Coverage for pygeodesy/ellipsoids.py: 96%

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« prev     ^ index     » next       coverage.py v7.2.2, created at 2024-01-10 13:43 -0500

1 

2# -*- coding: utf-8 -*- 

3 

4u'''Ellipsoidal and spherical earth models. 

5 

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}. 

9 

10See module L{datums} for L{Datum} and L{Transform} information and other details. 

11 

12Following is the list of predefined L{Ellipsoid}s, all instantiated lazily. 

13 

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=10001867.62764495, 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=10001970.42122641, 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=10001964.63159782, 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=10002012.0443814, R1=6371037.76231981, R2=6371036.17227197, R3=6371029.78316994, Rbiaxial=6367483.11833616, Rtriaxial=6372826.29723739) 

59@var Ellipsoids.WGS66: Ellipsoid(name='WGS66', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e21=0.99330546, e22=0.00673966, e32=0.00335851, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802667, Rbiaxial=6367461.36258457, Rtriaxial=6372805.42010473) 

60@var Ellipsoids.WGS72: Ellipsoid(name='WGS72', a=6378135, b=6356750.52001609, f_=298.26, f=0.00335278, f2=0.00336406, n=0.0016792, e=0.08181881, e2=0.00669432, e21=0.99330568, e22=0.00673943, e32=0.0033584, A=6367447.24862383, L=10001962.74919858, R1=6371006.84000536, R2=6371005.24953886, R3=6370998.8587507, Rbiaxial=6367451.7372317, Rtriaxial=6372795.60727472) 

61@var Ellipsoids.WGS84: Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e21=0.99330562, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594) 

62''' 

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

64from __future__ import division as _; del _ # PYCHOK semicolon 

65 

66from pygeodesy.basics import copysign0, isbool, isint 

67from pygeodesy.constants import EPS, EPS0, EPS02, EPS1, INF, NINF, PI4, PI_2, PI_3, R_M, R_MA, R_FM, \ 

68 _EPSqrt, _EPStol as _TOL, _floatuple as _T, _isfinite, _SQRT2_2, \ 

69 _0_0s, _0_0, _0_5, _1_0, _1_EPS, _2_0, _4_0, _90_0, \ 

70 _0_25, _3_0 # PYCHOK used! 

71from pygeodesy.errors import _AssertionError, IntersectionError, _ValueError, _xattr, _xkwds_not 

72from pygeodesy.fmath import cbrt, cbrt2, fdot, Fhorner, fpowers, Fsum, hypot, hypot_, \ 

73 hypot1, hypot2, sqrt3 

74# from pygeodesy.fsums import Fsum # from .fmath 

75from pygeodesy.interns import NN, _a_, _Airy1830_, _AiryModified_, _b_, _Bessel1841_, _beta_, \ 

76 _Clarke1866_, _Clarke1880IGN_, _DOT_, _f_, _GRS80_, _height_, \ 

77 _Intl1924_, _incompatible_, _invalid_, _Krassovski1940_, \ 

78 _Krassowsky1940_, _meridional_, _lat_, _negative_, _not_, \ 

79 _not_finite_, _prime_vertical_, _radius_, _Sphere_, _SPACE_, \ 

80 _vs_, _WGS72_, _WGS84_ 

81# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .named 

82from pygeodesy.named import _lazyNamedEnumItem as _lazy, _NamedEnum, _NamedEnumItem, \ 

83 _NamedTuple, _Pass, _ALL_LAZY, _MODS 

84from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple 

85from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_, \ 

86 deprecated_property_RO, property_RO 

87from pygeodesy.streprs import Fmt, fstr, instr, strs, unstr 

88from pygeodesy.units import Bearing_, Distance, Float, Float_, Height, Lam_, Lat, Meter, \ 

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

90from pygeodesy.utily import atan1, atan1d, atan2b, degrees90, m2radians, radians2m, sincos2d 

91 

92from math import asinh, atan, atanh, cos, degrees, exp, fabs, radians, sin, sinh, sqrt, tan 

93 

94__all__ = _ALL_LAZY.ellipsoids 

95__version__ = '23.12.02' 

96 

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 

101 

102 

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__) 

107 

108 

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 

122 

123 

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}. 

127 

128 @see: Class L{Ellipsoid2}. 

129 ''' 

130 _Names_ = (_a_, _f_) # name 'f' not 'f_' 

131 _Units_ = (_Pass, _Pass) 

132 

133 def __new__(cls, a, f, **name): 

134 '''New L{a_f2Tuple} ellipsoid specification. 

135 

136 @arg a: Equatorial radius (C{scalar} > 0). 

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

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

139 

140 @return: An L{a_f2Tuple}C{(a, f)} instance. 

141 

142 @raise UnitError: Invalid B{C{a}} or B{C{f}}. 

143 

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) 

152 

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 

158 

159 def ellipsoid(self, name=NN): 

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

161 

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

163 same B{C{name}} already exists. 

164 ''' 

165 return Ellipsoid(self.a, f=self.f, name=name or self.name) # PYCHOK .a and .f 

166 

167 @Property_RO 

168 def f_(self): 

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

170 ''' 

171 return f2f_(self.f) # PYCHOK .f 

172 

173 

174class Circle4Tuple(_NamedTuple): 

175 '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height}, 

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

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

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

179 

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

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

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

183 equals C{lat}. 

184 ''' 

185 _Names_ = (_radius_, _height_, _lat_, _beta_) 

186 _Units_ = ( Radius, Height, Lat, Lat) 

187 

188 

189class Curvature2Tuple(_NamedTuple): 

190 '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in 

191 C{meter}, conventionally. 

192 ''' 

193 _Names_ = (_meridional_, _prime_vertical_) 

194 _Units_ = ( Meter, Meter) 

195 

196 @property_RO 

197 def transverse(self): 

198 '''Get this I{prime_vertical}, aka I{transverse} radius of curvature. 

199 ''' 

200 return self.prime_vertical 

201 

202 

203class Ellipsoid(_NamedEnumItem): 

204 '''Ellipsoid with I{equatorial} and I{polar} radii, I{flattening}, I{inverse 

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

206 I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models. 

207 ''' 

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

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

210 _f = 0 # (1st) flattening: (a - b) / a 

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

212 

213 _geodsolve = NN # means, use PYGEODESY_GEODSOLVE 

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

215 _rhumbsolve = NN # means, use PYGEODESY_RHUMBSOLVE 

216 

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

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

219 the I{polar} radius or I{inverse flattening} or I{flattening}. 

220 

221 @arg a: Equatorial radius, semi-axis (C{meter}). 

222 @arg b: Optional polar radius, semi-axis (C{meter}). 

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

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

225 spherical). 

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

227 

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

229 

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

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

232 

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

234 to M{1 / f_ = 0}, spherical. 

235 ''' 

236 ff_ = f, f_ # assertion below 

237 try: 

238 a = Radius_(a=a) # low=EPS 

239 if not _isfinite(a): 

240 raise ValueError(_SPACE_(_a_, _not_finite_)) 

241 

242 if b: # not in (_0_0, None) 

243 b = Radius_(b=b) # low=EPS 

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

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

246 elif f is not None: 

247 f = Float(f=f) 

248 b = a_f2b(a, f) 

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

250 elif f_: 

251 f_ = Float(f_=f_) 

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

253 f = f_2f(f_) 

254 else: # only a, spherical 

255 f_ = f = 0 

256 b = a # superfluous 

257 

258 if not f < _1_0: # sanity check, see .ecef.Ecef.__init__ 

259 raise ValueError(_SPACE_(_f_, _invalid_)) 

260 if not _isfinite(b): 

261 raise ValueError(_SPACE_(_b_, _not_finite_)) 

262 

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

264 b = a 

265 f = _f_0_0 

266 f_ = _f__0_0 

267 

268 except (TypeError, ValueError) as x: 

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

270 t = instr(self, a=a, name=name, **d) 

271 raise _ValueError(t, cause=x) 

272 

273 self._a = a 

274 self._b = b 

275 self._f = f 

276 self._f_ = f_ 

277 

278 self._register(Ellipsoids, name) 

279 

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

281 d = dict(eps=_TOL) 

282 if None in ff_: # both f_ and f given 

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

284 self._assert(_1_0 / f, f_=f_, **d) 

285 self._assert(_1_0 / f_, f =f, **d) 

286 self._assert(self.b2_a2, e21=self.e21, eps=EPS) 

287 

288 def __eq__(self, other): 

289 '''Compare this and an other ellipsoid. 

290 

291 @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). 

292 

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

294 ''' 

295 return self is other or (isinstance(other, Ellipsoid) and 

296 self.a == other.a and 

297 (self.f == other.f or self.b == other.b)) 

298 

299 def __hash__(self): 

300 return self._hash # memoized 

301 

302 @Property_RO 

303 def a(self): 

304 '''Get the I{equatorial} radius, semi-axis (C{meter}). 

305 ''' 

306 return self._a 

307 

308 equatoradius = a # = Requatorial 

309 

310 @Property_RO 

311 def a2(self): 

312 '''Get the I{equatorial} radius I{squared} (C{meter} I{squared}), M{a**2}. 

313 ''' 

314 return Meter2(a2=self.a**2) 

315 

316 @Property_RO 

317 def a2_(self): 

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

319 ''' 

320 return Float(a2_=_1_0 / self.a2) 

321 

322 @Property_RO 

323 def a_b(self): 

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

325 ''' 

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

327 

328 @Property_RO 

329 def a2_b(self): 

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

331 

332 @see: U{Radii of Curvature 

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

334 and U{Moritz, H. (1980), Geodetic Reference System 1980 

335 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}. 

336 

337 @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature 

338 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{c2} 

339 and L{R2} or L{Rauthalic}. 

340 ''' 

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

342 

343 @Property_RO 

344 def a2_b2(self): 

345 '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}), 

346 M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e21}. 

347 ''' 

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

349 

350 @Property_RO 

351 def a_f(self): 

352 '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}), see method C{toEllipsoid2}. 

353 ''' 

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

355 

356 @Property_RO 

357 def A(self): 

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

359 

360 @see: I{Meridian arc unit} U{Q<https://StudyLib.net/doc/7443565/>}. 

361 ''' 

362 A, n = self.a, self.n 

363 if n: 

364 d = (n + _1_0) * 1048576 / A 

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

366 # <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html> 

367 # <https://GeographicLib.SourceForge.io/C++/doc/transversemercator.html> and 

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

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

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

371 return A 

372 

373 @Property_RO 

374 def _albersCyl(self): 

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

376 ''' 

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

378 

379 @Property_RO 

380 def AlphaKs(self): 

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

382 ''' 

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

384 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 

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

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

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

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

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

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

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

392 _T(1424729850961/743921418240)) # PYCHOK unaligned 

393 

394 @Property_RO 

395 def area(self): 

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

397 

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

399 L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}. 

400 ''' 

401 return Meter2(area=self.c2 * PI4) 

402 

403 @Property_RO 

404 def areax(self): 

405 '''Get the ellipsoid's surface area (C{meter} I{squared}), M{4 * PI * c2x}, more 

406 accurate for very I{oblate} ellipsoids. 

407 

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

409 functions L{ellipsoidalExact.areaOf} and L{ellipsoidalKarney.areaOf}. 

