# Coverage for pygeodesy/ellipsoids.py : 95%

## 612 statements 581 run 31 missing 36 excluded

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1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 | # -*- coding: utf-8 -*- u'''Classes L{a_f2Tuple}, L{Ellipsoid} and L{Ellipsoid2}, an L{Ellipsoids} registry and a dozen functions to convert I{equatorial} radius, I{polar} radius, I{eccentricities}, I{flattenings} and I{inverse flattening}. See module L{datums} for more information and other details. @newfield example: Example, Examples @var Ellipsoids.Airy1830: Ellipsoid(name='Airy1830', a=6377563.396, b=6356256.90923729, f_=299.3249646, f=0.00334085, f2=0.00335205, n=0.00167322, e=0.08167337, e2=0.00667054, e22=0.00671533, e32=0.00334643, A=6366914.60892522, L=10001126.0807165, R1=6370461.23374576, R2=6370459.65470808, R3=6370453.30994572) @var Ellipsoids.AiryModified: Ellipsoid(name='AiryModified', a=6377340.189, b=6356034.44793853, f_=299.3249646, f=0.00334085, f2=0.00335205, n=0.00167322, e=0.08167337, e2=0.00667054, e22=0.00671533, e32=0.00334643, A=6366691.77461988, L=10000776.05340819, R1=6370238.27531284, R2=6370236.69633043, R3=6370230.35179012) @var Ellipsoids.Australia1966: Ellipsoid(name='Australia1966', a=6378160, b=6356774.71919531, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367471.84853228, L=10002001.39064442, R1=6371031.5730651, R2=6371029.9824858, R3=6371023.59124343) @var Ellipsoids.Bessel1841: Ellipsoid(name='Bessel1841', a=6377397.155, b=6356078.962818, f_=299.1528128, f=0.00334277, f2=0.00335398, n=0.00167418, e=0.08169683, e2=0.00667437, e22=0.00671922, e32=0.00334836, A=6366742.52023395, L=10000855.76443237, R1=6370291.09093933, R2=6370289.51012659, R3=6370283.15821522) @var Ellipsoids.CPM1799: Ellipsoid(name='CPM1799', a=6375738.7, b=6356671.92557493, f_=334.39, f=0.00299052, f2=0.00299949, n=0.0014975, e=0.07727934, e2=0.0059721, e22=0.00600798, e32=0.00299499, A=6366208.88184784, L=10000017.52721564, R1=6369383.10852498, R2=6369381.8434158, R3=6369376.76247021) @var Ellipsoids.Clarke1866: Ellipsoid(name='Clarke1866', a=6378206.4, b=6356583.8, f_=294.97869821, f=0.00339008, f2=0.00340161, n=0.00169792, e=0.08227185, e2=0.00676866, e22=0.00681478, e32=0.00339582, A=6367399.68916978, L=10001888.04298286, R1=6370998.86666667, R2=6370997.240633, R3=6370990.70659881) @var Ellipsoids.Clarke1880: Ellipsoid(name='Clarke1880', a=6378249.145, b=6356514.86954978, f_=293.465, f=0.00340756, f2=0.00341921, n=0.00170669, e=0.0824834, e2=0.00680351, e22=0.00685012, e32=0.00341337, A=6367386.64398051, L=10001867.55164747, R1=6371004.38651659, R2=6371002.74366963, R3=6370996.1419165) @var Ellipsoids.Clarke1880IGN: Ellipsoid(name='Clarke1880IGN', a=6378249.2, b=6356515, f_=293.46602129, f=0.00340755, f2=0.0034192, n=0.00170668, e=0.08248326, e2=0.00680349, e22=0.00685009, e32=0.00341336, A=6367386.73667336, L=10001867.69724907, R1=6371004.46666667, R2=6371002.82383112, R3=6370996.22212394) @var Ellipsoids.Clarke1880Mod: Ellipsoid(name='Clarke1880Mod', a=6378249.145, b=6356514.96582849, f_=293.4663, f=0.00340755, f2=0.0034192, n=0.00170668, e=0.08248322, e2=0.00680348, e22=0.00685009, e32=0.00341335, A=6367386.69207875, L=10001867.62720001, R1=6371004.4186095, R2=6371002.77577708, R3=6370996.17408252) @var Ellipsoids.Delambre1810: Ellipsoid(name='Delambre1810', a=6376428, b=6355957.92616372, f_=311.5, f=0.00321027, f2=0.00322061, n=0.00160772, e=0.08006397, e2=0.00641024, e22=0.0064516, e32=0.00321543, A=6366197.07684334, L=9999998.98395793, R1=6369604.64205457, R2=6369603.18419749, R3=6369597.32739068) @var Ellipsoids.Engelis1985: Ellipsoid(name='Engelis1985', a=6378136.05, b=6356751.32272154, f_=298.2566, f=0.00335282, f2=0.0033641, n=0.00167922, e=0.08181928, e2=0.00669439, e22=0.00673951, e32=0.00335844, A=6367448.17507971, L=10001964.20447208, R1=6371007.80757385, R2=6371006.21707085, R3=6370999.82613572) @var Ellipsoids.Everest1969: Ellipsoid(name='Everest1969', a=6377295.664, b=6356094.667915, f_=300.8017, f=0.00332445, f2=0.00333554, n=0.00166499, e=0.08147298, e2=0.00663785, e22=0.0066822, e32=0.00332998, A=6366699.57839501, L=10000788.3115495, R1=6370228.665305, R2=6370227.10178537, R3=6370220.81951617) @var Ellipsoids.Fisher1968: Ellipsoid(name='Fisher1968', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367463.65604381, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995034) @var Ellipsoids.GEM10C: Ellipsoid(name='GEM10C', a=6378137, b=6356752.31424783, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14582474, L=10001965.7293148, R1=6371008.77141594, R2=6371007.18091936, R3=6371000.79001004) @var Ellipsoids.GRS67: Ellipsoid(name='GRS67', a=6378160, b=6356774.51609071, f_=298.24716743, f=0.00335292, f2=0.0033642, n=0.00167928, e=0.08182057, e2=0.00669461, e22=0.00673973, e32=0.00335854, A=6367471.74706533, L=10002001.2312605, R1=6371031.50536357, R2=6371029.91475409, R3=6371023.52339014) @var Ellipsoids.GRS80: Ellipsoid(name='GRS80', a=6378137, b=6356752.31414035, f_=298.2572221, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14577104, L=10001965.72923046, R1=6371008.77138012, R2=6371007.18088351, R3=6371000.78997413) @var Ellipsoids.Helmert1906: Ellipsoid(name='Helmert1906', a=6378200, b=6356818.16962789, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367513.57227074, L=10002066.93013953, R1=6371072.7232093, R2=6371071.13315272, R3=6371064.74401563) @var Ellipsoids.IERS1989: Ellipsoid(name='IERS1989', a=6378136, b=6356751.30156878, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.13949125, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669088, R3=6370999.78577296) @var Ellipsoids.IERS1992TOPEX: Ellipsoid(name='IERS1992TOPEX', a=6378136.3, b=6356751.61659215, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.44699641, L=10001964.63159783, R1=6371008.07219738, R2=6371006.48170097, R3=6371000.09079235) @var Ellipsoids.IERS2003: Ellipsoid(name='IERS2003', a=6378136.6, b=6356751.85797165, f_=298.25642, f=0.00335282, f2=0.0033641, n=0.00167922, e=0.0818193, e2=0.0066944, e22=0.00673951, e32=0.00335844, A=6367448.71771058, L=10001965.05683465, R1=6371008.35265722, R2=6371006.76215217, R3=6371000.37120876) @var Ellipsoids.Intl1924: Ellipsoid(name='Intl1924', a=6378388, b=6356911.94612795, f_=297, f=0.003367, f2=0.00337838, n=0.00168634, e=0.08199189, e2=0.00672267, e22=0.00676817, e32=0.00337267, A=6367654.50005758, L=10002288.29898944, R1=6371229.31537598, R2=6371227.71133444, R3=6371221.26587487) @var Ellipsoids.Intl1967: Ellipsoid(name='Intl1967', a=6378157.5, b=6356772.2, f_=298.24961539, f=0.0033529, f2=0.00336418, n=0.00167926, e=0.08182023, e2=0.00669455, e22=0.00673967, e32=0.00335852, A=6367469.33894446, L=10001997.44859308, R1=6371029.06666667, R2=6371027.47608389, R3=6371021.08482752) @var Ellipsoids.Krassovski1940: Ellipsoid(name='Krassovski1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367558.49687498, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439) @var Ellipsoids.Krassowsky1940: Ellipsoid(name='Krassowsky1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367558.49687498, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439) @var Ellipsoids.Maupertuis1738: Ellipsoid(name='Maupertuis1738', a=6397300, b=6363806.28272251, f_=191, f=0.0052356, f2=0.00526316, n=0.00262467, e=0.10219488, e2=0.01044379, e22=0.01055402, e32=0.00524931, A=6380564.13011837, L=10022566.69846922, R1=6386135.42757417, R2=6386131.54144847, R3=6386115.88628229) @var Ellipsoids.Mercury1960: Ellipsoid(name='Mercury1960', a=6378166, b=6356784.28360711, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367479.62923643, L=10002013.61254591, R1=6371038.76120237, R2=6371037.17115427, R3=6371030.78205124) @var Ellipsoids.Mercury1968Mod: Ellipsoid(name='Mercury1968Mod', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367463.65604381, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995034) @var Ellipsoids.NWL1965: Ellipsoid(name='NWL1965', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802666) @var Ellipsoids.OSU86F: Ellipsoid(name='OSU86F', a=6378136.2, b=6356751.51693008, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.3471653, L=10001964.47478349, R1=6371007.97231003, R2=6371006.38181364, R3=6370999.99090512) @var Ellipsoids.OSU91A: Ellipsoid(name='OSU91A', a=6378136.3, b=6356751.6165948, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.44699773, L=10001964.6315999, R1=6371008.07219827, R2=6371006.48170186, R3=6371000.09079324) @var Ellipsoids.Plessis1817: Ellipsoid(name='Plessis1817', a=6376523, b=6355862.93325557, f_=308.64, f=0.00324002, f2=0.00325055, n=0.00162264, e=0.08043347, e2=0.00646954, e22=0.00651167, e32=0.00324527, A=6366197.15710739, L=9999999.11003639, R1=6369636.31108519, R2=6369634.82608583, R3=6369628.85999667) @var Ellipsoids.SGS85: Ellipsoid(name='SGS85', a=6378136, b=6356751.30156878, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.13949125, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669087, R3=6370999.78577296) @var Ellipsoids.SoAmerican1969: Ellipsoid(name='SoAmerican1969', a=6378160, b=6356774.71919531, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367471.84853228, L=10002001.39064442, R1=6371031.5730651, R2=6371029.98248581, R3=6371023.59124343) @var Ellipsoids.Sphere: Ellipsoid(name='Sphere', a=6371008.771415, b=6371008.771415, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6371008.771415, L=10007557.17611675, R1=6371008.771415, R2=6371008.771415, R3=6371008.771415) @var Ellipsoids.SphereAuthalic: Ellipsoid(name='SphereAuthalic', a=6371000, b=6371000, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6371000, L=10007543.39801029, R1=6371000, R2=6371000, R3=6371000) @var Ellipsoids.SpherePopular: Ellipsoid(name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6378137, L=10018754.17139462, R1=6378137, R2=6378137, R3=6378137) @var Ellipsoids.Struve1860: Ellipsoid(name='Struve1860', a=6378298.3, b=6356657.14266956, f_=294.73, f=0.00339294, f2=0.00340449, n=0.00169935, e=0.0823065, e2=0.00677436, e22=0.00682056, e32=0.00339869, A=6367482.31832549, L=10002017.83655714, R1=6371084.58088985, R2=6371082.95208988, R3=6371076.40691418) @var Ellipsoids.WGS60: Ellipsoid(name='WGS60', a=6378165, b=6356783.28695944, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367478.63091189, L=10002012.0443814, R1=6371037.76231981, R2=6371036.17227197, R3=6371029.78316993) @var Ellipsoids.WGS66: Ellipsoid(name='WGS66', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802666) @var Ellipsoids.WGS72: Ellipsoid(name='WGS72', a=6378135, b=6356750.52001609, f_=298.26, f=0.00335278, f2=0.00336406, n=0.0016792, e=0.08181881, e2=0.00669432, e22=0.00673943, e32=0.0033584, A=6367447.24862383, L=10001962.74919858, R1=6371006.84000536, R2=6371005.24953886, R3=6370998.85875069) @var Ellipsoids.WGS84: Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000915) ''' # make sure int/int division yields float quotient, see .basics from __future__ import division from pygeodesy.basics import copysign, isfinite, isint, _xinstanceof from pygeodesy.errors import _AssertionError, _ValueError from pygeodesy.fmath import cbrt, cbrt2, fdot, fhorner, fpowers, Fsum, fsum_, \ hypot1, hypot2, sqrt3 from pygeodesy.interns import EPS, EPS0, EPS1, INF, NN, PI4, PI_2, R_M, _a_, _A_, \ _Airy1830_, _AiryModified_, _Bessel1841_, _Clarke1866_, \ _Clarke1880IGN_, _DOT_, _e_, _1_EPS, _EPStol as _TOL, \ _EPS0__2, _f_, _finite_, _float as _F, _floatuple as _T, \ _GRS80_, _height_, _Intl1924_, _Krassovski1940_, _vs_, \ _Krassowsky1940_, _lat_, _meridional_, _n_, _negative_, \ _not_, _prime_vertical_, _radius_, _Sphere_, _SPACE_, \ _WGS72_, _WGS84_, _0_0, _0_5, _1_0, _2_0, _4_0, _90_0 from pygeodesy.lazily import _ALL_LAZY from pygeodesy.named import _lazyNamedEnumItem as _lazy, _NamedEnum, \ _NamedEnumItem, _NamedTuple, _Pass from pygeodesy.namedTuples import Distance2Tuple from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_ from pygeodesy.streprs import Fmt, fstr, instr, strs, unstr from pygeodesy.units import Bearing_, Distance, Float, Float_, Height, Lam_, Lat, \ Meter, Meter2, Meter3, Phi, Phi_, Radius, Radius_, Scalar from pygeodesy.utily import atand, atan2b, atan2d, degrees90, m2km, m2NM, m2SM, \ m2radians, radians2m, sincos2d from math import asinh, atan, atanh, cos, degrees, exp, hypot, radians, \ sin, sinh, sqrt, tan R_M = Radius(R_M =R_M) # mean (spherical) earth radius (C{meter}) R_MA = Radius(R_MA=_F(6378137.0)) # equatorial earth radius (C{meter}), WGS84, EPSG:3785 R_MB = Radius(R_MB=_F(6356752.3)) # polar earth radius (C{meter}), WGS84, EPSG:3785 R_KM = Radius(R_KM=_F(m2km(R_M))) # mean (spherical) earth radius (C{KM}, kilo meter) R_NM = Radius(R_NM=_F(m2NM(R_M))) # mean (spherical) earth radius (C{NM}, nautical miles) R_SM = Radius(R_SM=_F(m2SM(R_M))) # mean (spherical) earth radius (C{SM}, statute miles) # See <https://www.EdWilliams.org/avform.htm>, # <https://www.DTIC.mil/dtic/tr/fulltext/u2/a216843.pdf> and # <https://GitHub.com/NASA/MultiDop/blob/master/src/share/man/man3/geog_lib.3> # based on International Standard Nautical Mile of 1,852 meter (1' latitude) R_FM = Radius(R_FM=_F(6371000.0)) # former FAI Sphere earth radius (C{meter}) R_GM = Radius(R_GM=_F(6371230.0)) # Avg. radius, distance to geoid surface (C{meter}) R_VM = Radius(R_VM=_F(6366707.0194937)) # Aviation/Navigation earth radius (C{meter}) # R_ = Radius(R_ =_F(6372797.560856)) # XXX some other earth radius??? __all__ = _ALL_LAZY.ellipsoids __version__ = '21.02.19' _f_0_0 = Float(f =_0_0) _f__0_0 = Float(f_=_0_0) def _aux(lat, inverse, auxLat, clip=90): '''Return named auxiliary latitude in C{degrees}. ''' return Lat(lat, clip=clip, name=_lat_ if inverse else auxLat) def _4Ecef(this, Ecef): '''Return an ECEF converter. ''' from pygeodesy.ecef import EcefKarney, EcefVeness, EcefYou if Ecef is None: Ecef = EcefKarney else: _xinstanceof(EcefKarney, EcefVeness, EcefYou, Ecef=Ecef) return Ecef(this, name=this.name) # datum or ellipsoid def _s2_c2(phi): '''(INTERNAL) Return 2-tuple C{(sin(B{phi})**2, cos(B{phi})**2)}. ''' if phi: s2 = sin(phi)**2 if s2 > 0: c2 = _1_0 - s2 if c2 > 0: if c2 < _1_0: return s2, c2 else: return _1_0, _0_0 return _0_0, _1_0 class a_f2Tuple(_NamedTuple): '''2-Tuple C{(a, f)} specifying an ellipsoid by I{equatorial} radius C{a} in C{meter} and scalar I{flattening} C{f}. @see: Class L{Ellipsoid2}. ''' _Names_ = (_a_, _f_) # 'f' not 'f_' _Units_ = (_Pass, _Pass) def __new__(cls, a, f, **name): '''New L{a_f2Tuple} ellipsoid specification. @arg a: Equatorial radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: An L{a_f2Tuple}C{(a, f)} instance. @raise UnitError: Invalid B{C{a}} or B{C{f}}. @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. Negative C{B{f}} produces a I{prolate} ellipsoid. ''' a = Radius_(a=a) f = Float_( f=f, low=None, high=EPS1) if abs(f) < EPS: # force spherical f = _f_0_0 return _NamedTuple.__new__(cls, a, f, **name) @Property_RO def b(self): '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}. ''' return a_f2b(self.a, self.f) # PYCHOK .a and .f @Property_RO def f_(self): '''Get the I{inverse} flattening (C{float}), M{1 / f} == M{a / (a - b)}. ''' return f2f_(self.f) # PYCHOK .f class Circle4Tuple(_NamedTuple): '''4-Tuple C{(radius, height, lat, beta)} of the C{radius} and C{height}, both conventionally in C{meter} of a parallel I{circle of latitude} at (geodetic) latitude C{lat} and the I{parametric (or reduced) auxiliary latitude} C{beta}, both in C{degrees90}. The C{height} is the (signed) distance along the z-axis between the parallel and the equator. At near-polar C{lat}s, the C{radius} is C{0}, the C{height} is the ellipsoid's (signed) polar radius and C{beta} equals C{lat}. ''' _Names_ = (_radius_, _height_, _lat_, 'beta') _Units_ = ( Radius, Height, Lat, Lat) class Curvature2Tuple(_NamedTuple): '''2-Tuple C{(meridional, prime_vertical)} of radii of curvature, both in C{meter}, conventionally. ''' _Names_ = (_meridional_, _prime_vertical_) _Units_ = ( Meter, Meter) class Ellipsoid(_NamedEnumItem): '''Ellipsoid with I{equatorial} and I{polar} radius, I{flattening}, I{inverse flattening} and other, often used, cached attributes, supporting I{spherical} and I{oblate} and I{prolate} ellipsoidal models. ''' _a = 0 # equatorial radius, semi-axis (C{meter}) _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f _f = 0 # (1st) flattening: (a - b) / a _f_ = 0 # inverse flattening: 1 / f = a / (a - b) _KsOrder = 8 # Krüger series order (4, 6 or 8) _Math = None # cached karney._wrapped.Math module def __init__(self, a, b=None, f_=None, name=NN): '''New L{Ellipsoid} from I{equatorial} and I{polar} radius or I{equatorial} radius and I{inverse flattening}. @arg a: Equatorial radius, semi-axis (C{meter}). @arg b: Optional, polar radius, semi-axis (C{meter}). @arg f_: Inverse flattening: M{a / (a - b)} (C{float} >>> 1.0). @kwarg name: Optional, unique name (C{str}). @raise NameError: Ellipsoid with that B{C{name}} already exists. @raise ValueError: Invalid B{C{a}}, B{C{b}} or B{C{f_}}. @note: M{abs(f_) > 1 / EPS} or M{abs(1 / f_) < EPS} is forced to M{1 / f_ = 0}, spherical. ''' try: a = Radius_(a=a) # low=EPS if not isfinite(a): raise ValueError(_not_(_finite_)) if b: b = Radius_(b=b) # low=EPS f = a_b2f(a, b) f_ = f2f_(f) if f_ is None else Float(f_=f_) elif f_: b = a_f_2b(a, f_) # a * (f_ - 1) / f_ f = a_b2f(a, b) f_ = Float(f_=f_) else: # only a, spherical f = f_ = 0 b = a # superfluous if not isfinite(b): raise ValueError(_not_(_finite_)) if abs(f) < EPS or a == b or not f_: # spherical b = a f = _f_0_0 f_ = _f__0_0 elif f > EPS1: # sanity check, see .ecef.Ecef.