The result is positive for I{oblate}, negative for I{prolate}
1951 or C{0} for I{near-spherical} ellipsoids.
1952 '''
1953 return Float(e22=_0_0 if _spherical_a_b(a, b) else ((a - b) * (a + b) / b**2))
1956def a_b2e32(a, b):
1957 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius.
1959 @arg a: Equatorial radius (C{scalar} > 0).
1960 @arg b: Polar radius (C{scalar} > 0).
1962 @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}),
1963 M{(a**2 - b**2) / (a**2 + b**2)}.
1965 @note: The result is positive for I{oblate}, negative for I{prolate}
1966 or C{0} for I{near-spherical} ellipsoids.
1967 '''
1968 a2, b2 = a**2, b**2
1969 return Float(e32=_0_0 if _spherical_a_b(a2, b2) else ((a2 - b2) / (a2 + b2)))
1972def a_b2f(a, b):
1973 '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius.
1975 @arg a: Equatorial radius (C{scalar} > 0).
1976 @arg b: Polar radius (C{scalar} > 0).
1978 @return: The flattening (C{scalar} or C{0}), M{(a - b) / a}.
1980 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1981 for I{near-spherical} ellipsoids.
1982 '''
1983 f = 0 if _spherical_a_b(a, b) else ((a - b) / a)
1984 return _f_0_0 if _spherical_f(f) else Float(f=f)
1987def a_b2f_(a, b):
1988 '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius.
1990 @arg a: Equatorial radius (C{scalar} > 0).
1991 @arg b: Polar radius (C{scalar} > 0).
1993 @return: The inverse flattening (C{scalar} or C{0}), M{a / (a - b)}.
1995 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
1996 for I{near-spherical} ellipsoids.
1997 '''
1998 f_ = 0 if _spherical_a_b(a, b) else (a / float(a - b))
1999 return _f__0_0 if _spherical_f_(f_) else Float(f_=f_)
2002def a_b2f2(a, b):
2003 '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius.
2005 @arg a: Equatorial radius (C{scalar} > 0).
2006 @arg b: Polar radius (C{scalar} > 0).
2008 @return: The I{signed}, 2nd flattening (C{scalar} or C{0}), M{(a - b) / b}.
2010 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0}
2011 for I{near-spherical} ellipsoids.
2012 '''
2013 t = 0 if _spherical_a_b(a, b) else float(a - b)
2014 return Float(f2=_0_0 if fabs(t) < EPS0 else (t / b))
2017def a_b2n(a, b):
2018 '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius.
2020 @arg a: Equatorial radius (C{scalar} > 0).
2021 @arg b: Polar radius (C{scalar} > 0).
2023 @return: The I{signed}, 3rd flattening (C{scalar} or C{0}), M{(a - b) / (a + b)}.
2025 @note: The result is positive for I{oblate}, negative for I{prolate}
2026 or C{0} for I{near-spherical} ellipsoids.
2027 '''
2028 t = 0 if _spherical_a_b(a, b) else float(a - b)
2029 return Float(n=_0_0 if fabs(t) < EPS0 else (t / (a + b)))
2032def a_f2b(a, f):
2033 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}.
2035 @arg a: Equatorial radius (C{scalar} > 0).
2036 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2038 @return: The polar radius (C{float}), M{a * (1 - f)}.
2039 '''
2040 b = a if _spherical_f(f) else (a * (_1_0 - f))
2041 return Radius_(b=a if _spherical_a_b(a, b) else b)
2044def a_f_2b(a, f_):
2045 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}.
2047 @arg a: Equatorial radius (C{scalar} > 0).
2048 @arg f_: Inverse flattening (C{scalar} >>> 1).
2050 @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}.
2051 '''
2052 b = a if _spherical_f_(f_) else (a * (f_ - _1_0) / f_)
2053 return Radius_(b=a if _spherical_a_b(a, b) else b)
2056def b_f2a(b, f):
2057 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}.
2059 @arg b: Polar radius (C{scalar} > 0).
2060 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2062 @return: The equatorial radius (C{float}), M{b / (1 - f)}.