410 ''' 

411 return Meter2(areax=self.c2x * PI4) 

412 

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

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

415 ''' 

416 for n, v in name_value.items(): 

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

418 t = (v, _vs_, val) 

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

420 t = Fmt.EQUAL(self._DOT_(n), t) 

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

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

423 raise Error(unstr(self._DOT_(self._assert.__name__), val, 

424 eps=eps, f0=f0, **name_value)) 

425 

426 def auxAuthalic(self, lat, inverse=False): 

427 '''Compute the I{authalic} auxiliary latitude or the I{inverse} thereof. 

428 

429 @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}). 

430 @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and 

431 return the geodetic latitude (C{bool}). 

432 

433 @return: The I{authalic} (or geodetic) latitude in C{degrees90}. 

434 

435 @see: U{Inverse-/AuthalicLatitude<https://GeographicLib.SourceForge.io/ 

436 html/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude 

437 <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and 

438 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16. 

439 ''' 

440 if self.f: 

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

442 lat = atan1d(f(tan(Phi_(lat)))) # PYCHOK attr 

443 return _aux(lat, inverse, Ellipsoid.auxAuthalic) 

444 

445 def auxConformal(self, lat, inverse=False): 

446 '''Compute the I{conformal} auxiliary latitude or the I{inverse} thereof. 

447 

448 @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}). 

449 @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and 

450 return the geodetic latitude (C{bool}). 

451 

452 @return: The I{conformal} (or geodetic) latitude in C{degrees90}. 

453 

454 @see: U{Inverse-/ConformalLatitude<https://GeographicLib.SourceForge.io/ 

455 html/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude 

456 <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and 

457 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16. 

458 ''' 

459 if self.f: 

460 f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr 

461 lat = atan1d(f(tan(Phi_(lat)))) # PYCHOK attr 

462 return _aux(lat, inverse, Ellipsoid.auxConformal) 

463 

464 def auxGeocentric(self, lat, inverse=False): 

465 '''Compute the I{geocentric} auxiliary latitude or the I{inverse} thereof. 

466 

467 @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}). 

468 @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and 

469 return the I{geocentric} latitude (C{bool}). 

470 

471 @return: The I{geocentric} (or geodetic) latitude in C{degrees90}. 

472 

473 @see: U{Inverse-/GeocentricLatitude<https://GeographicLib.SourceForge.io/ 

474 html/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude 

475 <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and 

476 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18. 

477 ''' 

478 if self.f: 

479 f = self.a2_b2 if inverse else self.b2_a2 

480 lat = atan1d(f * tan(Phi_(lat))) 

481 return _aux(lat, inverse, Ellipsoid.auxGeocentric) 

482 

483 def auxIsometric(self, lat, inverse=False): 

484 '''Compute the I{isometric} auxiliary latitude or the I{inverse} thereof. 

485 

486 @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}). 

487 @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and 

488 return the geodetic latitude (C{bool}). 

489 

490 @return: The I{isometric} (or geodetic) latitude in C{degrees}. 

491 

492 @note: The I{isometric} latitude for geodetic C{+/-90} is far 

493 outside the C{[-90..+90]} range but the inverse 

494 thereof is the original geodetic latitude. 

495 

496 @see: U{Inverse-/IsometricLatitude<https://GeographicLib.SourceForge.io/ 

497 html/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude 

498 <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and 

499 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16. 

500 ''' 

501 if self.f: 

502 r = Phi_(lat, clip=0) 

503 lat = degrees(atan1(self.es_tauf(sinh(r))) if inverse else 

504 asinh(self.es_taupf(tan(r)))) 

505 # clip=0, since auxIsometric(+/-90) is far outside [-90..+90] 

506 return _aux(lat, inverse, Ellipsoid.auxIsometric, clip=0) 

507 

508 def auxParametric(self, lat, inverse=False): 

509 '''Compute the I{parametric} auxiliary latitude or the I{inverse} thereof. 

510 

511 @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}). 

512 @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and 

513 return the geodetic latitude (C{bool}). 

514 

515 @return: The I{parametric} (or geodetic) latitude in C{degrees90}. 

516 

517 @see: U{Inverse-/ParametricLatitude<https://GeographicLib.SourceForge.io/ 

518 html/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude 

519 <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>}, 

520 and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18. 

521 ''' 

522 if self.f: 

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

524 return _aux(lat, inverse, Ellipsoid.auxParametric) 

525 

526 auxReduced = auxParametric # synonymous 

527 

528 def auxRectifying(self, lat, inverse=False): 

529 '''Compute the I{rectifying} auxiliary latitude or the I{inverse} thereof. 

530 

531 @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}). 

532 @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and 

533 return the geodetic latitude (C{bool}). 

534 

535 @return: The I{rectifying} (or geodetic) latitude in C{degrees90}. 

536 

537 @see: U{Inverse-/RectifyingLatitude<https://GeographicLib.SourceForge.io/ 

538 html/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude 

539 <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and 

540 U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17. 

541 ''' 

542 if self.f: 

543 lat = Lat(lat) 

544 if 0 < fabs(lat) < _90_0: 

545 if inverse: 

546 e = self._elliptic_e22 

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

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

549 else: 

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

551 return _aux(lat, inverse, Ellipsoid.auxRectifying) 

552 

553 @Property_RO 

554 def b(self): 

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

556 ''' 

557 return self._b 

558 

559 polaradius = b # = Rpolar 

560 

561 @Property_RO 

562 def b_a(self): 

563 '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a == f1 == 1 - f}. 

564 

565 @see: Property L{f1}. 

566 ''' 

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

568 

569 @Property_RO 

570 def b2(self): 

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

572 ''' 

573 return Meter2(b2=self.b**2) 

574 

575 @Property_RO 

576 def b2_a(self): 

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

578 

579 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. 

580 ''' 

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

582 

583 @Property_RO 

584 def b2_a2(self): 

585 '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2} 

586 == M{(1 - f)**2} == M{1 - e**2} == C{e21}. 

587 ''' 

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

589 

590 def _beta(self, lat, inverse=False): 

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

592 ''' 

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

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

595 return atan1d(s, c) 

596 

597 @Property_RO 

598 def BetaKs(self): 

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

600 ''' 

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

602 # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 

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

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

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

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

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

608 _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne 

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

610 _T(191773887257/3719607091200)) # PYCHOK unaligned 

611 

612 @deprecated_Property_RO 

613 def c(self): # PYCHOK no cover 

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

615 return self.R2 

616 

617 @Property_RO 

618 def c2(self): 

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

620 

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

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

623 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>}, 

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

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

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

627 ''' 

628 return self._c2f(False) 

629 

630 @Property_RO 

631 def c2x(self): 

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

633 ellipsoids. 

634 

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

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

637 ''' 

638 return self._c2f(True) 

639 

640 def _c2f(self, c2x): 

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

642 ''' 

643 f, c2 = self.f, self.b2 

644 if f: 

645 e = self.e 

646 if e > EPS0: 

647 if f > 0: # .isOblate 

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

649 elif f < 0: # .isProlate 

650 c2 *= atan1(e) / e # XXX asin? 

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

652 return c2 

653 

654 def circle4(self, lat): 

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

656 

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

658 

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

660 instance. 

661 

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

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

664 

665 @raise TypeError: Invalid B{C{lat}}. 

666 

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

668 

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

670 <https://WikiPedia.org/wiki/Latitude>}, I{Karney's} C++ U{CircleRadius and CircleHeight 

671 <https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1Ellipsoid.html>} 

672 and method C{Rlat}. 

673 ''' 

674 lat = Lat(lat) 

675 if lat: 

676 b = lat 

677 if fabs(lat) < _90_0: 

678 if self.f: 

679 b = self._beta(lat) 

680 z, r = sincos2d(b) 

681 r *= self.a 

682 z *= self.b 

683 else: # near-polar 

684 r, z = _0_0, copysign0(self.b, lat) 

685 else: # equator 

686 r = self.a 

687 z = lat = b = _0_0 

688 return Circle4Tuple(r, z, lat, b) 

689 

690 def degrees2m(self, deg, lat=0): 

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

692 along a parallel of (geodetic) latitude. 

693 

694 @arg deg: The angle (C{degrees}). 

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

696 

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

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

699 

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

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

702 

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

704 ''' 

705 return self.radians2m(radians(deg), lat=lat) 

706 

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

708 '''I{Approximate} the distance and (initial) bearing between 

709 two points based on the U{local, flat earth approximation 

710 <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny 

711 <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. 

712 

713 I{Suitable only for distances of several hundred Km or Miles 

714 and only between points not near-polar}. 

715 

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

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

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

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

720 

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

722 in same units as this ellipsoid's axes. 

723 

724 @note: The meridional and prime_vertical radii of curvature are 

725 taken and scaled I{at the initial latitude}, see C{roc2}. 

726 

727 @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}. 

728 ''' 

729 phi0 = Phi_(lat0=lat0) 

730 m, n = self.roc2_(phi0, scaled=True) 

731 m *= Phi_(lat1=lat1) - phi0 

732 n *= Lam_(lon1=lon1) - Lam_(lon0=lon0) 

733 return Distance2Tuple(hypot(m, n), atan2b(n, m)) 

734 

735 @Property_RO 

736 def e(self): 

737 '''Get the I{unsigned, (1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{a_b2e}. 

738 

739 @see: Property L{es}. 

740 ''' 

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

742 

743 @deprecated_Property_RO 

744 def e12(self): # see property ._e12 

745 '''DEPRECATED, use property C{e21}.''' 

746 return self.e21 

747 

748# @Property_RO 

749# def _e12(self): # see property ._elliptic_e12 

750# # (INTERNAL) until e12 above can be replaced with e21. 

751# return self.e2 / (_1_0 - self.e2) # see I{Karney}'s Ellipsoid._e12 = e2 / (1 - e2) 

752 

753 @Property_RO 

754 def e2(self): 

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

756 == 1 - (b / a)**2}, see C{a_b2e2}. 

757 ''' 

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

759 

760 @Property_RO 

761 def e2abs(self): 

762 '''Get the I{unsigned, (1st) eccentricity squared} (C{float}). 

763 ''' 

764 return fabs(self.e2) 

765 

766 @Property_RO 

767 def e21(self): 

768 '''Get 1 less I{1st eccentricity squared} (C{float}), M{1 - e**2} 

769 == M{1 - e2} == M{(1 - f)**2} == M{b**2 / a**2}, see C{b2_a2}. 

770 ''' 

771 return self._assert((_1_0 - self.f)**2, e21=_1_0 - self.e2, f0=_1_0) 

772 

773# _e2m = e21 # see I{Karney}'s Ellipsoid._e2m = 1 - _e2 

774 _1_e21 = a2_b2 # == M{1 / e21} == M{1 / (1 - e**2)} 

775 

776 @Property_RO 

777 def e22(self): 

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

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

780 ''' 

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

782 

783 @Property_RO 

784 def e22abs(self): 

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

786 ''' 

787 return fabs(self.e22) 

788 

789 @Property_RO 

790 def e32(self): 

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

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

793 ''' 

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

795 

796 @Property_RO 

797 def e32abs(self): 

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

799 ''' 

800 return fabs(self.e32) 

801 

802 @Property_RO 

803 def e4(self): 

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

805 ''' 

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

807 

808 eccentricity = e # eccentricity 

809# eccentricity2 = e2 # eccentricity squared 

810 eccentricity1st2 = e2 # first eccentricity squared 

811 eccentricity2nd2 = e22 # second eccentricity squared 

812 eccentricity3rd2 = e32 # third eccentricity squared 

813 

814 def ecef(self, Ecef=None): 

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

816 

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

818 

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

820 

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

822 

823 @see: Module L{pygeodesy.ecef}. 