__init__ raise _ValueError(f=f) except (TypeError, ValueError) as x: t = instr(self, a=a, b=b, f_=f_, name=name) raise _ValueError(t, txt=str(x)) self._a = a self._b = b self._f = f self._f_ = f_ self._register(Ellipsoids, name) if f and f_: # see .test/testEllipsoidal.py self._assert(_1_0 / f, f_=f_, eps=_TOL) self._assert(_1_0 / f_, f =f, eps=_TOL) self._assert(self.b2_a2, e12=self.e12, eps=EPS) def __eq__(self, other): '''Compare this and an other ellipsoid. @arg other: The other ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). @return: C{True} if equal, C{False} otherwise. ''' return self is other or (isinstance(other, Ellipsoid) and self.a == other.a and (self.b == other.b or self.f == other.f)) @Property_RO def a(self): '''Get the I{equatorial} radius, semi-axis (C{meter}). ''' return self._a equatoradius = a # = Requatorial @Property_RO def a2(self): '''Get the I{equatorial} radius I{squared} (C{float}), M{a**2}. ''' return Meter2(a2=self.a**2) @Property_RO def a2_(self): '''Get the inverse of the I{equatorial} radius I{squared} (C{float}), M{1 / a**2}. ''' return Float(a2_=_1_0 / self.a2) @Property_RO def a_b(self): '''Get the ratio I{equatorial} over I{polar} radius (C{float}), M{a / b} == M{1 / (1 - f)}. ''' return Float(a_b=self.a / self.b if self.f else _1_0) @Property_RO def a2_b(self): '''Get the I{polar} meridional radius of curvature (C{meter}), M{a**2 / b}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} and U{Moritz, H. (1980), Geodetic Reference System 1980 <https://WikiPedia.org/wiki/Earth_radius#cite_note-Moritz-2>}. @note: Symbol C{c} is used by IUGG and IERS for the U{polar radius of curvature<https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar @Property_RO def a2_b2(self): '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}), M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / e12}. ''' return Float(a2_b2=self.a_b**2 if self.f else _1_0) @Property_RO def a_f(self): '''Get the I{equatorial} radius and I{flattening} (L{a_f2Tuple}). ''' return a_f2Tuple(self.a, self.f, name=self.name) @Property_RO def A(self): '''Get the UTM I{meridional (or rectifying)} radius (C{meter}). ''' A, n = self.a, self.n if n: n1 = _1_0 + n if n1: # use 6 n**2 terms, half-way between the _KsOrder's 4, 6, 8 # <https://GeographicLib.SourceForge.io/html/tmseries30.html> # <https://GeographicLib.SourceForge.io/html/transversemercator.html> and # <https://www.MyGeodesy.id.AU/documents/Karney-Krueger%20equations.pdf> (3) A = Radius(A=A / n1 * (fhorner(n**2, 1048576, 262144, 16384, 4096, 1600, 784, 441) / 1048576)) return A @Property_RO def _albersCyl(self): '''(INTERNAL) Helper for C{auxAuthalic}. ''' from pygeodesy.albers import AlbersEqualAreaCylindrical return AlbersEqualAreaCylindrical(datum=self, name=self.name) @Property_RO def AlphaKs(self): '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/html/tmseries30.html>} (L{KsOrder}C{-tuple}). ''' return self._Kseries( # XXX int/int quotients may require from __future__ import division # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200), _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned _T(1424729850961/743921418240)) # PYCHOK unaligned @Property_RO def area(self): '''Get the ellipsoid's surface area (C{meter**2}), M{4 * PI * R2**2}. @see: L{c2} and L{R2}. ''' return Meter2(area=self.c2 * PI4) def _assert(self, val, eps=_TOL, f0=_0_0, **name_value): '''(INTERNAL) Assert a C{name=value} vs C{val}. ''' for n, v in name_value.items(): if abs(v - val) > eps: t = (v, _vs_, val) t = _SPACE_.join(strs(t, prec=12, fmt=Fmt.g)) t = Fmt.EQUAL(self._DOT_(n), t) raise _AssertionError(t, txt=Fmt.exceeds_eps(eps)) return Float(v if self.f else f0, name=n) raise _AssertionError(unstr(self._DOT_(self._assert.__name__), val, eps=eps, f0=f0, **name_value)) def auxAuthalic(self, lat, inverse=False): '''Compute the I{authalic} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{authalic}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{authalic} and return the geodetic latitude (C{bool}). @return: The I{authalic} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/AuthalicLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic latitude <https://WikiPedia.org/wiki/Latitude#Authalic_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 16. ''' if self.f: f = self._albersCyl._tanf if inverse else self._albersCyl._txif # PYCHOK attr lat = atand(f(tan(Phi_(lat)))) # PYCHOK attr return _aux(lat, inverse, Ellipsoid.auxAuthalic.__name__) def auxConformal(self, lat, inverse=False): '''Compute the I{conformal} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{conformal}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{conformal} and return the geodetic latitude (C{bool}). @return: The I{conformal} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/ConformalLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Conformal latitude <https://WikiPedia.org/wiki/Latitude#Conformal_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16. ''' if self.f: f = self.es_tauf if inverse else self.es_taupf # PYCHOK attr lat = atand(f(tan(Phi_(lat)))) # PYCHOK attr return _aux(lat, inverse, Ellipsoid.auxConformal.__name__) def auxGeocentric(self, lat, inverse=False): '''Compute the I{geocentric} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{geocentric}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the geocentric and return the I{geocentric} latitude (C{bool}). @return: The I{geocentric} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/GeocentricLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Geocentric latitude <https://WikiPedia.org/wiki/Latitude#Geocentric_latitude>}, and U{Snyder<<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 17-18. ''' if self.f: f = self.a2_b2 if inverse else self.b2_a2 lat = atand(f * tan(Phi_(lat))) return _aux(lat, inverse, Ellipsoid.auxGeocentric.__name__) def auxIsometric(self, lat, inverse=False): '''Compute the I{isometric} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{isometric}) latitude (C{degrees}). @kwarg inverse: If C{True}, B{C{lat}} is the I{isometric} and return the geodetic latitude (C{bool}). @return: The I{isometric} (or geodetic) latitude in C{degrees}. @note: The I{isometric} latitude for geodetic C{+/-90} is far outside the C{[-90..+90]} range but the inverse thereof is the original geodetic latitude. @see: U{Inverse-/IsometricLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Isometric latitude <https://WikiPedia.org/wiki/Latitude#Isometric_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 15-16. ''' if self.f: r = Phi_(lat, clip=0) lat = degrees( atan(self.es_tauf(sinh(r))) if inverse else asinh(self.es_taupf(tan(r)))) # clip=0, since auxIsometric(+/-90) is far outside [-90..+90] return _aux(lat, inverse, Ellipsoid.auxIsometric.__name__, clip=0) def auxParametric(self, lat, inverse=False): '''Compute the I{parametric} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{parametric}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{parametric} and return the geodetic latitude (C{bool}). @return: The I{parametric} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/ParametricLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Parametric latitude <https://WikiPedia.org/wiki/Latitude#Parametric_(or_reduced)_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, p 18. ''' if self.f: lat = self._beta(Lat(lat), inverse=inverse) return _aux(lat, inverse, Ellipsoid.auxParametric.