2063 '''
2064 t = _1_0 - f
2065 a = b if fabs(t) < EPS0 else (b / t)
2066 return Radius_(a=b if _spherical_a_b(a, b) else a)
2069def b_f_2a(b, f_):
2070 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}.
2072 @arg b: Polar radius (C{scalar} > 0).
2073 @arg f_: Inverse flattening (C{scalar} >>> 1).
2075 @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}.
2076 '''
2077 t = f_ - _1_0
2078 a = b if _spherical_f_(f_) or fabs(t - f_) < EPS0 \
2079 or fabs(t) < EPS0 else (b * f_ / t)
2080 return Radius_(a=b if _spherical_a_b(a, b) else a)
2083def e2f(e):
2084 '''Return C{f}, the I{flattening} for a given I{1st eccentricity}.
2086 @arg e: The (1st) eccentricity (0 <= C{float} < 1)
2088 @return: The flattening (C{scalar} or C{0}).
2090 @see: Function L{e22f}.
2091 '''
2092 return e22f(e**2)
2095def e22f(e2):
2096 '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}.
2098 @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1)
2100 @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(e2 - 1) + 1)}.
2101 '''
2102 return Float(f=e2 / (sqrt(_1_0 - e2) + _1_0)) if e2 else _f_0_0
2105def f2e2(f):
2106 '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}.
2108 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2110 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1),
2111 M{f * (2 - f)}.
2113 @note: The result is positive for I{oblate}, negative for I{prolate}
2114 or C{0} for I{near-spherical} ellipsoids.
2116 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2117 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2118 <https://WikiPedia.org/wiki/Flattening>}.
2119 '''
2120 return Float(e2=_0_0 if _spherical_f(f) else (f * (_2_0 - f)))
2123def f2e22(f):
2124 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}.
2126 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2128 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or
2129 C{INF}), M{f * (2 - f) / (1 - f)**2}.
2131 @note: The result is positive for I{oblate}, negative for I{prolate}
2132 or C{0} for near-spherical ellipsoids.
2134 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2135 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.
2136 '''
2137 # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2
2138 t = (_1_0 - f)**2
2139 return Float(e22=INF if t < EPS0 else (f2e2(f) / t)) # PYCHOK type
2142def f2e32(f):
2143 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}.
2145 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2147 @return: The I{signed}, 3rd eccentricity I{squared} (C{float}),
2148 M{f * (2 - f) / (1 + (1 - f)**2)}.
2150 @note: The result is positive for I{oblate}, negative for I{prolate}
2151 or C{0} for I{near-spherical} ellipsoids.
2153 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2154 C++/doc/classGeographicLib_1_1Ellipsoid.html>}.
2155 '''
2156 # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2)
2157 e2 = f2e2(f)
2158 return Float(e32=e2 / (_2_0 - e2))
2161def f_2f(f_):
2162 '''Return C{f}, the I{flattening} for a given I{inverse flattening}.
2164 @arg f_: Inverse flattening (C{scalar} >>> 1).
2166 @return: The flattening (C{scalar} or C{0}), M{1 / f_}.
2168 @note: The result is positive for I{oblate}, negative for I{prolate}
2169 or C{0} for I{near-spherical} ellipsoids.
2170 '''
2171 f = 0 if _spherical_f_(f_) else _1_0 / f_
2172 return _f_0_0 if _spherical_f(f) else Float(f=f) # PYCHOK type
2175def f2f_(f):
2176 '''Return C{f_}, the I{inverse flattening} for a given I{flattening}.
2178 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2180 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2182 @note: The result is positive for I{oblate}, negative for I{prolate}
2183 or C{0} for I{near-spherical} ellipsoids.
2184 '''
2185 f_ = 0 if _spherical_f(f) else _1_0 / f
2186 return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) # PYCHOK type
2189def f2f2(f):
2190 '''Return C{f2}, the I{2nd flattening} for a given I{flattening}.
2192 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2194 @return: The I{signed}, 2nd flattening (C{scalar} or C{INF}), M{f / (1 - f)}.
2196 @note: The result is positive for I{oblate}, negative for I{prolate}
2197 or C{0} for I{near-spherical} ellipsoids.