824 ''' 

825 return _MODS.ecef._4Ecef(self, Ecef) 

826 

827 @Property_RO 

828 def _elliptic_e12(self): # see I{Karney}'s Ellipsoid._e12 

829 '''(INTERNAL) Elliptic helper for C{Rhumb}. 

830 ''' 

831 e12 = self.e2 / (_1_0 - self.e2) # NOT DEPRECATED .e12! 

832 return _MODS.elliptic.Elliptic(-e12) 

833 

834 @Property_RO 

835 def _elliptic_e22(self): # aka ._elliptic_ep2 

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

837 ''' 

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

839 

840 equatoradius = a # Requatorial 

841 

842 def e2s(self, s): 

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

844 

845 @arg s: Sine value (C{scalar}). 

846 

847 @return: Norm (C{float}). 

848 

849 @raise ValueError: Invalid B{C{s}}. 

850 ''' 

851 return sqrt(self.e2s2(s)) if self.e2 else _1_0 

852 

853 def e2s2(self, s): 

854 '''Compute M{1 - e2 * s**2}. 

855 

856 @arg s: Sine value (C{scalar}). 

857 

858 @return: Result (C{float}). 

859 

860 @raise ValueError: Invalid B{C{s}}. 

861 ''' 

862 r = _1_0 

863 if self.e2: 

864 try: 

865 r -= self.e2 * Scalar(s=s)**2 

866 if r < 0: 

867 raise ValueError(_negative_) 

868 except (TypeError, ValueError) as x: 

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

870 raise _ValueError(t, s, cause=x) 

871 return r 

872 

873 @Property_RO 

874 def es(self): 

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

876 

877 @see: Property L{e}. 

878 ''' 

879 # note, self.e is always non-negative 

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

881 

882 def es_atanh(self, x): 

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

884 for I{oblate} respectively I{prolate} ellipsoids where 

885 I{es} is the I{signed} (1st) eccentricity. 

886 

887 @raise ValueError: Invalid B{C{x}}. 

888 

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

890 html/classGeographicLib_1_1Math.html>}. 

891 ''' 

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

893 

894 def _es_atanh(self, x): # see .albers._atanhee, .AuxLat._atanhee 

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

896 ''' 

897 es = self.es # signOf(es) == signOf(f) 

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

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

900 _0_0)) # .isSpherical 

901 

902 @Property_RO 

903 def es_c(self): 

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

905 ''' 

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

907 

908 def es_tauf(self, taup): 

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

910 <https://ArXiv.org/abs/1002.1417>}. 

911 

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

913 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>} and 

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

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

916 ''' 

917 t = Scalar(taup=taup) 

918 if self.f: # .isEllipsoidal 

919 a = fabs(t) 

920 T = t * (self._exp_es_atanh_1 if a > 70 else self._1_e21) 

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

922 s = (a * _TOL) if a > _1_0 else _TOL 

923 for T, _, d in self._es_tauf3(t, T): # max 2 

924 if fabs(d) < s: 

925 break 

926 t = Scalar(tauf=T) 

927 return t 

928 

929 def _es_tauf3(self, taup, T, N=9): # in .utm.Utm._toLLEB 

930 '''(INTERNAL) Yield a 3-tuple C{(τi, iteration, delta)} for at most 

931 B{C{N}} Newton iterations, converging rapidly except when C{delta} 

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

933 ''' 

934 e = self._1_e21 

935 _F2_ = Fsum(T).fsum2_ # τ0 

936 _tf2 = self._es_taupf2 

937 for i in range(1, N + 1): 

938 a, h = _tf2(T) 

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

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

941 T, d = _F2_(d) # τi, (τi - τi-1) 

942 yield T, i, d 

943 

944 def es_taupf(self, tau): 

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

946 <https://ArXiv.org/abs/1002.1417>}. 

947 

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

949 SourceForge.io/C++/doc/classGeographicLib_1_1Math.html>}. 

950 ''' 

951 t = Scalar(tau=tau) 

952 if self.f: # .isEllipsoidal 

953 t, _ = self._es_taupf2(t) 

954 t = Scalar(taupf=t) 

955 return t 

956 

957 def _es_taupf2(self, tau): 

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

959 ''' 

960 if _isfinite(tau): 

961 h = hypot1(tau) 

962 s = sinh(self._es_atanh(tau / h)) 

963 a = hypot1(s) * tau - h * s 

964 else: 

965 a, h = tau, INF 

966 return a, h 

967 

968 @Property_RO 

969 def _exp_es_atanh_1(self): 

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

971 ''' 

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

973 

974 @Property_RO 

975 def f(self): 

976 '''Get the I{flattening} (C{scalar}), M{(a - b) / a}, C{0} for spherical, negative for prolate. 

977 ''' 

978 return self._f 

979 

980 @Property_RO 

981 def f_(self): 

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

983 ''' 

984 return self._f_ 

985 

986 @Property_RO 

987 def f1(self): 

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

989 

990 @see: Property L{b_a}. 

991 ''' 

992 return Float(f1=_1_0 - self.f) 

993 

994 @Property_RO 

995 def f2(self): 

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

997 ''' 

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

999 

1000 @deprecated_Property_RO 

1001 def geodesic(self): 

1002 '''DEPRECATED, use property C{geodesicw}.''' 

1003 return self.geodesicw 

1004 

1005 def geodesic_(self, exact=True): 

1006 '''Get the an I{exact} C{Geodesic...} instance for this ellipsoid. 

1007 

1008 @kwarg exact: If C{bool} return L{GeodesicExact}C{(exact=B{exact}, ...)}, 

1009 otherwise a L{Geodesic}, L{GeodesicExact} or L{GeodesicSolve} 

1010 instance for I{this} ellipsoid. 

1011 

1012 @return: The C{exact} geodesic (C{Geodesic...}). 

1013 

1014 @raise TypeError: Invalid B{C{exact}}. 

1015 

1016 @raise ValueError: Incompatible B{C{exact}} ellipsoid. 

1017 ''' 

1018 if isbool(exact): # for consistenccy with C{.rhumb_} 

1019 g = _MODS.geodesicx.GeodesicExact(self, C4order=30 if exact else 24, 

1020 name=self.name) 

1021 else: 

1022 g = exact 

1023 E = _xattr(g, ellipsoid=None) 

1024 if not (E is self and isinstance(g, self._Geodesics)): 

1025 raise _ValueError(exact=g, ellipsoid=E, txt=_not_(self.name)) 

1026 return g 

1027 

1028 @property_RO 

1029 def _Geodesics(self): 

1030 '''(INTERNAL) Get all C{Geodesic...} classes, I{once}. 

1031 ''' 

1032 Ellipsoid._Geodesics = t = (_MODS.geodesicw._wrapped.Geodesic, # overwrite property_RO 

1033 _MODS.geodesicx.GeodesicExact, 

1034 _MODS.geodsolve.GeodesicSolve) 

1035 return t 

1036 

1037 @property_RO 

1038 def geodesicw(self): 

1039 '''Get this ellipsoid's I{wrapped} U{geodesicw.Geodesic 

1040 <https://GeographicLib.SourceForge.io/Python/doc/code.html>}, provided 

1041 I{Karney}'s U{geographiclib<https://PyPI.org/project/geographiclib>} 

1042 package is installed. 

1043 ''' 

1044 # if not self.isEllipsoidal: 

1045 # raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

1046 return _MODS.geodesicw.Geodesic(self) 

1047 

1048 @property_RO 

1049 def geodesicx(self): 

1050 '''Get this ellipsoid's I{exact} L{GeodesicExact}. 

1051 ''' 

1052 # if not self.isEllipsoidal: 

1053 # raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

1054 return _MODS.geodesicx.GeodesicExact(self, name=self.name) 

1055 

1056 @property 

1057 def geodsolve(self): 

1058 '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility 

1059 U{GeodSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}, 

1060 provided the path to the C{GeodSolve} executable is specified with env 

1061 variable C{PYGEODESY_GEODSOLVE} or re-/set with this property.. 

1062 ''' 

1063 # if not self.isEllipsoidal: 

1064 # raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

1065 return _MODS.geodsolve.GeodesicSolve(self, path=self._geodsolve, name=self.name) 

1066 

1067 @geodsolve.setter # PYCHOK setter! 

1068 def geodsolve(self, path): 

1069 '''Re-/set the (fully qualified) path to the U{GeodSolve 

1070 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable, 

1071 overriding env variable C{PYGEODESY_GEODSOLVE} (C{str}). 

1072 ''' 

1073 self._geodsolve = path 

1074 

1075 def hartzell4(self, pov, los=None): 

1076 '''Compute the intersection of this ellipsoid's surface and a Line-Of-Sight 

1077 from a Point-Of-View in space. 

1078 

1079 @arg pov: Point-Of-View outside this ellipsoid (C{Cartesian}, L{Ecef9Tuple} 

1080 or L{Vector3d}). 

1081 @kwarg los: Line-Of-Sight, I{direction} to this ellipsoid (L{Vector3d}) or 

1082 C{None} to point to this ellipsoid's center. 

1083 

1084 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x}, 

1085 C{y} and C{z} of the projection on or the intersection with this 

1086 ellipsoid and the I{distance} C{h} from B{C{pov}} to C{(x, y, z)} 

1087 along B{C{los}}, all in C{meter}, conventionally. 

1088 

1089 @raise IntersectionError: Null B{C{pov}} or B{C{los}} vector, or B{C{pov}} 

1090 is inside this ellipsoid or B{C{los}} points 

1091 outside this ellipsoid or points in an opposite 

1092 direction. 

1093 

1094 @raise TypeError: Invalid B{C{pov}} or B{C{los}}. 

1095 

1096 @see: U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell. 

1097 Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>} and 

1098 methods L{Ellipsoid.height4} and L{Triaxial.hartzell4}. 

1099 ''' 

1100 try: 

1101 v, d = _MODS.triaxials._hartzell2(pov, los, self._triaxial) 

1102 except Exception as x: 

1103 raise IntersectionError(pov=pov, los=los, cause=x) 

1104 return Vector4Tuple(v.x, v.y, v.z, d, name=self.hartzell4.__name__) 

1105 

1106 @Property_RO 

1107 def _hash(self): 

1108 return hash((self.a, self.f)) 

1109 

1110 def height4(self, xyz, normal=True): 

1111 '''Compute the projection on and the height of a cartesian above or below 

1112 this ellipsoid's surface. 

1113 

1114 @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d}, 

1115 L{Vector3Tuple} or L{Vector4Tuple}). 

1116 @kwarg normal: If C{True}, the projection is perpendicular to (the nearest 

1117 point on) this ellipsoid's surface, otherwise the C{radial} 

1118 line to this ellipsoid's center (C{bool}). 

1119 

1120 @return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates C{x}, 

1121 C{y} and C{z} of the projection on and with the height C{h} above 

1122 the ellipsoid's surface, all in C{meter}, conventionally. 

1123 

1124 @raise ValueError: Null B{C{xyz}}. 

1125 

1126 @raise TypeError: Non-cartesian B{C{xyz}}. 

1127 

1128 @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>} 

1129 and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse and 

1130 methods L{Ellipsoid.hartzell4} and L{Triaxial.height4}. 