__name__) auxReduced = auxParametric # synonyms def auxRectifying(self, lat, inverse=False): '''Compute the I{rectifying} auxiliary latitude or inverse thereof. @arg lat: The geodetic (or I{rectifying}) latitude (C{degrees90}). @kwarg inverse: If C{True}, B{C{lat}} is the I{rectifying} and return the geodetic latitude (C{bool}). @return: The I{rectifying} (or geodetic) latitude in C{degrees90}. @see: U{Inverse-/RectifyingLatitude<https://geographiclib.sourceforge.io/ html/classGeographicLib_1_1Ellipsoid.html>}, U{Rectifying latitude <https://WikiPedia.org/wiki/Latitude#Rectifying_latitude>}, and U{Snyder<https://Pubs.USGS.gov/pp/1395/report.pdf>}, pp 16-17. ''' if self.f: lat = Lat(lat) if 0 < abs(lat) < _90_0: if inverse: e = self._elliptic_e22 lat = degrees90(e.fEinv(e.cE * lat / _90_0)) lat = self.auxParametric(lat, inverse=True) else: lat = _90_0 * self.Llat(lat) / self.L return _aux(lat, inverse, Ellipsoid.auxRectifying.__name__) @Property_RO def b(self): '''Get the I{polar} radius, semi-axis (C{meter}). ''' return self._b polaradius = b # = Rpolar @Property_RO def b_a(self): '''Get the ratio I{polar} over I{equatorial} radius (C{float}), M{b / a} == M{1 - f}. ''' return self._assert(self.b / self.a, b_a=_1_0 - self.f, f0=_1_0) @Property_RO def b2(self): '''Get the I{polar} radius I{squared} (C{float}), M{b**2}. ''' return Meter2(b2=self.b**2) @Property_RO def b2_a(self): '''Get the I{equatorial} meridional radius of curvature (C{meter}), M{b**2 / a}, see L{rocMeridional}C{(0)}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return Radius(b2_a=self.b2 / self.a if self.f else self.b) @Property_RO def b2_a2(self): '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2} == M{(1 - f)**2} == M{1 - e**2} == L{e12}. ''' return Float(b2_a2=self.b_a**2 if self.f else _1_0) def _beta(self, lat, inverse=False): '''(INTERNAL) Get the I{parametric (or reduced) auxiliary latitude} or inverse thereof. ''' s, c = sincos2d(lat) # like Karney's tand(lat) _f_1 = self.a_b if inverse else self.b_a return atan2d(_f_1 * s, c) # == atand(_f_1 * s / c) if c else copysign(_90_0, lat) @Property_RO def BetaKs(self): '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/html/tmseries30.html>} (L{KsOrder}C{-tuple}). ''' return self._Kseries( # XXX int/int quotients may require from __future__ import division # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600), _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned _T(191773887257/3719607091200)) # PYCHOK unaligned @deprecated_Property_RO def c(self): '''DEPRECATED, use property L{R2} or L{Rauthalic}.''' return self.R2 @Property_RO def c2(self): '''Get the I{authalic} earth radius I{squared} (C{meter**2}), see L{area}, L{R2}. @see: I{Karney's} U{equation 60<https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area()<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Ellipsoid.html>}, U{Authalic radius<https://WikiPedia.org/wiki/Earth_radius#Authalic_radius>}, U{Surface area <https://WikiPedia.org/wiki/Ellipsoid>} and U{surface area <https://www.Numericana.com/answer/geometry.htm#oblate>}. ''' f = self.f if f: c2, e = self.b2, self.e if e > EPS0: if f > 0: # .isOblate c2 *= atanh(e) / e elif f < 0: # .isProlate c2 *= atan(e) / e # XXX asin? c2 = Meter2(c2=(self.a2 + c2) * _0_5) else: c2 = self.a2 return c2 def circle4(self, lat): '''Get the equatorial or a parallel I{circle of latitude}. @arg lat: Geodetic latitude (C{degrees90}, C{str}). @return: A L{Circle4Tuple}C{(radius, height, lat, beta)} instance. @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise TypeError: Invalid B{C{lat}}. @raise ValueError: Invalid B{C{lat}}. @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude}<https://WikiPedia.org/wiki/Latitude>} and I{Karney's} C++ U{CircleRadius and CircleHeight <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Ellipsoid.html>}. ''' lat = Lat(lat) if lat: b = lat if abs(lat) < _90_0: if self.f: b = self._beta(lat) z, r = sincos2d(b) r *= self.a z *= self.b else: # near-polar r, z = _0_0, copysign(self.b, lat) else: # equator r = self.a z = lat = b = _0_0 return Circle4Tuple(r, z, lat, b) def degrees2m(self, deg, lat=0): '''Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. @arg deg: The angle (C{degrees}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Distance (C{meter}, same units as the equatorial and polar radii) or C{0} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise ValueError: Invalid B{C{deg}} or B{C{lat}}. ''' return self.radians2m(radians(deg), lat=lat) def distance2(self, lat0, lon0, lat1, lon1): '''I{Approximate} the distance and (initial) bearing between two points based on the U{local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. I{Suitable only for distances of several hundred Km or Miles and only between points not near-polar}. @arg lat0: From latitude (C{degrees}). @arg lon0: From longitude (C{degrees}). @arg lat1: To latitude (C{degrees}). @arg lon1: To longitude (C{degrees}). @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance} in same units as this ellipsoid's axes. @note: The meridional and prime_vertical radii of curvature are taken and scaled I{at the initial latitude}, see C{roc2}. @see: Function L{flatLocal}/L{hubeny}. ''' phi0 = Phi_(lat0=lat0) m, n = self.roc2_(phi0, scaled=True) m *= Phi_(lat1=lat1) - phi0 n *= Lam_(lon1=lon1) - Lam_(lon0=lon0) return Distance2Tuple(hypot(m, n), atan2b(n, m)) @Property_RO def e(self): '''Get the I{(1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see L{a_b2e}. ''' return Float(e=sqrt(self.e2abs) if self.e2 else _0_0) eccentricity = e # eccentricity @Property_RO def e2(self): '''Get the I{(1st) eccentricity squared} (C{float}), M{f * (2 - f) == 1 - (b / a)**2}, see L{a_b2e2}. ''' return self._assert(a_b2e2(self.a, self.b), e2=f2e2(self.f)) eccentricity2 = e2 # eccentricity squared @Property_RO def e2abs(self): '''Get C{abs} value of the I{(1st) eccentricity squared} (C{float}). ''' return abs(self.e2) @Property_RO def e12(self): '''Get 1 less I{(1st) eccentricity squared} (C{float}), M{1 - e**2} == M{(1 - f)**2} == M{b**2 / a**2}, see L{b2_a2}. ''' return self._assert((_1_0 - self.f)**2, e12=_1_0 - self.e2, f0=_1_0) # 1 - e2 @Property_RO def e22(self): '''Get the I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2) == M{e2 / (1 - f)**2} == (a / b)**2 - 1}, see L{a_b2e22}. ''' return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f)) eccentricity2nd2 = e22 # second eccentricity squared @Property_RO def e22abs(self): '''Get C{abs} value of the I{(2nd) eccentricity squared} (C{float}). ''' return abs(self._e22) @Property_RO def e32(self): '''Get the I{3rd eccentricity squared} (C{float}), M{e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2)}, see L{a_b2e32}. ''' return self._assert(a_b2e32(self.a, self.b), e32=f2e32(self.f)) eccentricity3rd2 = e32 # third eccentricity squared @Property_RO def e32abs(self): '''Get C{abs} value of the I{(3rd) eccentricity squared} (C{float}). ''' return abs(self.e32) @Property_RO def e4(self): '''Get the I{(1st) eccentricity} to 4th power (C{float}), M{e**4 == e2**2}. ''' return Float(e4=self.e2**2 if self.e2 else _0_0) def ecef(self, Ecef=None): '''Return U{ECEF<https://WikiPedia.org/wiki/ECEF>} converter. @kwarg Ecef: ECEF class to use (L{EcefKarney}, L{EcefVeness} or L{EcefYou}). @return: An ECEF converter for this C{ellipsoid} (L{EcefKarney}, L{EcefVeness} or L{EcefYou}). @raise TypeError: Invalid B{C{Ecef}}. ''' return _4Ecef(self, Ecef) @Property_RO def _elliptic_e22(self): '''(INTERNAL) Elliptic helper for C{auxRectifying}, C{L}, C{Llat}. ''' from pygeodesy.elliptic import Elliptic return Elliptic(-abs(self.e22)) def e2s(self, s): '''Compute norm M{sqrt(1 - e2 * s**2)}. @arg s: Sine value (C{scalar}). @return: Norm (C{float}). @raise ValueError: Invalid B{C{s}}. ''' return sqrt(self.e2s2(s)) if self.e2 else _1_0 def e2s2(self, s): '''Compute M{1 - e2 * s**2}. @arg s: S value (C{scalar}). @return: Result (C{float}). @raise ValueError: Invalid B{C{s}}. ''' r = _1_0 if self.e2: try: r -= self.e2 * Scalar(s=s)**2 if r < 0: raise ValueError(_negative_) except (TypeError, ValueError) as x: t = self._DOT_(Ellipsoid.e2s2.