2199 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2200 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2201 <https://WikiPedia.org/wiki/Flattening>}.
2202 '''
2203 t = _1_0 - f
2204 return Float(f2=_0_0 if _spherical_f(f) else (INF if fabs(t) < EPS
2205 else (f / t))) # PYCHOK type
2208def f2n(f):
2209 '''Return C{n}, the I{3rd flattening} for a given I{flattening}.
2211 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}).
2213 @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1),
2214 M{f / (2 - f)}.
2216 @note: The result is positive for I{oblate}, negative for I{prolate}
2217 or C{0} for I{near-spherical} ellipsoids.
2219 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2220 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2221 <https://WikiPedia.org/wiki/Flattening>}.
2222 '''
2223 return Float(n=_0_0 if _spherical_f(f) else (f / float(_2_0 - f)))
2226def n2e2(n):
2227 '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}.
2229 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2231 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF),
2232 M{4 * n / (1 + n)**2}.
2234 @note: The result is positive for I{oblate}, negative for I{prolate}
2235 or C{0} for I{near-spherical} ellipsoids.
2237 @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}.
2238 '''
2239 t = (n + _1_0)**2
2240 return Float(e2=_0_0 if fabs(n) < EPS0 else
2241 (NINF if t < EPS0 else (_4_0 * n / t)))
2244def n2f(n):
2245 '''Return C{f}, the I{flattening} for a given I{3rd flattening}.
2247 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2249 @return: The flattening (C{scalar} or NINF), M{2 * n / (1 + n)}.
2251 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/
2252 C++/doc/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening
2253 <https://WikiPedia.org/wiki/Flattening>}.
2254 '''
2255 t = n + _1_0
2256 f = 0 if fabs(n) < EPS0 else (NINF if t < EPS0 else (_2_0 * n / t))
2257 return _f_0_0 if _spherical_f(f) else Float(f=f)
2260def n2f_(n):
2261 '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}.
2263 @arg n: The 3rd flattening (-1 <= C{scalar} < 1).
2265 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}.
2267 @see: L{n2f} and L{f2f_}.
2268 '''
2269 return f2f_(n2f(n))
2272def _normalTo3(px, py, E, eps=EPS): # in .height4 above
2273 '''(INTERNAL) Nearest point on a 2-D ellipse in 1st quadrant.
2275 @see: Functions C{pygeodesy.triaxial._normalTo4} and C{-To5}.
2276 '''
2277 a, b, e0 = E.a, E.b, EPS0
2278 if min(px, py, a, b) < e0:
2279 raise _AssertionError(px=px, py=py, a=a, b=b, E=E)
2281 a2 = a - b * E.b_a
2282 b2 = b - a * E.a_b
2283 tx = ty = _SQRT2_2
2284 _a, _h = fabs, hypot
2285 for i in range(16): # max 5
2286 ex = a2 * tx**3
2287 ey = b2 * ty**3
2289 qx = px - ex
2290 qy = py - ey
2291 q = _h(qx, qy)
2292 if q < e0: # PYCHOK no cover
2293 break
2294 r = _h(ex - tx * a,
2295 ey - ty * b) / q
2297 sx, tx = tx, min(_1_0, max(0, (ex + qx * r) / a))
2298 sy, ty = ty, min(_1_0, max(0, (ey + qy * r) / b))
2299 t = _h(ty, tx)
2300 if t < e0: # PYCHOK no cover
2301 break
2302 tx = tx / t # /= chokes PyChecker
2303 ty = ty / t
2304 if _a(sx - tx) < eps and \
2305 _a(sy - ty) < eps:
2306 break
2308 tx *= a / px
2309 ty *= b / py
2310 return tx, ty, i # x and y as fractions
2313class Ellipsoids(_NamedEnum):
2314 '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class
2315 to accommodate the L{_LazyNamedEnumItem} properties.
2316 '''
2317 def _Lazy(self, a, b, f_, **kwds):
2318 '''(INTERNAL) Instantiate the L{Ellipsoid}.
2319 '''
2320 return Ellipsoid(a, b=b, f_=f_, **kwds)
2322Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton
2323'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.'''