1131 ''' 

1132 v = _MODS.vector3d._otherV3d(xyz=xyz) 

1133 r = v.length 

1134 

1135 a, b, i = self.a, self.b, None 

1136 if r < EPS0: # EPS 

1137 v = v.times(_0_0) 

1138 h = -a 

1139 

1140 elif self.isSpherical: 

1141 v = v.times(a / r) 

1142 h = r - a 

1143 

1144 elif normal: # perpendicular to ellipsoid 

1145 x, y = hypot(v.x, v.y), fabs(v.z) 

1146 if x < EPS0: # PYCHOK no cover 

1147 z = copysign0(b, v.z) 

1148 v = Vector3Tuple(v.x, v.y, z) 

1149 h = y - b # polar 

1150 elif y < EPS0: # PYCHOK no cover 

1151 t = a / r 

1152 v = v.times_(t, t, 0) # force z=0.0 

1153 h = x - a # equatorial 

1154 else: # normal in 1st quadrant 

1155 x, y, i = _normalTo3(x, y, self) 

1156 t, v = v, v.times_(x, x, y) 

1157 h = t.minus(v).length 

1158 

1159 else: # radial to ellipsoid's center 

1160 h = hypot_(a * v.z, b * v.x, b * v.y) 

1161 t = (a * b / h) if h > EPS0 else _0_0 # EPS 

1162 v = v.times(t) 

1163 h = r * (_1_0 - t) 

1164 

1165 return Vector4Tuple(v.x, v.y, v.z, h, iteration=i, 

1166 name=self.height4.__name__) 

1167 

1168 def _hubeny_2(self, phi2, phi1, lam21, scaled=True, squared=True): 

1169 '''(INTERNAL) like function C{pygeodesy.flatLocal_}/C{pygeodesy.hubeny_}, 

1170 returning the I{angular} distance in C{radians squared} or C{radians} 

1171 ''' 

1172 m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=scaled) 

1173 h, r = (hypot2, self.a2_) if squared else (hypot, _1_0 / self.a) 

1174 return h(m * (phi2 - phi1), n * lam21) * r 

1175 

1176 @Property_RO 

1177 def isEllipsoidal(self): 

1178 '''Is this model I{ellipsoidal} (C{bool})? 

1179 ''' 

1180 return self.f != 0 

1181 

1182 @Property_RO 

1183 def isOblate(self): 

1184 '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or 

1185 spherical otherwise. 

1186 ''' 

1187 return self.f > 0 

1188 

1189 @Property_RO 

1190 def isProlate(self): 

1191 '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or 

1192 spherical otherwise. 

1193 ''' 

1194 return self.f < 0 

1195 

1196 @Property_RO 

1197 def isSpherical(self): 

1198 '''Is this ellipsoid I{spherical} (C{bool})? 

1199 ''' 

1200 return self.f == 0 

1201 

1202 def _Kseries(self, *AB8Ks): 

1203 '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha 

1204 or Beta series coefficients per I{Karney}'s U{equations (35) 

1205 and (36)<https://ArXiv.org/pdf/1002.1417v3.pdf>}. 

1206 

1207 @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series 

1208 coefficient tuples. 

1209 

1210 @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}). 

1211 

1212 @see: I{Karney}'s 30-th order U{TMseries30 

1213 <https://GeographicLib.SourceForge.io/C++/doc/tmseries30.html>}. 

1214 ''' 

1215 k = self.KsOrder 

1216 if self.n: 

1217 ns = fpowers(self.n, k) 

1218 ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k)) 

1219 else: 

1220 ks = _0_0s(k) 

1221 return ks 

1222 

1223 @property_doc_(''' the I{Krüger} series' order (C{int}), see properties C{AlphaKs}, C{BetaKs}.''') 

1224 def KsOrder(self): 

1225 '''Get the I{Krüger} series' order (C{int} 4, 6 or 8). 

1226 ''' 

1227 return self._KsOrder 

1228 

1229 @KsOrder.setter # PYCHOK setter! 

1230 def KsOrder(self, order): 

1231 '''Set the I{Krüger} series' order (C{int} 4, 6 or 8). 

1232 

1233 @raise ValueError: Invalid B{C{order}}. 

1234 ''' 

1235 if not (isint(order) and order in (4, 6, 8)): 

1236 raise _ValueError(order=order) 

1237 if self._KsOrder != order: 

1238 Ellipsoid.AlphaKs._update(self) 

1239 Ellipsoid.BetaKs._update(self) 

1240 self._KsOrder = order 

1241 

1242 @Property_RO 

1243 def L(self): 

1244 '''Get the I{quarter meridian} C{L}, aka the C{polar distance} 

1245 along a meridian between the equator and a pole (C{meter}), 

1246 M{b * Elliptic(-e2 / (1 - e2)).cE} or M{b * PI / 2}. 

1247 ''' 

1248 r = self._elliptic_e22.cE if self.f else PI_2 

1249 return Distance(L=self.b * r) 

1250 

1251 def Llat(self, lat): 

1252 '''Return the I{meridional length}, the distance along a meridian 

1253 between the equator and a (geodetic) latitude, see C{L}. 

1254 

1255 @arg lat: Geodetic latitude (C{degrees90}). 

1256 

1257 @return: The meridional length at B{C{lat}}, negative on southern 

1258 hemisphere (C{meter}). 

1259 ''' 

1260 r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phi_(lat) 

1261 return Distance(Llat=self.b * r) 

1262 

1263 Lmeridian = Llat # meridional distance 

1264 

1265 @property_RO 

1266 def _Lpd(self): 

1267 '''Get the I{quarter meridian} per degree (C{meter}), M{self.L / 90}. 

1268 ''' 

1269 return Meter(_Lpd=self.L / _90_0) 

1270 

1271 @property_RO 

1272 def _Lpr(self): 

1273 '''Get the I{quarter meridian} per radian (C{meter}), M{self.L / PI_2}. 

1274 ''' 

1275 return Meter(_Lpr=self.L / PI_2) 

1276 

1277 @deprecated_Property_RO 

1278 def majoradius(self): # PYCHOK no cover 

1279 '''DEPRECATED, use property C{a} or C{Requatorial}.''' 

1280 return self.a 

1281 

1282 def m2degrees(self, distance, lat=0): 

1283 '''Convert a distance to an angle along the equator or 

1284 along a parallel of (geodetic) latitude. 

1285 

1286 @arg distance: Distance (C{meter}). 

1287 @kwarg lat: Parallel latitude (C{degrees90}, C{str}). 

1288 

1289 @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}. 

1290 

1291 @raise RangeError: Latitude B{C{lat}} outside valid range and 

1292 L{pygeodesy.rangerrors} set to C{True}. 

1293 

1294 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}. 

1295 ''' 

1296 return degrees(self.m2radians(distance, lat=lat)) 

1297 

1298 def m2radians(self, distance, lat=0): 

1299 '''Convert a distance to an angle along the equator or 

1300 along a parallel of (geodetic) latitude. 

1301 

1302 @arg distance: Distance (C{meter}). 

1303 @kwarg lat: Parallel latitude (C{degrees90}, C{str}). 

1304 

1305 @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}. 

1306 

1307 @raise RangeError: Latitude B{C{lat}} outside valid range and 

1308 L{pygeodesy.rangerrors} set to C{True}. 

1309 

1310 @raise ValueError: Invalid B{C{distance}} or B{C{lat}}. 

1311 ''' 

1312 r = self.circle4(lat).radius if lat else self.a 

1313 return m2radians(distance, radius=r, lat=0) 

1314 

1315 @deprecated_Property_RO 

1316 def minoradius(self): # PYCHOK no cover 

1317 '''DEPRECATED, use property C{b}, C{polaradius} or C{Rpolar}.''' 

1318 return self.b 

1319 

1320 @Property_RO 

1321 def n(self): 

1322 '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}. 

1323 ''' 

1324 return self._assert(a_b2n(self.a, self.b), n=f2n(self.f)) 

1325 

1326 flattening = f 

1327 flattening1st = f 

1328 flattening2nd = f2 

1329 flattening3rd = n 

1330 

1331 polaradius = b # Rpolar 

1332 

1333# @Property_RO 

1334# def Q(self): 

1335# '''Get the I{meridian arc unit} C{Q}, the mean, meridional length I{per radian} C({float}). 

1336# 

1337# @note: C{Q * PI / 2} ≈ C{L}, the I{quarter meridian}. 

1338# 

1339# @see: Property C{A} and U{Engsager, K., Poder, K.<https://StudyLib.net/doc/7443565/ 

1340# a-highly-accurate-world-wide-algorithm-for-the-transverse...>}. 

1341# ''' 

1342# n = self.n 

1343# d = (n + _1_0) / self.a 

1344# return Float(Q=Fhorner(n**2, _1_0, _0_25, _1_16th, _0_25).fover(d) if d else self.b) 

1345 

1346# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf> 

1347# # Q = (1 - 3/4 * e'2 + 45/64 * e'4 - 175/256 * e'6 + 11025/16384 * e'8) * rocPolar 

1348# # = (4 + e'2 * (-3 + e'2 * (45/16 + e'2 * (-175/64 + e'2 * 11025/4096)))) * rocPolar / 4 

1349# return Fhorner(self.e22, 4, -3, 45 / 16, -175 / 64, 11025 / 4096).fover(4 / self.rocPolar) 

1350 

1351 @deprecated_Property_RO 

1352 def quarteradius(self): # PYCHOK no cover 

1353 '''DEPRECATED, use property C{L} or method C{Llat}.''' 

1354 return self.L 

1355 

1356 @Property_RO 

1357 def R1(self): 

1358 '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3 == a * (1 - f / 3)}. 

1359 

1360 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} 

1361 and method C{Rgeometric}. 

1362 ''' 

1363 r = Fsum(self.a, self.a, self.b).fover(_3_0) if self.f else self.a 

1364 return Radius(R1=r) 

1365 

1366 Rmean = R1 

1367 

1368 @Property_RO 

1369 def R2(self): 

1370 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}. 

1371 

1372 @see: C{R2x}, C{c2}, C{area} and U{Earth radius 

1373 <https://WikiPedia.org/wiki/Earth_radius>}. 

1374 ''' 

1375 return Radius(R2=sqrt(self.c2) if self.f else self.a) 

1376 

1377 Rauthalic = R2 

1378 

1379# @Property_RO 

1380# def R2(self): 

1381# # Moritz, H. <https://Geodesy.Geology.Ohio-State.EDU/course/refpapers/00740128.pdf> 

1382# # R2 = (1 - 2/3 * e'2 + 26/45 * e'4 - 100/189 * e'6 + 7034/14175 * e'8) * rocPolar 

1383# # = (3 + e'2 * (-2 + e'2 * (26/15 + e'2 * (-100/63 + e'2 * 7034/4725)))) * rocPolar / 3 

1384# return Fhorner(self.e22, 3, -2, 26 / 15, -100 / 63, 7034 / 4725).fover(3 / self.rocPolar) 

1385 

1386 @Property_RO 

1387 def R2x(self): 

1388 '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}. 

1389 

1390 @see: C{R2}, C{c2x} and C{areax}. 

1391 ''' 

1392 return Radius(R2x=sqrt(self.c2x) if self.f else self.a) 

1393 

1394 Rauthalicx = R2x 

1395 

1396 @Property_RO 

1397 def R3(self): 

1398 '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}. 

1399 

1400 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}. 

1401 ''' 

1402 r = (cbrt(self.b_a) * self.a) if self.f else self.a 

1403 return Radius(R3=r) 

1404 

1405 Rvolumetric = R3 

1406 

1407 def radians2m(self, rad, lat=0): 

1408 '''Convert an angle to the distance along the equator or 

1409 along a parallel of (geodetic) latitude. 