__name__) raise _ValueError(t, s, txt=str(x)) return r @Property_RO def es(self): '''Get the I{(1st) eccentricity signed} (C{float}). ''' # note, self.e is always non-negative return Float(es=copysign(self.e, self.f)) # see .ups def es_atanh(self, x): '''Compute M{es * atanh(es * x)} where I{es} is the I{signed} (1st) eccentricity. @raise ValueError: Invalid B{C{x}}. @see: Function U{Math::eatanhe<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Math.html>}. ''' # note, self.e is always non-negative if self.f > 0: # .isOblate r = atanh(self.e * Scalar(x=x)) * self.e elif self.f < 0: # .isProlate r = -atan(self.e * Scalar(x=x)) * self.e else: # .isSpherical r = _0_0 return r @Property_RO def es_c(self): '''Get M{(1 - f) * exp(es_atanh(1))} (C{float}), M{b_a * exp(es_atanh(1))}. ''' return Float(es_c=(self.b_a * exp(self.es_atanh(_1_0))) if self.f else _1_0) def es_tauf(self, taup): '''Compute I{Karney}'s U{equations (19), (20) and (21) <https://ArXiv.org/abs/1002.1417>}. @see: U{Math::tauf<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Math.html>}. ''' tp = Scalar(taup=taup) if not self.f: # .isSpherical return tp tol = max(abs(tp), _1_0) * _TOL # To lowest order in e^2, taup = (1 - e^2) * tau, so use starting # guess tau = taup / (1 - e^2) = taup * a^2 / b^2. This starting # guess is the geocentric latitude which, to first order in the # flattening, is equal to the conformal auxiliary latitude. e = self.a2_b2 # == _1_0 / self.e12 t = tp * e T = Fsum(t) for _ in range(9): a, h = self._es_taupf2(t) d = (tp - a) * (e + t**2) / (h * hypot1(a)) t, d = T.fsum2_(d) if abs(d) < tol: break return t def es_taupf(self, tau): '''Compute I{Karney}'s U{equations (7), (8) and (9) <https://ArXiv.org/abs/1002.1417>}. @see: U{Math::taupf<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Math.html>}. ''' t = Scalar(tau=tau) if self.f: t, _ = self._es_taupf2(t) return t def _es_taupf2(self, t): '''(INTERNAL) Return C{(es_taupf(t), hypot1(t))}. ''' h = hypot1(t) s = sinh(self.es_atanh(t / h)) a = hypot1(s) * t - h * s return a, h @Property_RO def f(self): '''Get the I{flattening} (C{float}), M{(a - b) / a}, C{0} for spherical, negative for prolate. ''' return self._f flattening = f @Property_RO def f_(self): '''Get the I{inverse flattening} (C{float}), M{1 / f} == M{a / (a - b)}, C{0} for spherical, see L{a_b2f_}. ''' return self._f_ @Property_RO def f2(self): '''Get the I{2nd flattening} (C{float}), M{(a - b) / b == f / (1 - f)}, C{0} for spherical, see L{a_b2f2}. ''' return self._assert(self.a_b - _1_0, f2=f2f2(self.f)) flattening2nd = f2 @Property_RO def geodesic(self): '''Get this ellipsoid's I{wrapped Karney} U{Geodesic <https://GeographicLib.SourceForge.io/html/python/code.html>}, provided the U{geographiclib<https://PyPI.org/project/geographiclib>} package is installed. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) from pygeodesy.karney import _wrapped Ellipsoid._Math = _wrapped.Math # hold return _wrapped.Geodesic(self.a, self.f) @Property_RO def _geodesic_Math2(self): '''(INTERNAL) Get this ellipsoid's I{wrapped Karney} C{Geodesic} and I{Karney}'s C{Math} class, see L{geodesic}. ''' g = self.geodesic return g, Ellipsoid._Math def _hubeny2_(self, phi2, phi1, lam21): '''(INTERNAL) like function L{flatLocal_}/L{hubeny_} but returning the I{angular} distance in C{radians squared}. ''' m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=True) return hypot2(m * (phi2 - phi1), n * lam21) * self.a2_ @Property_RO def isEllipsoidal(self): '''Is this model I{ellipsoidal} (C{bool})? ''' return self.f != 0 @Property_RO def isOblate(self): '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or spherical otherwise. ''' return self.f > 0 @Property_RO def isProlate(self): '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or spherical otherwise. ''' return self.f < 0 @Property_RO def isSpherical(self): '''Is this model I{spherical} (C{bool})? ''' return self.f == 0 def _Kseries(self, *AB8Ks): '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha or Beta series coefficients per I{Karney}'s U{equations 35 and 36<https://Arxiv.org/pdf/1002.1417v3.pdf>}. @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series coefficient tuples. @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}). @see: I{Karney}s 30-th order U{TMseries30 <https://GeographicLib.SourceForge.io/html/tmseries30.html>}. ''' k = self.KsOrder if self.n: ns = fpowers(self.n, k) ks = tuple(fdot(AB8Ks[i][:k-i], *ns[i:]) for i in range(k)) else: ks = (_0_0,) * k return ks @property_doc_(''' the I{Krüger} series' order (C{int}), see L{AlphaKs}, L{BetaKs}.''') def KsOrder(self): '''Get the Krüger series order (C{int} 4, 6 or 8), see L{AlphaKs}, L{BetaKs}. ''' return self._KsOrder @KsOrder.setter # PYCHOK setter! def KsOrder(self, order): '''Set the I{Krüger} series' order, see L{AlphaKs}, L{BetaKs}. @arg order: New I{Krüger} series' order (C{int} 4, 6 or 8). @raise ValueError: Invalid B{C{order}}. ''' if not (isint(order) and order in (4, 6, 8)): raise _ValueError(order=order) if order != self._KsOrder: Ellipsoid.AlphaKs._update(self) Ellipsoid.BetaKs._update(self) self._KsOrder = order @Property_RO def L(self): '''Get the I{quarter meridian} C{L}, aka C{polar distance}, the distance along a meridian between the equator and a pole (C{meter}), M{b * Elliptic(-e2 / (1 - e2)).E} or M{a * PI / 2}. ''' if self.f: # complete integral 2nd ... # kind: Elliptic(-e2 / (1 - e2)).E r = self._elliptic_e22.cE else: # spherical r = PI_2 return Distance(L=self.b * r) def Llat(self, lat): '''Return the I{meridional length}, the distance along a meridian between the equator and a (geodetic) latitude, see C{L}. @arg lat: Geodetic latitude (C{degrees90}). @return: The meridional length at B{C{lat}}, negative on southern hemisphere (C{meter}). ''' r = self._elliptic_e22.fEd(self.auxParametric(lat)) if self.f else Phi_(lat) return Distance(Llat=self.b * r) Lmeridian = Llat # meridional distance @Property_RO def Mabcd(self): '''Get the OSGR meridional coefficients (C{4-Tuple}), C{Airy130} only. ''' if self.n: n1, n2, n3 = fpowers(self.n, 3) # PYCHOK false! Mabcd = (fsum_(4, 4 * n1, 5 * n2, 5 * n3) / 4, fsum_( 24 * n1, 24 * n2, 21 * n3) / 8, fsum_( 15 * n2, 15 * n3) / 8, (35 * n3) / 24) else: Mabcd = _1_0, _0_0, _0_0, _0_0 return Mabcd @deprecated_Property_RO def majoradius(self): # PYCHOK no cover '''DEPRECATED, use property L{a} or L{Requatorial}.''' return self.a def m2degrees(self, distance, lat=0): '''Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. @arg distance: Distance (C{meter}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise ValueError: Invalid B{C{distance}} or B{C{lat}}. ''' return degrees(self.m2radians(distance, lat=lat)) def m2radians(self, distance, lat=0): '''Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. @arg distance: Distance (C{meter}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise ValueError: Invalid B{C{distance}} or B{C{lat}}. ''' r = self.circle4(lat).radius if lat else self.a return m2radians(distance, radius=r, lat=0) @deprecated_Property_RO def minoradius(self): # PYCHOK no cover '''DEPRECATED, use property L{b} or C{Rpolar}.''' return self.b @Property_RO def n(self): '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see L{a_b2n}. ''' return self._assert(a_b2n(self.a, self.b), n=f2n(self.f)) flattening3rd = n @deprecated_Property_RO def quarteradius(self): # PYCHOK no cover '''DEPRECATED, use property L{L} or L{Llat}.''' return self.L @Property_RO def R1(self): '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and method C{Rgeometric}. ''' r = (fsum_(self.a, self.a, self.b) / 3) if self.f else self.a return Radius(R1=r) Rmean = R1 @Property_RO def R2(self): '''Get the I{authalic} earth radius (C{meter}), M{sqrt((a**2 + b**2 * atan[h](e) / e) / 2)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}, L{area} and L{c2}. @note: Symbol C{c} is also used by IUGG and IERS for the U{polar radius of curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}, see L{rocPolar}. ''' return Radius(R2=sqrt(self.c2) if self.f else self.a) Rauthalic = R2 @Property_RO def R3(self): '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and L{volume}. ''' return Radius(R3=cbrt(self._v3) if self.f else self.a) Rvolumetric = R3 def radians2m(self, rad, lat=0): '''Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. @arg rad: The angle (C{radians}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Distance (C{meter}, same units as the equatorial and polar radii) or C{0} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise ValueError: Invalid B{C{rad}} or B{C{lat}}. ''' r = self.circle4(lat).radius if lat else self.a return radians2m(rad, radius=r, lat=0) @Property_RO def Rbiaxial(self): '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}. ''' b = sqrt((self.a2 + self.b2) * _0_5) if self.f else self.a return Radius(Rbiaxial=b) Requatorial = a # for consistent naming def Rgeocentric(self, lat): '''Compute the I{geocentric} earth radius of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Geocentric earth radius (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Geocentric Radius <https://WikiPedia.org/wiki/Earth_radius#Geocentric_radius>} ''' r, a = self.a, Phi_(lat) if a and self.f: if abs(a) < PI_2: s2, c2 = _s2_c2(a) b2_a2_s2 = self.b2_a2 * s2 # R == sqrt((a2**2 * c2 + b2**2 * s2) / (a2 * c2 + b2 * s2)) # == sqrt(a2**2 * (c2 + (b2 / a2)**2 * s2) / (a2 * (c2 + b2 / a2 * s2))) # == sqrt(a2 * (c2 + (b2 / a2)**2 * s2) / (c2 + (b2 / a2) * s2)) # == a * sqrt((c2 + b2_a2 * b2_a2 * s2) / (c2 + b2_a2 * s2)) # == a * sqrt((c2 + b2_a2 * b2_a2_s2) / (c2 + b2_a2_s2)) r *= sqrt((c2 + b2_a2_s2 * self.b2_a2) / (c2 + b2_a2_s2)) else: r = self.b return Radius(Rgeocentric=r) @Property_RO def Rgeometric(self): '''Get the I{geometric} mean earth radius (C{meter}), M{sqrt(a * b)}. @see: L{R1}. ''' g = sqrt(self.a * self.b) if self.f else self.a return Radius(Rgeometric=g) def Rlat(self, lat): '''I{Approximate} the earth radius of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Approximate earth radius (C{meter}). @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise TypeError: Invalid B{C{lat}}. @raise ValueError: Invalid B{C{lat}}. @note: C{Rlat(B{90})} equals C{Rpolar}. @see: Method C{Rparallel}. ''' # r = a - (a - b) * |lat| / 90 r = self.a if self.f and lat: # .isEllipsoidal r -= (r - self.b) * abs(Lat(lat)) / _90_0 r = Radius(Rlat=r) return r Rpolar = b # for consistent naming Rquadratic = Rbiaxial # synonyms @deprecated_Property_RO def Rr(self): # PYCHOK no cover '''DEPRECATED, use property L{Rrectifying}.''' return self.Rrectifying @Property_RO def Rrectifying(self): '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}. ''' r = cbrt2((sqrt3(self.a) + sqrt3(self.b)) * _0_5) if self.f else self.a return Radius(Rrectifying=r) @deprecated_Property_RO def Rs(self): # PYCHOK no cover '''DEPRECATED, use property L{Rgeometric}.''' return self.Rgeometric def roc1_(self, sa, ca=None): '''Compute the I{prime-vertical}, I{normal} radius of curvature of (geodetic) latitude, I{unscaled}. @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]). @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0]) to use an alternate formula. @return: The prime-vertical radius of curvature (C{float}). @note: The delta between both formulae with C{Ellipsoids.WGS84} is less than 2 nanometer over the entire latitude range. @see: Method L{roc2_} and class L{EcefYou}. ''' if not self.f: # .isSpherical n = self.a elif ca is None: r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward n = (self.a / sqrt(r)) if r > _EPS0__2 else _0_0 elif ca: # derived from EcefYou.forward h = hypot(ca, self.b_a * sa) if sa else abs(ca) n = self.a / h elif sa: n = self.a2_b / abs(sa) else: n = self.a return n def roc2(self, lat, scaled=False): '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @kwarg scaled: Scale prime_vertical by B{C{cos(radians(lat))}} (C{bool}). @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the radii of curvature. @raise ValueError: Invalid B{C{lat}}. @see: Methods L{roc2_} and L{roc1_} and U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>} and meridional and prime vertical U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat), scaled=scaled) def roc2_(self, phi, scaled=False): '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of curvature of (geodetic) latitude. @arg phi: Latitude (C{radians}). @kwarg scaled: Scale prime_vertical by B{C{cos(phi)}} (C{bool}). @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the radii of curvature. @raise ValueError: Invalid B{C{phi}}. @see: Methods L{roc2} and L{roc1_} and U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>} and the meridional and prime vertical U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' a = abs(Phi(phi)) if self.f: r = self.e2s2(sin(a)) if r > _EPS0__2: n = self.a / sqrt(r) m = n * self.e12 / r # PYCHOK attr else: m = n = _0_0 # PYCHOK attr else: m = n = self.a if scaled and a: n *= cos(a) if a < PI_2 else _0_0 return Curvature2Tuple(Radius(rocMeridional=m), Radius(rocPrimeVertical=n)) def rocBearing(self, lat, bearing): '''Compute the I{directional} radius of curvature of (geodetic) latitude and compass direction. @arg lat: Latitude (C{degrees90}). @arg bearing: Direction (compass C{degrees360}). @return: Directional radius of curvature (C{meter}). @raise RangeError: Latitude B{C{lat}} outside valid range and L{rangerrors} set to C{True}. @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} ''' if self.f: s2, c2 = _s2_c2(Bearing_(bearing)) m, n = self.roc2_(Phi_(lat)) if n < m: # == n / (c2 * n / m + s2) c2 *= n / m elif m < n: # == m / (c2 + s2 * m / n) s2 *= m / n n = m b = n / (c2 + s2) # == 1 / (c2 / m + s2 / n) else: b = self.b # == self.a return Radius(rocBearing=b) def rocGauss(self, lat): '''Compute the I{Gaussian} radius of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Gaussian radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} ''' # using ... # m, n = self.roc2_(Phi_(lat)) # return sqrt(m * n) # ... requires 1 or 2 sqrt if self.f: s2, c2 = _s2_c2(Phi_(lat)) g = self.b / (c2 + self.b2_a2 * s2) else: g = self.b return Radius(rocGauss=g) def rocMean(self, lat): '''Compute the I{mean} radius of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Mean radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} ''' if self.f: m, n = self.roc2_(Phi_(lat)) m *= 2 * n / (m + n) # == 2 / (1 / m + 1 / n) else: m = self.a return Radius(rocMean=m) def rocMeridional(self, lat): '''Compute the I{meridional} radius of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @return: Meridional radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: Methods L{roc2} and L{roc2_} and U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat)).meridional rocPolar = a2_b # synonymous def rocPrimeVertical(self, lat): '''Compute the I{prime-vertical}, I{normal} radius of curvature of (geodetic) latitude, aka the transverse radius of curvature. @arg lat: Latitude (C{degrees90}). @return: Prime-vertical radius of curvature (C{meter}). @raise ValueError: Invalid B{C{lat}}. @see: Methods L{roc2}, L{roc2_} and L{roc1_} and U{Local, flat earth approximation<https://www.EdWilliams.org/avform.htm#flat>} and U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return self.roc2_(Phi_(lat)).prime_vertical rocTransverse = rocPrimeVertical # synonyms def toStr(self, prec=8): # PYCHOK expected '''Return this ellipsoid as a text string. @kwarg prec: Optional number of decimals, unstripped (C{int}). @return: Ellipsoid attributes (C{str}). ''' return self._instr(prec, _a_, 'b', 'f_', _f_, 'f2', _n_, _e_, 'e2', 'e22', 'e32', _A_, 'L', 'R1', 'R2', 'R3') @Property_RO def _v3(self): '''(INTERNAL) Get the I{volumetric} earth radius I{cubed} (C{meter**3}). ''' return self.a2 * self.b @Property_RO def volume(self): '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}. @see: L{R3}. ''' return Meter3(volume=self._v3 * PI4 / 3) class Ellipsoid2(Ellipsoid): '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}. ''' def __init__(self, a, f, name=NN): '''New L{Ellipsoid2}. @arg a: Equatorial radius, semi-axis (C{meter}). @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}). @kwarg name: Optional, unique name (C{str}). @raise NameError: Ellipsoid with that B{C{name}} already exists. @raise ValueError: Invalid B{C{a}} or B{C{f}}. @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. Negative C{B{f}} produces a I{prolate} ellipsoid. ''' try: a = Radius_(a=a, low=EPS, high=None) # like Radius_ f = Float_( f=f, low=None, high=EPS1) except (TypeError, ValueError) as x: t = instr(self, a=a, f=f, name=name) # PYCHOK no cover raise _ValueError(t, txt=str(x)) Ellipsoid.