2324# <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html>
2325# <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS>
2326# <https://kb.OSU.edu/dspace/handle/1811/77986>
2327# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids>
2328# <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html>
2329# <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java>
2330Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid>
2331 Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909
2332 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448
2333# APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965
2334# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid
2335# AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America
2336 Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192
2337 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System"
2338 Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)),
2339 BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)),
2340# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)),
2341 CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC)
2342# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)),
2343 Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)),
2344 Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)),
2345 Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)),
2346 Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc
2347 CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures
2348 Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium
2349 Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)),
2350# Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)),
2351# Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)),
2352# Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)),
2353 Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)),
2354 Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)),
2355 Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)),
2356# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)),
2357 GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)),
2358 GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid"
2359 GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516
2360# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)),
2361 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.25722210088)), # IUGG, ITRS, ETRS89
2362# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297
2363 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)),
2364# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)),
2365 IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union
2366 IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302
2367 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy
2368 IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)),
2369 Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795
2370 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l
2371 Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2372 Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling
2373# Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961
2374# Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979
2375 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France
2376 Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)),
2377 Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)),
2378# MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983
2379# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab.
2380 NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab.
2381# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965
2382 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)),
2383 OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)),
2384# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect?
2385 Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972
2386# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)),
2387 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11
2388# SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia
2389 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System
2390 SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American
2391 Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo
2392 SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo
2393 SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)), # EPSG:3857 Spheroid
2394 Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)),
2395# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)),
2396# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)),
2397 WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)),
2398 WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)),
2399 WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52
2400 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # GPS b=6356752.3142451793
2401# U{NOAA/NOS/NGS/inverse<https://GitHub.com/noaa-ngs/inverse/blob/main/invers3d.f>}
2402 WGS84_NGS = _lazy('WGS84_NGS', *_T(R_MA, _0_0, 298.257222100882711243162836600094))
2403)
2405_EWGS84 = Ellipsoids.WGS84 # (INTERNAL) shared
2407if __name__ == '__main__':
2409 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_
2410 from pygeodesy import nameof, printf
2412 for E in (_EWGS84, Ellipsoids.GRS80, # NAD83,
2413 Ellipsoids.Sphere, Ellipsoids.SpherePopular,
2414 Ellipsoid(_EWGS84.b, _EWGS84.a, name='_Prolate')):
2415 e = f2n(E.f) - E.n
2416 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10), nl=1)
2417 printf('# e=%s, f_=%s, f=%s, n=%s (%s)', fstr(E.e, prec=13, fmt=Fmt.e),
2418 fstr(E.f_, prec=13, fmt=Fmt.e),
2419 fstr(E.f, prec=13, fmt=Fmt.e),
2420 fstr(E.n, prec=13, fmt=Fmt.e),
2421 fstr(e, prec=9, fmt=Fmt.e))
2422 printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20))
2423 printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20))
2424 printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property
2426 # __doc__ of this file, force all into registry
2427 t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_)
2428 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t))
2430# % python3 -m pygeodesy.ellipsoids
2432# 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
2433# e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00)
2434# AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2435# BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2436# KsOrder 8
2438# 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
2439# e=8.1819191042833e-02, f_=2.9825722210088e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00)
2440# AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0
2441# BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0
2442# KsOrder 8
2444# 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
2445# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2446# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2447# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2448# KsOrder 8
2450# 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
2451# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00)
2452# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2453# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0
2454# KsOrder 8
2456# 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
2457# e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00)
2458# AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0
2459# BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0
2460# KsOrder 8
2462# **) MIT License
2463#
2464# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
2465#
2466# Permission is hereby granted, free of charge, to any person obtaining a
2467# copy of this software and associated documentation files (the "Software"),
2468# to deal in the Software without restriction, including without limitation
2469# the rights to use, copy, modify, merge, publish, distribute, sublicense,
2470# and/or sell copies of the Software, and to permit persons to whom the
2471# Software is furnished to do so, subject to the following conditions:
2472#
2473# The above copyright notice and this permission notice shall be included
2474# in all copies or substantial portions of the Software.
2475#
2476# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
2477# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
2478# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
2479# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
2480# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
2481# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
2482# OTHER DEALINGS IN THE SOFTWARE.