1410 

1411 @arg rad: The angle (C{radians}). 

1412 @kwarg lat: Parallel latitude (C{degrees90}, C{str}). 

1413 

1414 @return: Distance (C{meter}, same units as the equatorial 

1415 and polar radii) or C{0} for near-polar B{C{lat}}. 

1416 

1417 @raise RangeError: Latitude B{C{lat}} outside valid range and 

1418 L{pygeodesy.rangerrors} set to C{True}. 

1419 

1420 @raise ValueError: Invalid B{C{rad}} or B{C{lat}}. 

1421 ''' 

1422 r = self.circle4(lat).radius if lat else self.a 

1423 return radians2m(rad, radius=r, lat=0) 

1424 

1425 @Property_RO 

1426 def Rbiaxial(self): 

1427 '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}. 

1428 

1429 @see: C{Rtriaxial} 

1430 ''' 

1431 a, b = self.a, self.b 

1432 if b < a: 

1433 b = sqrt(_0_5 + self.b2_a2 * _0_5) * a 

1434 elif b > a: 

1435 b *= sqrt(_0_5 + self.a2_b2 * _0_5) 

1436 return Radius(Rbiaxial=b) 

1437 

1438 Requatorial = a # for consistent naming 

1439 

1440 def Rgeocentric(self, lat): 

1441 '''Compute the I{geocentric} earth radius of (geodetic) latitude. 

1442 

1443 @arg lat: Latitude (C{degrees90}). 

1444 

1445 @return: Geocentric earth radius (C{meter}). 

1446 

1447 @raise ValueError: Invalid B{C{lat}}. 

1448 

1449 @see: U{Geocentric Radius 

1450 <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>} 

1451 ''' 

1452 r, a = self.a, Phi_(lat) 

1453 if a and self.f: 

1454 if fabs(a) < PI_2: 

1455 s2, c2 = _s2_c2(a) 

1456 b2_a2_s2 = self.b2_a2 * s2 

1457 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2)) 

1458 # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2))) 

1459 # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2)) 

1460 # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2)) 

1461 # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2)) 

1462 r *= sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2)) 

1463 else: 

1464 r = self.b 

1465 return Radius(Rgeocentric=r) 

1466 

1467 @Property_RO 

1468 def Rgeometric(self): 

1469 '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}. 

1470 

1471 @see: C{R1}. 

1472 ''' 

1473 g = sqrt(self.a * self.b) if self.f else self.a 

1474 return Radius(Rgeometric=g) 

1475 

1476 def rhumb_(self, exact=True): 

1477 '''Get the an I{exact} C{Rhumb...} instance for this ellipsoid. 

1478 

1479 @kwarg exact: If C{bool} or C{None} return L{Rhumb}C{(exact=B{exact}, ...)}, 

1480 otherwise a L{Rhumb}, L{RhumbAux} or L{RhumbSolve} instance 

1481 for I{this} ellipsoid. 

1482 

1483 @return: The C{exact} rhumb (C{Rhumb...}). 

1484 

1485 @raise TypeError: Invalid B{C{exact}}. 

1486 

1487 @raise ValueError: Incompatible B{C{exact}} ellipsoid. 

1488 ''' 

1489 if isbool(exact): # use Rhumb for backward compatibility 

1490 r = _MODS.rhumb.ekx.Rhumb(self, exact=exact, name=self.name) 

1491 else: 

1492 r = exact 

1493 E = _xattr(r, ellipsoid=None) 

1494 if not (E is self and isinstance(r, self._Rhumbs)): 

1495 raise _ValueError(exact=r, ellipsosid=E, txt=_not_(self.name)) 

1496 return r 

1497 

1498 @property_RO 

1499 def rhumbaux(self): 

1500 '''Get this ellipsoid's I{Auxiliary} C{rhumb.RhumbAux}. 

1501 ''' 

1502 # if not self.isEllipsoidal: 

1503 # raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

1504 return _MODS.rhumb.aux_.RhumbAux(self, name=self.name) 

1505 

1506 @property_RO 

1507 def rhumbekx(self): 

1508 '''Get this ellipsoid's I{Elliptic, Krüger} C{rhumb.Rhumb}. 

1509 ''' 

1510 # if not self.isEllipsoidal: 

1511 # raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

1512 return _MODS.rhumb.ekx.Rhumb(self, name=self.name) 

1513 

1514 @property_RO 

1515 def _Rhumbs(self): 

1516 '''(INTERNAL) Get all C{Rhumb...} classes, I{once}. 

1517 ''' 

1518 p = _MODS.rhumb 

1519 Ellipsoid._Rhumbs = t = (p.aux_.RhumbAux, # overwrite property_RO 

1520 p.ekx.Rhumb, p.solve.RhumbSolve) 

1521 return t 

1522 

1523 @property 

1524 def rhumbsolve(self): 

1525 '''Get this ellipsoid's L{RhumbSolve}, the I{wrapper} around utility 

1526 U{RhumbSolve<https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>}, 

1527 provided the path to the C{RhumbSolve} executable is specified with env 

1528 variable C{PYGEODESY_RHUMBSOLVE} or re-/set with this property. 

1529 ''' 

1530 # if not self.isEllipsoidal: 

1531 # raise _IsnotError(_ellipsoidal_, ellipsoid=self) 

1532 return _MODS.rhumb.solve.RhumbSolve(self, path=self._rhumbsolve, name=self.name) 

1533 

1534 @rhumbsolve.setter # PYCHOK setter! 

1535 def rhumbsolve(self, path): 

1536 '''Re-/set the (fully qualified) path to the U{RhumbSolve 

1537 <https://GeographicLib.SourceForge.io/C++/doc/GeodSolve.1.html>} executable, 

1538 overriding env variable C{PYGEODESY_RHUMBSOLVE} (C{str}). 

1539 ''' 

1540 self._rhumbsolve = path 

1541 

1542 @deprecated_property_RO 

1543 def rhumbx(self): 

1544 '''DEPRECATED on 2023.11.28, use property C{rhumbekx}. ''' 

1545 return self.rhumbekx 

1546 

1547 def Rlat(self, lat): 

1548 '''I{Approximate} the earth radius of (geodetic) latitude. 

1549 

1550 @arg lat: Latitude (C{degrees90}). 

1551 

1552 @return: Approximate earth radius (C{meter}). 

1553 

1554 @raise RangeError: Latitude B{C{lat}} outside valid range and 

1555 L{pygeodesy.rangerrors} set to C{True}. 

1556 

1557 @raise TypeError: Invalid B{C{lat}}. 

1558 

1559 @raise ValueError: Invalid B{C{lat}}. 

1560 

1561 @note: C{Rlat(B{90})} equals C{Rpolar}. 

1562 

1563 @see: Method C{circle4}. 

1564 ''' 

1565 # r = a - (a - b) * |lat| / 90 

1566 r = self.a 

1567 if self.f and lat: # .isEllipsoidal 

1568 r -= (r - self.b) * fabs(Lat(lat)) / _90_0 

1569 r = Radius(Rlat=r) 

1570 return r 

1571 

1572 Rpolar = b # for consistent naming 

1573 

1574 def roc1_(self, sa, ca=None): 

1575 '''Compute the I{prime-vertical}, I{normal} radius of curvature 

1576 of (geodetic) latitude, I{unscaled}. 

1577 

1578 @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]). 

1579 @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0]) 

1580 to use an alternate formula. 

1581 

1582 @return: The prime-vertical radius of curvature (C{float}). 

1583 

1584 @note: The delta between both formulae with C{Ellipsoids.WGS84} 

1585 is less than 2 nanometer over the entire latitude range. 

1586 

1587 @see: Method L{roc2_} and class L{EcefYou}. 

1588 ''' 

1589 if not self.f: # .isSpherical 

1590 n = self.a 

1591 elif ca is None: 

1592 r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward 

1593 n = sqrt(self.a2 / r) if r > EPS02 else _0_0 

1594 elif ca: # derived from EcefYou.forward 

1595 h = hypot(ca, self.b_a * sa) if sa else fabs(ca) 

1596 n = self.a / h 

1597 elif sa: 

1598 n = self.a2_b / fabs(sa) 

1599 else: 

1600 n = self.a 

1601 return n 

1602 

1603 def roc2(self, lat, scaled=False): 

1604 '''Compute the I{meridional} and I{prime-vertical}, I{normal} 

1605 radii of curvature of (geodetic) latitude. 

1606 

1607 @arg lat: Latitude (C{degrees90}). 

1608 @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}). 

1609 

1610 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with 

1611 the radii of curvature. 

1612 

1613 @raise ValueError: Invalid B{C{lat}}. 

1614 

1615 @see: Methods L{roc2_} and L{roc1_}, U{Local, flat earth approximation 

1616 <https://www.EdWilliams.org/avform.htm#flat>} and meridional and 

1617 prime vertical U{Radii of Curvature<https://WikiPedia.org/wiki/ 

1618 Earth_radius#Radii_of_curvature>}. 

1619 ''' 

1620 return self.roc2_(Phi_(lat), scaled=scaled) 

1621 

1622 def roc2_(self, phi, scaled=False): 

1623 '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of 

1624 curvature of (geodetic) latitude. 

1625 

1626 @arg phi: Latitude (C{radians}). 

1627 @kwarg scaled: Scale prime_vertical by C{cos(B{phi})} (C{bool}). 

1628 

1629 @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the 

1630 radii of curvature. 

1631 

1632 @raise ValueError: Invalid B{C{phi}}. 

1633 

1634 @see: Methods L{roc2} and L{roc1_}, property L{rocEquatorial2}, U{Local, 

1635 flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>} 

1636 and the meridional and prime vertical U{Radii of Curvature 

1637 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. 

1638 ''' 

1639 a = fabs(Phi(phi)) 

1640 if self.f: 

1641 r = self.e2s2(sin(a)) 

1642 if r > EPS02: 

1643 n = self.a / sqrt(r) 

1644 m = n * self.e21 / r # PYCHOK attr 

1645 else: 

1646 m = n = _0_0 # PYCHOK attr 

1647 else: 

1648 m = n = self.a 

1649 if scaled and a: 

1650 n *= cos(a) if a < PI_2 else _0_0 

1651 return Curvature2Tuple(Radius(rocMeridional=m), 

1652 Radius(rocPrimeVertical=n)) 

1653 

1654 def rocBearing(self, lat, bearing): 

1655 '''Compute the I{directional} radius of curvature of (geodetic) 

1656 latitude and compass direction. 

1657 

1658 @arg lat: Latitude (C{degrees90}). 

1659 @arg bearing: Direction (compass C{degrees360}). 

1660 

1661 @return: Directional radius of curvature (C{meter}). 

1662 

1663 @raise RangeError: Latitude B{C{lat}} outside valid range and 

1664 L{pygeodesy.rangerrors} set to C{True}. 

1665 

1666 @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}. 

1667 

1668 @see: U{Radii of Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} 

1669 ''' 

1670 if self.f: 

1671 s2, c2 = _s2_c2(Bearing_(bearing)) 

1672 m, n = self.roc2_(Phi_(lat)) 

1673 if n < m: # == n / (c2 * n / m + s2) 

1674 c2 *= n / m 

1675 elif m < n: # == m / (c2 + s2 * m / n) 

1676 s2 *= m / n 

1677 n = m 

1678 b = n / (c2 + s2) # == 1 / (c2 / m + s2 / n) 

1679 else: 

1680 b = self.b # == self.a 

1681 return Radius(rocBearing=b) 

1682 

1683 @Property_RO 

1684 def rocEquatorial2(self): 

1685 '''Get the I{meridional} and I{prime-vertical}, I{normal} radii of curvature 

1686 at the equator as L{Curvature2Tuple}C{(meridional, prime_vertical)}. 