__init__(self, a, a_f2b(a, f), name=name) def _spherical_a_b(a, b): '''(INTERNAL) C{True} for spherical or invalid C{a} or C{b}. ''' return a < EPS or b < EPS or abs(a - b) < EPS def _spherical_f(f): '''(INTERNAL) C{True} for spherical or invalid C{f}. ''' return abs(f) < EPS or f > EPS1 def _spherical_f_(f_): '''(INTERNAL) C{True} for spherical or invalid C{f_}. ''' return abs(f_) < EPS or abs(f_) > _1_EPS def a_b2e(a, b): '''Return C{e}, the I{1st eccentricity} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The (1st) eccentricity (C{float} or C{0}), M{sqrt(1 - (b / a)**2)}. @note: The result is always non-negative and C{0} for I{near-spherical} ellipsoids. ''' return Float(e=sqrt(abs(a_b2e2(a, b)))) def a_b2e2(a, b): '''Return C{e2}, the I{1st eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The (1st) eccentricity I{squared} (C{float} or C{0}), M{1 - (b / a)**2}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' return Float(e2=_0_0 if _spherical_a_b(a, b) else (_1_0 - (b / a)**2)) def a_b2e22(a, b): '''Return C{e22}, the I{2nd eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The 2nd eccentricity I{squared} (C{float} or C{0}), M{(a / b)**2 - 1}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' return Float(e22=_0_0 if _spherical_a_b(a, b) else ((a / b)**2 - _1_0)) def a_b2e32(a, b): '''Return C{e32}, the I{3rd eccentricity squared} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The 3rd eccentricity I{squared} (C{float} or C{0}), M{(a**2 - b**2) / (a**2 + b**2)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' a2, b2 = a**2, b**2 return Float(e32=_0_0 if _spherical_a_b(a2, b2) else ((a2 - b2) / (a2 + b2))) def a_b2f(a, b): '''Return C{f}, the I{flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The flattening (C{float} or C{0}), M{(a - b) / a}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' f = 0 if _spherical_a_b(a, b) else ((a - b) / a) return _f_0_0 if _spherical_f(f) else Float(f=f) def a_b2f_(a, b): '''Return C{f_}, the I{inverse flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The inverse flattening (C{float} or C{0}), M{a / (a - b)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' f_ = 0 if _spherical_a_b(a, b) else (a / float(a - b)) return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) def a_b2f2(a, b): '''Return C{f2}, the I{2nd flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The 2nd flattening (C{float} or C{0}), M{(a - b) / b}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' t = 0 if _spherical_a_b(a, b) else float(a - b) return Float(f2=_0_0 if abs(t) < EPS else (t / b)) def a_b2n(a, b): '''Return C{n}, the I{3rd flattening} for a given I{equatorial} and I{polar} radius. @arg a: Equatorial radius (C{scalar} > 0). @arg b: Polar radius (C{scalar} > 0). @return: The 3rd flattening (C{float} or C{0}), M{(a - b) / (a + b)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' t = 0 if _spherical_a_b(a, b) else float(a - b) return Float(n=_0_0 if abs(t) < EPS else (t / (a + b))) def a_f2b(a, f): '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{flattening}. @arg a: Equatorial radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The polar radius (C{float}), M{a * (1 - f)}. ''' b = a if _spherical_f(f) else (a * (_1_0 - f)) return Radius_(b=a if _spherical_a_b(a, b) else b) def a_f_2b(a, f_): '''Return C{b}, the I{polar} radius for a given I{equatorial} radius and I{inverse flattening}. @arg a: Equatorial radius (C{scalar} > 0). @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The polar radius (C{float}), M{a * (f_ - 1) / f_}. ''' b = a if _spherical_f_(f_) else (a * (f_ - _1_0) / f_) return Radius_(b=a if _spherical_a_b(a, b) else b) def b_f2a(b, f): '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{flattening}. @arg b: Polar radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The equatorial radius (C{float}), M{b / (1 - f)}. ''' t = _1_0 - f a = b if abs(t < EPS) else (b / t) return Radius_(a=b if _spherical_a_b(a, b) else a) def b_f_2a(b, f_): '''Return C{a}, the I{equatorial} radius for a given I{polar} radius and I{inverse flattening}. @arg b: Polar radius (C{scalar} > 0). @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The equatorial radius (C{float}), M{b * f_ / (f_ - 1)}. ''' t = f_ - _1_0 a = b if _spherical_f_(f_) or abs(t - f_) < EPS \ or abs(t) < EPS else (b * f_ / t) return Radius_(a=b if _spherical_a_b(a, b) else a) def f2e2(f): '''Return C{e2}, the I{1st eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The (1st) eccentricity I{squared} (C{float} < 1), M{f * (2 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' return Float(e2=_0_0 if _spherical_f(f) else (f * (_2_0 - f))) def f2e22(f): '''Return C{e22}, the I{2nd eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The 2nd eccentricity I{squared} (C{float} > -1 or C{INF}), M{f * (2 - f) / (1 - f)**2}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for near-spherical ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}. ''' # e2 / (1 - e2) == f * (2 - f) / (1 - f)**2 t = (_1_0 - f)**2 return Float(e22=INF if t < EPS else (f2e2(f) / t)) # PYCHOK type def f2e32(f): '''Return C{e32}, the I{3rd eccentricity squared} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The 3rd eccentricity I{squared} (C{float}), M{f * (2 - f) / (1 + (1 - f)**2)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>}. ''' # e2 / (2 - e2) == f * (2 - f) / (1 + (1 - f)**2) e2 = f2e2(f) return Float(e32=e2 / (_2_0 - e2)) def f_2f(f_): '''Return C{f}, the I{flattening} for a given I{inverse flattening}. @arg f_: Inverse flattening (C{scalar} >>> 1). @return: The flattening (C{float} or C{0}), M{1 / f_}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' f = 0 if _spherical_f_(f_) else _1_0 / f_ return _f_0_0 if _spherical_f(f) else Float(f=f) # PYCHOK type def f2f_(f): '''Return C{f_}, the I{inverse flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The inverse flattening (C{float} or C{0}), M{1 / f}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. ''' f_ = 0 if _spherical_f(f) else _1_0 / f return _f__0_0 if _spherical_f_(f_) else Float(f_=f_) # PYCHOK type def f2f2(f): '''Return C{f2}, the I{2nd flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The 2nd flattening (C{float} or C{INF}), M{f / (1 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' t = _1_0 - f return Float(f2=_0_0 if _spherical_f(f) else (INF if abs(t) < EPS else (f / t))) # PYCHOK type def f2n(f): '''Return C{n}, the I{3rd flattening} for a given I{flattening}. @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: The 3rd flattening (-1 <= C{float} < 1), M{f / (2 - f)}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' return Float(n=_0_0 if _spherical_f(f) else (f / float(_2_0 - f))) def n2e2(n): '''Return C{e2}, the I{1st eccentricity squared} for a given I{3rd flattening}. @arg n: The 3rd flattening (-1 <= C{scalar} < 1). @return: The (1st) eccentricity I{squared} (C{float} or -INF), M{4 * n / (1 + n)**2}. @note: The result is positive for I{oblate}, negative for I{prolate} or C{0} for I{near-spherical} ellipsoids. @see: U{Flattening<https://WikiPedia.org/wiki/Flattening>}. ''' t = (_1_0 + n)**2 return Float(e2=_0_0 if abs(n) < EPS else (-INF if t < EPS else (_4_0 * n / t))) def n2f(n): '''Return C{f}, the I{flattening} for a given I{3rd flattening}. @arg n: The 3rd flattening (-1 <= C{scalar} < 1). @return: The flattening (C{float} or -INF), M{2 * n / (1 + n)}. @see: U{Eccentricity conversions<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1Ellipsoid.html>} and U{Flattening <https://WikiPedia.org/wiki/Flattening>}. ''' t = n + _1_0 f = 0 if abs(n) < EPS else (-INF if t < EPS else _2_0 * n / t) return _f_0_0 if _spherical_f(f) else Float(f=f) def n2f_(n): '''Return C{f_}, the I{inverse flattening} for a given I{3rd flattening}. @arg n: The 3rd flattening (-1 <= C{scalar} < 1). @return: The inverse flattening (C{float} or C{0}), M{1 / f}. @see: L{n2f} and L{f2f_}. ''' return f2f_(n2f(n)) class Ellipsoids(_NamedEnum): '''(INTERNAL) L{Ellipsoid} registry, I{must} be a sub-class to accommodate the L{_LazyNamedEnumItem} properties. ''' def _Lazy(self, a, b, f_, **kwds): '''(INTERNAL) Instantiate the L{Ellipsoid}. ''' if b is not None: b = _F(b) return Ellipsoid(_F(a), b, f_, **kwds) Ellipsoids = Ellipsoids(Ellipsoid) # PYCHOK singleton '''Some pre-defined L{Ellipsoid}s, all I{lazily} instantiated.''' # <https://www.GNU.org/software/gama/manual/html_node/Supported-ellipsoids.html> # <https://w3.Energistics.org/archive/Epicentre/Epicentre_v3.0/DataModel/ # LogicalDictionary/StandardValues/ellipsoid.html> # <https://kb.OSU.edu/dspace/handle/1811/77986> Ellipsoids._assert( # <https://WikiPedia.org/wiki/Earth_ellipsoid> Airy1830 = _lazy(_Airy1830_, 6377563.396, None, 299.3249646), # b=6356256.909 AiryModified = _lazy(_AiryModified_, 6377340.189, None, 299.3249646), # b=6356034.448 # ANS = _lazy('ANS', 6378160.0, None, 298.25), # Australian Nat. Spheroid Australia1966 = _lazy('Australia1966', 6378160.0, None, 298.25), # b=6356774.719 # Bessel1841 = _lazy(_Bessel1841_, 6377397.155, 6356078.963, 299.152815351), Bessel1841 = _lazy(_Bessel1841_, 6377397.155, 6356078.962818, 299.152812797), Clarke1866 = _lazy(_Clarke1866_, 6378206.4, 6356583.8, 294.978698214), Clarke1880 = _lazy('Clarke1880', 6378249.145, 6356514.86954978, 293.465), Clarke1880IGN = _lazy(_Clarke1880IGN_, 6378249.2, 6356515.0, 293.466021294), Clarke1880Mod = _lazy('Clarke1880Mod', 6378249.145, 6356514.96582849, 293.4663), CPM1799 = _lazy('CPM1799', 6375738.7, 6356671.92557493, 334.39), # Comm. des Poids et Mesures Delambre1810 = _lazy('Delambre1810', 6376428.0, 6355957.92616372, 311.5), # Belgium Engelis1985 = _lazy('Engelis1985', 6378136.05, 6356751.32272154, 298.2566), Everest1969 = _lazy('Everest1969', 6377295.664, 6356094.667915, 300.801699997), Fisher1968 = _lazy('Fisher1968', 6378150.0, 6356768.33724438, 298.3), GEM10C = _lazy('GEM10C', 6378137.0, 6356752.31424783, 298.2572236), GRS67 = _lazy('GRS67', 6378160.0, None, 298.247167427), # Lucerne b=6356774.516 GRS80 = _lazy(_GRS80_, 6378137.0, 6356752.314140347, 298.257222101), # ITRS, ETRS89 Helmert1906 = _lazy('Helmert1906', 6378200.0, 6356818.16962789, 298.3), IERS1989 = _lazy('IERS1989', 6378136.0, None, 298.257), # b=6356751.302 IERS1992TOPEX = _lazy('IERS1992TOPEX', 6378136.3, 6356751.61659215, 298.257223563), # IERS/TOPEX/Poseidon/McCarthy IERS2003 = _lazy('IERS2003', 6378136.6, 6356751.85797165, 298.25642), Intl1924 = _lazy(_Intl1924_, 6378388.0, None, 297.0), # aka Hayford b=6356911.946 Intl1967 = _lazy('Intl1967', 6378157.5, 6356772.2, 298.24961539), # New Int'l Krassovski1940 = _lazy(_Krassovski1940_, 6378245.0, 6356863.01877305, 298.3), # spelling Krassowsky1940 = _lazy(_Krassowsky1940_, 6378245.0, 6356863.01877305, 298.3), # spelling Maupertuis1738 = _lazy('Maupertuis1738', 6397300.0, 6363806.28272251, 191.0), # France Mercury1960 = _lazy('Mercury1960', 6378166.0, 6356784.28360711, 298.3), Mercury1968Mod = _lazy('Mercury1968Mod', 6378150.0, 6356768.33724438, 298.3), NWL1965 = _lazy('NWL1965', 6378145.0, 6356759.76948868, 298.25), # Naval Weapons Lab. OSU86F = _lazy('OSU86F', 6378136.2, 6356751.51693008, 298.2572236), OSU91A = _lazy('OSU91A', 6378136.3, 6356751.6165948, 298.2572236), # Plessis1817 = _lazy('Plessis1817', 6397523.0, 6355863.0, 153.56512242), # XXX incorrect? Plessis1817 = _lazy('Plessis1817', 6376523.0, 6355862.93325557, 308.64), # XXX IGN France 1972 SGS85 = _lazy('SGS85', 6378136.0, 6356751.30156878, 298.257), # Soviet Geodetic System SoAmerican1969 = _lazy('SoAmerican1969', 6378160.0, 6356774.71919531, 298.25), # South American Struve1860 = _lazy('Struve1860', 6378298.3, 6356657.14266956, 294.73), WGS60 = _lazy('WGS60', 6378165.0, 6356783.28695944, 298.3), WGS66 = _lazy('WGS66', 6378145.0, 6356759.76948868, 298.25), WGS72 = _lazy(_WGS72_, 6378135.0, None, 298.26), # b=6356750.52 WGS84 = _lazy(_WGS84_, R_MA, None, 298.257223563), # GPS b=6356752.3142451793 # Prolate = _lazy('Prolate', 6356752.3, 6378137.0, None), Sphere = _lazy(_Sphere_, R_M, R_M, _0_0), # pseudo SphereAuthalic = _lazy('SphereAuthalic', R_FM, R_FM, _0_0), # pseudo SpherePopular = _lazy('SpherePopular', R_MA, R_MA, _0_0), # EPSG:3857 Spheroid ) if __name__ == '__main__': from pygeodesy.interns import _COMMA_, _NL_, _NL_hash_, _NL_var_ from pygeodesy.named import nameof for E in (Ellipsoids.WGS84, Ellipsoids.GRS80, # NAD83, Ellipsoids.Sphere, Ellipsoids.SpherePopular, Ellipsoid(Ellipsoids.WGS84.b, Ellipsoids.WGS84.a, name='_Prolate')): e = f2n(E.f) - E.n t = NN(_COMMA_, _NL_hash_, _SPACE_)(E.toStr(prec=10), # re-callable 'e=%s, f_=%s, f=%s, n=%s (%s)' % (fstr(E.e, prec=13, fmt=Fmt.e), fstr(E.f_, prec=13, fmt=Fmt.e), fstr(E.f, prec=13, fmt=Fmt.e), fstr(E.n, prec=13, fmt=Fmt.e), fstr(e, prec=9, fmt=Fmt.e),), '%s=(%s)' % (Ellipsoid.AlphaKs.name, fstr(E.AlphaKs, prec=20),), '%s= (%s)' % (Ellipsoid.BetaKs.name, fstr(E.BetaKs, prec=20),), '%s= %s' % (nameof(Ellipsoid.KsOrder), E.KsOrder), # property '%s= (%s)' % (Ellipsoid.Mabcd.name, fstr(E.Mabcd, prec=20),)) print('%s%s: %s' % (_NL_hash_, _DOT_(E.classname, E.name), t)) # __doc__ of this file, force all into registry t = [NN] + Ellipsoids.toRepr(all=True).split(_NL_) print(_NL_var_.join(i.strip(_COMMA_) for i in t)) # **) MIT License # # Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE. # % python -m pygeodesy.ellipsoids # Ellipsoid.WGS84: name='WGS84', a=6378137, b=6356752.3142451793, f_=298.257223563, f=0.0033528107, f2=0.0033640898, n=0.0016792204, e=0.0818191908, e2=0.00669438, e22=0.0067394967, e32=0.0033584313, A=6367449.1458234144, L=10001965.7293127235, R1=6371008.7714150595, R2=6371007.1809184747, R3=6371000.790009154, # e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00), # AlphaKs=(0.00083773182062447786, 0.00000076085277735726, 0.00000000119764550324, 0.00000000000242917068, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0), # BetaKs= (0.00083773216405795667, 0.0000000590587015222, 0.00000000016734826653, 0.00000000000021647981, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0), # KsOrder= 8, # Mabcd= (1.00168275103155868244, 0.00504613293193333871, 0.00000529596776243457, 0.00000000690525779769) # Ellipsoid.GRS80: name='GRS80', a=6378137, b=6356752.3141403468, f_=298.257222101, f=0.0033528107, f2=0.0033640898, n=0.0016792204, e=0.081819191, e2=0.00669438, e22=0.0067394968, e32=0.0033584313, A=6367449.1457710434, L=10001965.7292304579, R1=6371008.7713801153, R2=6371007.1808835147, R3=6371000.7899741307, # e=8.1819191042833e-02, f_=2.98257222101e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00), # AlphaKs=(0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0), # BetaKs= (0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0), # KsOrder= 8, # Mabcd= (1.00168275103983916985, 0.0050461329567537995, 0.00000529596781448937, 0.00000000690525789941) # Ellipsoid.Sphere: name='Sphere', a=6371008.7714149999, b=6371008.7714149999, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6371008.7714149999, L=10007557.1761167478, R1=6371008.7714149999, R2=6371008.7714149999, R3=6371008.7714149999, # e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00), # AlphaKs=(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # BetaKs= (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # KsOrder= 8, # Mabcd= (1.0, 0.0, 0.0, 0.0) # Ellipsoid.SpherePopular: name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6378137, L=10018754.171394622, R1=6378137, R2=6378137, R3=6378137, # e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00), # AlphaKs=(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # BetaKs= (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # KsOrder= 8, # Mabcd= (1.0, 0.0, 0.0, 0.0) # Ellipsoid._Prolate: name='_Prolate', a=6356752.3142451793, b=6378137, f_=-297.257223563, f=-0.0033640898, f2=-0.0033528107, n=-0.0016792204, e=0.0820944379, e2=-0.0067394967, e22=-0.00669438, e32=-0.0033584313, A=6367449.1458234154, L=10035500.5204500332, R1=6363880.5428301198, R2=6363878.9413582645, R3=6363872.5644020028, # e=8.2094437949696e-02, f_=-2.97257223563e+02, f=-3.3640898209765e-03, n=-1.6792203863837e-03 (0.0e+00), # AlphaKs=(-0.00084149152514366627, 0.00000076653480614871, -0.00000000120934503389, 0.0000000000024576225, -0.00000000000000578863, 0.00000000000000001502, -0.00000000000000000004, 0.0), # BetaKs= (-0.00084149187224351817, 0.00000005842735196773, -0.0000000001680487236, 0.00000000000021706261, -0.00000000000000038002, 0.00000000000000000073, -0.0, 0.0), # KsOrder= 8, # Mabcd= (0.99832429842120640195, -0.00502921424529705757, 0.00000527821138524052, -0.00000000690525779769)