1687 

1688 @see: Methods L{rocMeridional} and L{rocPrimeVertical}, properties L{b2_a}, 

1689 L{a2_b}, C{rocPolar} and polar and equatorial U{Radii of Curvature 

1690 <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. 

1691 ''' 

1692 return Curvature2Tuple(Radius(rocMeridional0=self.b2_a if self.f else self.a), 

1693 Radius(rocPrimeVertical0=self.a)) 

1694 

1695 def rocGauss(self, lat): 

1696 '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude. 

1697 

1698 @arg lat: Latitude (C{degrees90}). 

1699 

1700 @return: Gaussian radius of curvature (C{meter}). 

1701 

1702 @raise ValueError: Invalid B{C{lat}}. 

1703 

1704 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/ 

1705 Earth_radius#Radii_of_curvature>} 

1706 ''' 

1707 # using ... 

1708 # m, n = self.roc2_(Phi_(lat)) 

1709 # return sqrt(m * n) 

1710 # ... requires 1 or 2 sqrt 

1711 g = self.b 

1712 if self.f: 

1713 s2, c2 = _s2_c2(Phi_(lat)) 

1714 g = g / (c2 + self.b2_a2 * s2) 

1715 return Radius(rocGauss=g) 

1716 

1717 def rocMean(self, lat): 

1718 '''Compute the I{mean} radius of curvature of (geodetic) latitude. 

1719 

1720 @arg lat: Latitude (C{degrees90}). 

1721 

1722 @return: Mean radius of curvature (C{meter}). 

1723 

1724 @raise ValueError: Invalid B{C{lat}}. 

1725 

1726 @see: Non-directional U{Radii of Curvature<https://WikiPedia.org/wiki/ 

1727 Earth_radius#Radii_of_curvature>} 

1728 ''' 

1729 if self.f: 

1730 m, n = self.roc2_(Phi_(lat)) 

1731 m *= n * _2_0 / (m + n) # == 2 / (1 / m + 1 / n) 

1732 else: 

1733 m = self.a 

1734 return Radius(rocMean=m) 

1735 

1736 def rocMeridional(self, lat): 

1737 '''Compute the I{meridional} radius of curvature of (geodetic) latitude. 

1738 

1739 @arg lat: Latitude (C{degrees90}). 

1740 

1741 @return: Meridional radius of curvature (C{meter}). 

1742 

1743 @raise ValueError: Invalid B{C{lat}}. 

1744 

1745 @see: Methods L{roc2} and L{roc2_}, U{Local, flat earth approximation 

1746 <https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of 

1747 Curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. 

1748 ''' 

1749 return self.roc2_(Phi_(lat)).meridional if lat else \ 

1750 self.rocEquatorial2.meridional 

1751 

1752 rocPolar = a2_b # synonymous 

1753 

1754 def rocPrimeVertical(self, lat): 

1755 '''Compute the I{prime-vertical}, I{normal} radius of curvature of 

1756 (geodetic) latitude, aka the I{transverse} radius of curvature. 

1757 

1758 @arg lat: Latitude (C{degrees90}). 

1759 

1760 @return: Prime-vertical radius of curvature (C{meter}). 

1761 

1762 @raise ValueError: Invalid B{C{lat}}. 

1763 

1764 @see: Methods L{roc2}, L{roc2_} and L{roc1_}, U{Local, flat earth 

1765 approximation<https://www.EdWilliams.org/avform.htm#flat>} and 

1766 U{Radii of Curvature<https://WikiPedia.org/wiki/ 

1767 Earth_radius#Radii_of_curvature>}. 

1768 ''' 

1769 return self.roc2_(Phi_(lat)).prime_vertical if lat else \ 

1770 self.rocEquatorial2.prime_vertical 

1771 

1772 rocTransverse = rocPrimeVertical # synonymous 

1773 

1774 @deprecated_Property_RO 

1775 def Rquadratic(self): # PYCHOK no cover 

1776 '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.''' 

1777 return self.Rbiaxial 

1778 

1779 @deprecated_Property_RO 

1780 def Rr(self): # PYCHOK no cover 

1781 '''DEPRECATED, use property C{Rrectifying}.''' 

1782 return self.Rrectifying 

1783 

1784 @Property_RO 

1785 def Rrectifying(self): 

1786 '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}. 

1787 

1788 @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}. 

1789 ''' 

1790 r = (cbrt2((_1_0 + sqrt3(self.b_a)) * _0_5) * self.a) if self.f else self.a 

1791 return Radius(Rrectifying=r) 

1792 

1793 @deprecated_Property_RO 

1794 def Rs(self): # PYCHOK no cover 

1795 '''DEPRECATED, use property C{Rgeometric}.''' 

1796 return self.Rgeometric 

1797 

1798 @Property_RO 

1799 def Rtriaxial(self): 

1800 '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}. 

1801 

1802 @see: C{Rbiaxial} 

1803 ''' 

1804 a, b = self.a, self.b 

1805 q = (sqrt((_3_0 + self.b2_a2) * _0_25) * a) if a > b else ( 

1806 (sqrt((_3_0 * self.a2_b2 + _1_0) * _0_25) * b) if a < b else a) 

1807 return Radius(Rtriaxial=q) 

1808 

1809 def toEllipsoid2(self, name=NN): 

1810 '''Get a copy of this ellipsoid as an L{Ellipsoid2}. 

1811 

1812 @kwarg name: Optional, unique name (C{str}). 

1813 

1814 @see: Property C{a_f}. 

1815 ''' 

1816 return Ellipsoid2(self, None, name=name) 

1817 

1818 def toStr(self, prec=8, name=NN, **unused): # PYCHOK expected 

1819 '''Return this ellipsoid as a text string. 

1820 

1821 @kwarg prec: Number of decimal digits, unstripped (C{int}). 

1822 @kwarg name: Override name (C{str}) or C{None} to exclude 

1823 this ellipsoid's name. 

1824 

1825 @return: This C{Ellipsoid}'s attributes (C{str}). 

1826 ''' 

1827 E = Ellipsoid 

1828 t = E.a, E.b, E.f_, E.f, E.f2, E.n, E.e, E.e2, E.e21, E.e22, E.e32, \ 

1829 E.A, E.L, E.R1, E.R2, E.R3, E.Rbiaxial, E.Rtriaxial 

1830 if Fmt.form: # terse form[at] 

1831 t = t[:4] 

1832 return self._instr(name, prec, props=t) 

1833 

1834 def toTriaxial(self, name=NN): 

1835 '''Convert this ellipsoid to a L{Triaxial_}. 

1836 

1837 @return: A L{Triaxial_} or L{Triaxial} with the C{X} axis 

1838 pointing east and C{Z} pointing north. 

1839 

1840 @see: Method L{Triaxial_.toEllipsoid}. 

1841 ''' 

1842 T = self._triaxial 

1843 return T.copy(name=name) if name else T 

1844 

1845 @property_RO 

1846 def _triaxial(self): 

1847 '''(INTERNAL) Get this ellipsoid's un-/ordered C{Triaxial/_}. 

1848 ''' 

1849 a, b, m = self.a, self.b, _MODS.triaxials 

1850 T = m.Triaxial if a > b else m.Triaxial_ 

1851 return T(a, a, b, name=self.name) 

1852 

1853 @Property_RO 

1854 def volume(self): 

1855 '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}. 

1856 

1857 @see: C{R3}. 

1858 ''' 

1859 return Meter3(volume=self.a2 * self.b * PI_3 * _4_0) 

1860 

1861 

1862class Ellipsoid2(Ellipsoid): 

1863 '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}. 

1864 ''' 

1865 def __init__(self, a, f, name=NN): 

1866 '''New L{Ellipsoid2}. 

1867 

1868 @arg a: Equatorial radius, semi-axis (C{meter}). 

1869 @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}). 

1870 @kwarg name: Optional, unique name (C{str}). 

1871 

1872 @raise NameError: Ellipsoid with that B{C{name}} already exists. 

1873 

1874 @raise ValueError: Invalid B{C{a}} or B{C{f}}. 

1875 

1876 @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. 

1877 Negative C{B{f}} produces a I{prolate} ellipsoid. 

1878 ''' 

1879 if f is None and isinstance(a, Ellipsoid): 

1880 Ellipsoid.__init__(self, a.a, f =a.f, 

1881 b=a.b, f_=a.f_, name=name) 

1882 else: 

1883 Ellipsoid.__init__(self, a, f=f, name=name) 

1884 

1885 

1886def _spherical_a_b(a, b): 

1887 '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}. 

1888 ''' 

1889 return a < EPS0 or b < EPS0 or fabs(a - b) < EPS0 

1890 

1891 

1892def _spherical_f(f): 

1893 '''(INTERNAL) C{True} for spherical or invalid C{f}. 

1894 ''' 

1895 return fabs(f) < EPS or f > EPS1 

1896 

1897 

1898def _spherical_f_(f_): 

1899 '''(INTERNAL) C{True} for spherical or invalid C{f_}. 

1900 ''' 

1901 return fabs(f_) < EPS or fabs(f_) > _1_EPS 

1902 

1903 

1904def a_b2e(a, b): 

1905 '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius. 

1906 

1907 @arg a: Equatorial radius (C{scalar} > 0). 

1908 @arg b: Polar radius (C{scalar} > 0). 

1909 

1910 @return: The I{unsigned}, (1st) eccentricity (C{float} or C{0}), 

1911 M{sqrt(1 - (b / a)**2)}. 

1912 

1913 @note: The result is always I{non-negative} and C{0} for I{near-spherical} ellipsoids. 

1914 ''' 

1915 return Float(e=sqrt(fabs(a_b2e2(a, b)))) # == sqrt(fabs(a - b) * (a + b)) / a) 

1916 

1917 

1918def a_b2e2(a, b): 

1919 '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius. 

1920 

1921 @arg a: Equatorial radius (C{scalar} > 0). 

1922 @arg b: Polar radius (C{scalar} > 0). 

1923 

1924 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or C{0}), 

1925 M{1 - (b / a)**2}. 

1926 

1927 @note: The result is positive for I{oblate}, negative for I{prolate} 

1928 or C{0} for I{near-spherical} ellipsoids. 

1929 ''' 

1930 return Float(e2=_0_0 if _spherical_a_b(a, b) else ((a - b) * (a + b) / a**2)) 

1931 

1932 

1933def a_b2e22(a, b): 

1934 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius. 

1935 

1936 @arg a: Equatorial radius (C{scalar} > 0). 

1937 @arg b: Polar radius (C{scalar} > 0). 

1938 

1939 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} or C{0}), 

1940 M{(a / b)**2 - 1}. 

1941 

1942 @note: The result is positive for I{oblate}, negative for I{prolate} 

1943 or C{0} for I{near-spherical} ellipsoids. 

1944 ''' 

1945 return Float(e22=_0_0 if _spherical_a_b(a, b) else ((a - b) * (a + b) / b**2)) 

1946 

1947 

1948def a_b2e32(a, b): 

1949 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius. 

1950 

1951 @arg a: Equatorial radius (C{scalar} > 0). 

1952 @arg b: Polar radius (C{scalar} > 0). 

1953 

1954 @return: The I{signed}, 3rd eccentricity I{squared} (C{float} or C{0}), 

1955 M{(a**2 - b**2) / (a**2 + b**2)}. 

1956 

1957 @note: The result is positive for I{oblate}, negative for I{prolate} 

1958 or C{0} for I{near-spherical} ellipsoids. 

1959 ''' 

1960 a2, b2 = a**2, b**2 

1961 return Float(e32=_0_0 if _spherical_a_b(a2, b2) else ((a2 - b2) / (a2 + b2))) 

1962 

1963 

1964def a_b2f(a, b): 

1965 '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius. 

1966 

1967 @arg a: Equatorial radius (C{scalar} > 0). 

1968 @arg b: Polar radius (C{scalar} > 0). 

1969 

1970 @return: The flattening (C{scalar} or C{0}), M{(a - b) / a}. 

1971 

1972 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} 

1973 for I{near-spherical} ellipsoids. 

1974 ''' 

1975 f = 0 if _spherical_a_b(a, b) else ((a - b) / a) 

1976 return _f_0_0 if _spherical_f(f) else Float(f=f) 

1977 

1978 

1979def a_b2f_(a, b): 

1980 '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius. 

1981 

1982 @arg a: Equatorial radius (C{scalar} > 0). 

1983 @arg b: Polar radius (C{scalar} > 0). 

1984 

1985 @return: The inverse flattening (C{scalar} or C{0}), M{a / (a - b)}. 

1986 

1987 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} 

1988 for I{near-spherical} ellipsoids. 

1989 ''' 

1990 f_ = 0 if _spherical_a_b(a, b) else (a / float(a - b)) 

1991 return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) 

1992 

1993 

1994def a_b2f2(a, b): 

1995 '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius. 

1996 

1997 @arg a: Equatorial radius (C{scalar} > 0). 

1998 @arg b: Polar radius (C{scalar} > 0). 

1999 

2000 @return: The I{signed}, 2nd flattening (C{scalar} or C{0}), M{(a - b) / b}. 

2001 

2002 @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} 

2003 for I{near-spherical} ellipsoids. 

2004 ''' 

2005 t = 0 if _spherical_a_b(a, b) else float(a - b) 

2006 return Float(f2=_0_0 if fabs(t) < EPS0 else (t / b)) 

2007 

2008 

2009def a_b2n(a, b): 

2010 '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius. 

2011 

2012 @arg a: Equatorial radius (C{scalar} > 0). 

2013 @arg b: Polar radius (C{scalar} > 0). 

2014 

2015 @return: The I{signed}, 3rd flattening (C{scalar} or C{0}), M{(a - b) / (a + b)}. 

2016 

2017 @note: The result is positive for I{oblate}, negative for I{prolate} 

2018 or C{0} for I{near-spherical} ellipsoids. 

2019 ''' 

2020 t = 0 if _spherical_a_b(a, b) else float(a - b) 

2021 return Float(n=_0_0 if fabs(t) < EPS0 else (t / (a + b))) 

2022 

2023 

2024def a_f2b(a, f): 

2025 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}. 

2026 

2027 @arg a: Equatorial radius (C{scalar} > 0). 

2028 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). 

2029 

2030 @return: The polar radius (C{float}), M{a * (1 - f)}. 

2031 ''' 

2032 b = a if _spherical_f(f) else (a * (_1_0 - f)) 

2033 return Radius_(b=a if _spherical_a_b(a, b) else b) 

2034 

2035 

2036def a_f_2b(a, f_): 

2037 '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}. 

2038 

2039 @arg a: Equatorial radius (C{scalar} > 0). 

2040 @arg f_: Inverse flattening (C{scalar} >>> 1). 

2041 

2042 @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}. 

2043 ''' 

2044 b = a if _spherical_f_(f_) else (a * (f_ - _1_0) / f_) 

2045 return Radius_(b=a if _spherical_a_b(a, b) else b) 

2046 

2047 

2048def b_f2a(b, f): 

2049 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}. 

2050 

2051 @arg b: Polar radius (C{scalar} > 0). 

2052 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). 

2053 

2054 @return: The equatorial radius (C{float}), M{b / (1 - f)}. 

2055 ''' 

2056 t = _1_0 - f 

2057 a = b if fabs(t) < EPS0 else (b / t) 

2058 return Radius_(a=b if _spherical_a_b(a, b) else a) 

2059 

2060 

2061def b_f_2a(b, f_): 

2062 '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}. 

2063 

2064 @arg b: Polar radius (C{scalar} > 0). 

2065 @arg f_: Inverse flattening (C{scalar} >>> 1). 

2066 

2067 @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}. 

2068 ''' 

2069 t = f_ - _1_0 

2070 a = b if _spherical_f_(f_) or fabs(t - f_) < EPS0 \ 

2071 or fabs(t) < EPS0 else (b * f_ / t) 

2072 return Radius_(a=b if _spherical_a_b(a, b) else a) 

2073 

2074 

2075def e2f(e): 

2076 '''Return C{f}, the I{flattening} for a given I{1st eccentricity}. 

2077 

2078 @arg e: The (1st) eccentricity (0 <= C{float} < 1) 

2079 

2080 @return: The flattening (C{scalar} or C{0}). 

2081 

2082 @see: Function L{e22f}. 

2083 ''' 

2084 return e22f(e**2) 

2085 

2086 

2087def e22f(e2): 

2088 '''Return C{f}, the I{flattening} for a given I{1st eccentricity squared}. 

2089 

2090 @arg e2: The (1st) eccentricity I{squared}, I{signed} (L{NINF} < C{float} < 1) 

2091 

2092 @return: The flattening (C{float} or C{0}), M{e2 / (sqrt(e2 - 1) + 1)}. 

2093 ''' 

2094 return Float(f=e2 / (sqrt(_1_0 - e2) + _1_0)) if e2 else _f_0_0 

2095 

2096 

2097def f2e2(f): 

2098 '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}. 

2099 

2100 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). 

2101 

2102 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} < 1), 

2103 M{f * (2 - f)}. 

2104 

2105 @note: The result is positive for I{oblate}, negative for I{prolate} 

2106 or C{0} for I{near-spherical} ellipsoids. 

2107 

2108 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ 

2109 html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening 

2110 <https://WikiPedia.org/wiki/Flattening>}. 

2111 ''' 

2112 return Float(e2=_0_0 if _spherical_f(f) else (f * (_2_0 - f))) 

2113 

2114 

2115def f2e22(f): 

2116 '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}. 

2117 

2118 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). 

2119 

2120 @return: The I{signed}, 2nd eccentricity I{squared} (C{float} > -1 or 

2121 C{INF}), M{f * (2 - f) / (1 - f)**2}. 

2122 

2123 @note: The result is positive for I{oblate}, negative for I{prolate} 

2124 or C{0} for near-spherical ellipsoids. 

2125 

2126 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ 

2127 html/classGeographicLib_1_1Ellipsoid.html>}. 

2128 ''' 

2129 # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2 

2130 t = (_1_0 - f)**2 

2131 return Float(e22=INF if t < EPS0 else (f2e2(f) / t)) # PYCHOK type 

2132 

2133 

2134def f2e32(f): 

2135 '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}. 

2136 

2137 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). 

2138 

2139 @return: The I{signed}, 3rd eccentricity I{squared} (C{float}), 

2140 M{f * (2 - f) / (1 + (1 - f)**2)}. 

2141 

2142 @note: The result is positive for I{oblate}, negative for I{prolate} 

2143 or C{0} for I{near-spherical} ellipsoids. 

2144 

2145 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ 

2146 html/classGeographicLib_1_1Ellipsoid.html>}. 

2147 ''' 

2148 # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2) 

2149 e2 = f2e2(f) 

2150 return Float(e32=e2 / (_2_0 - e2)) 

2151 

2152 

2153def f_2f(f_): 

2154 '''Return C{f}, the I{flattening} for a given I{inverse flattening}. 

2155 

2156 @arg f_: Inverse flattening (C{scalar} >>> 1). 

2157 

2158 @return: The flattening (C{scalar} or C{0}), M{1 / f_}. 

2159 

2160 @note: The result is positive for I{oblate}, negative for I{prolate} 

2161 or C{0} for I{near-spherical} ellipsoids. 

2162 ''' 

2163 f = 0 if _spherical_f_(f_) else _1_0 / f_ 

2164 return _f_0_0 if _spherical_f(f) else Float(f=f) # PYCHOK type 

2165 

2166 

2167def f2f_(f): 

2168 '''Return C{f_}, the I{inverse flattening} for a given I{flattening}. 

2169 

2170 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). 

2171 

2172 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}. 

2173 

2174 @note: The result is positive for I{oblate}, negative for I{prolate} 

2175 or C{0} for I{near-spherical} ellipsoids. 

2176 ''' 

2177 f_ = 0 if _spherical_f(f) else _1_0 / f 

2178 return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) # PYCHOK type 

2179 

2180 

2181def f2f2(f): 

2182 '''Return C{f2}, the I{2nd flattening} for a given I{flattening}. 

2183 

2184 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). 

2185 

2186 @return: The I{signed}, 2nd flattening (C{scalar} or C{INF}), M{f / (1 - f)}. 

2187 

2188 @note: The result is positive for I{oblate}, negative for I{prolate} 

2189 or C{0} for I{near-spherical} ellipsoids. 

2190 

2191 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ 

2192 html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening 

2193 <https://WikiPedia.org/wiki/Flattening>}. 

2194 ''' 

2195 t = _1_0 - f 

2196 return Float(f2=_0_0 if _spherical_f(f) else (INF if fabs(t) < EPS 

2197 else (f / t))) # PYCHOK type 

2198 

2199 

2200def f2n(f): 

2201 '''Return C{n}, the I{3rd flattening} for a given I{flattening}. 

2202 

2203 @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). 

2204 

2205 @return: The I{signed}, 3rd flattening (-1 <= C{float} < 1), 

2206 M{f / (2 - f)}. 

2207 

2208 @note: The result is positive for I{oblate}, negative for I{prolate} 

2209 or C{0} for I{near-spherical} ellipsoids. 

2210 

2211 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ 

2212 html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening 

2213 <https://WikiPedia.org/wiki/Flattening>}. 

2214 ''' 

2215 return Float(n=_0_0 if _spherical_f(f) else (f / float(_2_0 - f))) 

2216 

2217 

2218def n2e2(n): 

2219 '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}. 

2220 

2221 @arg n: The 3rd flattening (-1 <= C{scalar} < 1). 

2222 

2223 @return: The I{signed}, (1st) eccentricity I{squared} (C{float} or NINF), 

2224 M{4 * n / (1 + n)**2}. 

2225 

2226 @note: The result is positive for I{oblate}, negative for I{prolate} 

2227 or C{0} for I{near-spherical} ellipsoids. 

2228 

2229 @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}. 

2230 ''' 

2231 t = (n + _1_0)**2 

2232 return Float(e2=_0_0 if fabs(n) < EPS0 else 

2233 (NINF if t < EPS0 else (_4_0 * n / t))) 

2234 

2235 

2236def n2f(n): 

2237 '''Return C{f}, the I{flattening} for a given I{3rd flattening}. 

2238 

2239 @arg n: The 3rd flattening (-1 <= C{scalar} < 1). 

2240 

2241 @return: The flattening (C{scalar} or NINF), M{2 * n / (1 + n)}. 

2242 

2243 @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ 

2244 html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening 

2245 <https://WikiPedia.org/wiki/Flattening>}. 

2246 ''' 

2247 t = n + _1_0 

2248 f = 0 if fabs(n) < EPS0 else (NINF if t < EPS0 else (_2_0 * n / t)) 

2249 return _f_0_0 if _spherical_f(f) else Float(f=f) 

2250 

2251 

2252def n2f_(n): 

2253 '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}. 

2254 

2255 @arg n: The 3rd flattening (-1 <= C{scalar} < 1). 

2256 

2257 @return: The inverse flattening (C{scalar} or C{0}), M{1 / f}. 

2258 

2259 @see: L{n2f} and L{f2f_}. 

2260 ''' 

2261 return f2f_(n2f(n)) 

2262 

2263 

2264def _normalTo3(px, py, E): # in .height4 above 

2265 '''(INTERNAL) Nearest point on a 2-D ellipse in 1st quadrant. 

2266 

2267 @see: Functions C{.triaxial._normalTo4} and C{-To5}. 

2268 ''' 

2269 a, b = E.a, E.b 

2270 if min(px, py, a, b) < EPS0: 

2271 raise _AssertionError(px=px, py=py, a=a, b=b, E=E) 

2272 

2273 a2 = a - b * E.b_a 

2274 b2 = b - a * E.a_b 

2275 tx = ty = _SQRT2_2 

2276 for i in range(16): # max 5 

2277 ex = a2 * tx**3 

2278 ey = b2 * ty**3 

2279 

2280 qx = px - ex 

2281 qy = py - ey 

2282 q = hypot(qx, qy) 

2283 if q < EPS0: # PYCHOK no cover 

2284 break 

2285 r = hypot(ex - tx * a, ey - ty * b) / q 

2286 

2287 sx, tx = tx, min(_1_0, max(0, (ex + qx * r) / a)) 

2288 sy, ty = ty, min(_1_0, max(0, (ey + qy * r) / b)) 

2289 t = hypot(ty, tx) 

2290 if t < EPS0: # PYCHOK no cover 

2291 break 

2292 tx = tx / t # /= chokes PyChecker 

2293 ty = ty / t 

2294 if fabs(sx - tx) < EPS and \ 

2295 fabs(sy - ty) < EPS: 

2296 break 

2297 

2298 tx *= a / px 

2299 ty *= b / py 

2300 return tx, ty, i # x and y as fractions 

2301 

2302 

2303class Ellipsoids(_NamedEnum): 

2304 '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class 

2305 to accommodate the L{_LazyNamedEnumItem} properties. 

2306 ''' 

2307 def _Lazy(self, a, b, f_, **kwds): 

2308 '''(INTERNAL) Instantiate the L{Ellipsoid}. 

2309 ''' 

2310 return Ellipsoid(a, b=b, f_=f_, **kwds) 

2311 

2312Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton 

2313'''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.''' 

2314# <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html> 

2315# <https://GSSC.ESA.int/navipedia/index.php/Reference_Frames_in_GNSS> 

2316# <https://kb.OSU.edu/dspace/handle/1811/77986> 

2317# <https://www.IBM.com/docs/en/db2/11.5?topic=systems-supported-spheroids> 

2318# <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/LogicalDictionary/StandardValues/ellipsoid.html> 

2319# <https://GitHub.com/locationtech/proj4j/blob/master/src/main/java/org/locationtech/proj4j/datum/Ellipsoid.java> 

2320Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid> 

2321 Airy1830 = _lazy(_Airy1830_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909 

2322 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448 

2323# APL4_9 = _lazy('APL4_9', *_T(6378137.0, _0_0, 298.24985392)), # Appl. Phys. Lab. 1965 

2324# ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid 

2325# AN_SA96 = _lazy('AN_SA96', *_T(6378160.0, _0_0, 298.24985392)), # Australian Nat. South America 

2326 Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192 

2327 ATS1977 = _lazy('ATS1977', *_T(6378135.0, _0_0, 298.257)), # "Average Terrestrial System" 

2328 Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)), 

2329 BesselModified = _lazy('BesselModified', *_T(6377492.018, _0_0, 299.1528128)), 

2330# BesselNamibia = _lazy('BesselNamibia', *_T(6377483.865, _0_0, 299.1528128)), 

2331 CGCS2000 = _lazy('CGCS2000', *_T(R_MA, _0_0, 298.257222101)), # BeiDou Coord System (BDC) 

2332# Clarke1858 = _lazy('Clarke1858', *_T(6378293.639, _0_0, 294.260676369)), 

2333 Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)), 

2334 Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)), 

2335 Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)), 

2336 Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96639549, 293.466307656)), # aka Clarke1880Arc 

2337 CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures 

2338 Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium 

2339 Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)), 

2340# Everest1830 = _lazy('Everest1830', *_T(6377276.345, _0_0, 300.801699997)), 

2341# Everest1948 = _lazy('Everest1948', *_T(6377304.063, _0_0, 300.801699997)), 

2342# Everest1956 = _lazy('Everest1956', *_T(6377301.243, _0_0, 300.801699997)), 

2343 Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)), 

2344 Everest1975 = _lazy('Everest1975', *_T(6377299.151, 6356098.14512013, 300.8017255)), 

2345 Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)), 

2346# Fisher1968Mod = _lazy('Fisher1968Mod', *_T(6378155.0, _0_0, 298.3)), 

2347 GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)), 

2348 GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid" 

2349 GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516 

2350# GRS67Truncated = _lazy('GRS67Truncated', *_T(6378160.0, _0_0, 298.25)), 

2351 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.25722210088)), # IUGG, ITRS, ETRS89 

2352# Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, None)), # aka Intl1924 f_=297 

2353 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)), 

2354# Hough1960 = _lazy('Hough1960', *_T(6378270.0, _0_0, 297.0)), 

2355 IAU76 = _lazy('IAU76', *_T(6378140.0, _0_0, 298.257)), # Int'l Astronomical Union 

2356 IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302 

2357 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy 

2358 IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)), 

2359 Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795 

2360 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l 

2361 Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling 

2362 Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling 

2363# Kaula = _lazy('Kaula', *_T(6378163.0, _0_0, 298.24)), # Kaula 1961 

2364# Lerch = _lazy('Lerch', *_T(6378139.0, _0_0, 298.257)), # Lerch 1979 

2365 Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France 

2366 Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)), 

2367 Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)), 

2368# MERIT = _lazy('MERIT', *_T(6378137.0, _0_0, 298.257)), # MERIT 1983 

2369# NWL10D = _lazy('NWL10D', *_T(6378135.0, _0_0, 298.26)), # Naval Weapons Lab. 

2370 NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab. 

2371# NWL9D = _lazy('NWL9D', *_T(6378145.0, 6356759.76948868, 298.25)), # NWL1965 

2372 OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)), 

2373 OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)), 

2374# Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect? 

2375 Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972 

2376 PZ90 = _lazy('PZ90', *_T(6378136.0, _0_0, 298.257839303)), # GLOSNASS PZ-90 and PZ-90.11 

2377# SEAsia = _lazy('SEAsia', *_T(6378155.0, _0_0, 298.3)), # SouthEast Asia 

2378 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System 

2379 SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American 

2380 Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)), 

2381# Walbeck = _lazy('Walbeck', *_T(6376896.0, _0_0, 302.78)), 

2382# WarOffice = _lazy('WarOffice', *_T(6378300.0, _0_0, 296.0)), 

2383 WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)), 

2384 WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)), 

2385 WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52 

2386 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f__WGS84)), # GPS b=6356752.3142451793 

2387# Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)), 

2388 Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo 

2389 SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo 

2390 SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)) # EPSG:3857 Spheroid 

2391) 

2392 

2393_EWGS84 = Ellipsoids.WGS84 # (INTERNAL) shared 

2394 

2395if __name__ == '__main__': 

2396 

2397 from pygeodesy.interns import _COMMA_, _NL_, _NLATvar_ 

2398 from pygeodesy import nameof, printf 

2399 

2400 for E in (_EWGS84, Ellipsoids.GRS80, # NAD83, 

2401 Ellipsoids.Sphere, Ellipsoids.SpherePopular, 

2402 Ellipsoid(_EWGS84.b, _EWGS84.a, name='_Prolate')): 

2403 e = f2n(E.f) - E.n 

2404 printf('# %s: %s', _DOT_('Ellipsoids', E.name), E.toStr(prec=10), nl=1) 

2405 printf('# e=%s, f_=%s, f=%s, n=%s (%s)', fstr(E.e, prec=13, fmt=Fmt.e), 

2406 fstr(E.f_, prec=13, fmt=Fmt.e), 

2407 fstr(E.f, prec=13, fmt=Fmt.e), 

2408 fstr(E.n, prec=13, fmt=Fmt.e), 

2409 fstr(e, prec=9, fmt=Fmt.e)) 

2410 printf('# %s %s', Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20)) 

2411 printf('# %s %s', Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20)) 

2412 printf('# %s %s', nameof(Ellipsoid.KsOrder), E.KsOrder) # property 

2413 

2414 # __doc__ of this file, force all into registry 

2415 t = [NN] + Ellipsoids.toRepr(all=True, asorted=True).split(_NL_) 

2416 printf(_NLATvar_.join(i.strip(_COMMA_) for i in t)) 

2417 

2418# % python3 -m pygeodesy.ellipsoids 

2419 

2420# 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=10001965.7293127216, R1=6371008.7714150595, R2=6371007.1809184738, R3=6371000.7900091587, Rbiaxial=6367453.6345163295, Rtriaxial=6372797.5559594007 

2421# e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00) 

2422# AlphaKs 0.00083773182062446994, 0.00000076085277735725, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0 

2423# BetaKs 0.00083773216405794875, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0 

2424# KsOrder 8 

2425 

2426# 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=10001965.7292304579, R1=6371008.7713801153, R2=6371007.1808835147, R3=6371000.7899741363, Rbiaxial=6367453.6344640013, Rtriaxial=6372797.5559332585 

2427# e=8.1819191042833e-02, f_=2.9825722210088e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00) 

2428# AlphaKs 0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0 

2429# BetaKs 0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0 

2430# KsOrder 8 

2431 

2432# 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 

2433# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00) 

2434# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 

2435# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 

2436# KsOrder 8 

2437 

2438# 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 

2439# e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00) 

2440# AlphaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 

2441# BetaKs 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 

2442# KsOrder 8 

2443 

2444# 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=10035500.5204500332, R1=6363880.5428301189, R2=6363878.9413582645, R3=6363872.5644020075, Rbiaxial=6367453.6345163295, Rtriaxial=6362105.2243882557 

2445# e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00) 

2446# AlphaKs -0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0 

2447# BetaKs -0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0 

2448# KsOrder 8 

2449 

2450# **) MIT License 

2451# 

2452# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved. 

2453# 

2454# Permission is hereby granted, free of charge, to any person obtaining a 

2455# copy of this software and associated documentation files (the "Software"), 

2456# to deal in the Software without restriction, including without limitation 

2457# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

2458# and/or sell copies of the Software, and to permit persons to whom the 

2459# Software is furnished to do so, subject to the following conditions: 

2460# 

2461# The above copyright notice and this permission notice shall be included 

2462# in all copies or substantial portions of the Software. 

2463# 

2464# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

2465# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

2466# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

2467# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

2468# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

2469# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

2470# OTHER DEALINGS IN THE SOFTWARE.