# Coverage for pygeodesy/ellipsoids.py : 95%

## 676 statements 640 run 36 missing 49 excluded

Hot-keys on this page

r m x p toggle line displays

j k next/prev highlighted chunk

0 (zero) top of page

1 (one) first highlighted chunk

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 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 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 | # -*- coding: utf-8 -*- u'''Ellipsoidal and spherical earth models. 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. @var Ellipsoids.Airy1830: Ellipsoid(name='Airy1830', a=6377563.396, b=6356256.90923729, f_=299.3249646, f=0.00334085, f2=0.00335205, n=0.00167322, e=0.08167337, e2=0.00667054, e22=0.00671533, e32=0.00334643, A=6366914.60892522, L=10001126.0807165, R1=6370461.23374576, R2=6370459.65470808, R3=6370453.30994572, Rbiaxial=6366919.065224, Rtriaxial=6372243.45317691) @var Ellipsoids.AiryModified: Ellipsoid(name='AiryModified', a=6377340.189, b=6356034.44793853, f_=299.3249646, f=0.00334085, f2=0.00335205, n=0.00167322, e=0.08167337, e2=0.00667054, e22=0.00671533, e32=0.00334643, A=6366691.77461988, L=10000776.05340819, R1=6370238.27531284, R2=6370236.69633043, R3=6370230.35179013, Rbiaxial=6366696.2307627, Rtriaxial=6372020.43236847) @var Ellipsoids.Australia1966: Ellipsoid(name='Australia1966', a=6378160, b=6356774.71919531, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367471.84853228, L=10002001.39064442, R1=6371031.5730651, R2=6371029.9824858, R3=6371023.59124344, Rbiaxial=6367476.337459, Rtriaxial=6372820.40754721) @var Ellipsoids.Bessel1841: Ellipsoid(name='Bessel1841', a=6377397.155, b=6356078.962818, f_=299.1528128, f=0.00334277, f2=0.00335398, n=0.00167418, e=0.08169683, e2=0.00667437, e22=0.00671922, e32=0.00334836, A=6366742.52023395, L=10000855.76443237, R1=6370291.09093933, R2=6370289.51012659, R3=6370283.15821523, Rbiaxial=6366746.98155108, Rtriaxial=6372074.29334012) @var Ellipsoids.CPM1799: Ellipsoid(name='CPM1799', a=6375738.7, b=6356671.92557493, f_=334.39, f=0.00299052, f2=0.00299949, n=0.0014975, e=0.07727934, e2=0.0059721, e22=0.00600798, e32=0.00299499, A=6366208.88184784, L=10000017.52721564, R1=6369383.10852498, R2=6369381.8434158, R3=6369376.76247022, Rbiaxial=6366212.45090321, Rtriaxial=6370977.3559758) @var Ellipsoids.Clarke1866: Ellipsoid(name='Clarke1866', a=6378206.4, b=6356583.8, f_=294.97869821, f=0.00339008, f2=0.00340161, n=0.00169792, e=0.08227185, e2=0.00676866, e22=0.00681478, e32=0.00339582, A=6367399.68916978, L=10001888.04298286, R1=6370998.86666667, R2=6370997.240633, R3=6370990.70659881, Rbiaxial=6367404.2783313, Rtriaxial=6372807.62791066) @var Ellipsoids.Clarke1880: Ellipsoid(name='Clarke1880', a=6378249.145, b=6356514.86954978, f_=293.465, f=0.00340756, f2=0.00341921, n=0.00170669, e=0.0824834, e2=0.00680351, e22=0.00685012, e32=0.00341337, A=6367386.64398051, L=10001867.55164747, R1=6371004.38651659, R2=6371002.74366963, R3=6370996.1419165, Rbiaxial=6367391.2806777, Rtriaxial=6372822.52526083) @var Ellipsoids.Clarke1880IGN: Ellipsoid(name='Clarke1880IGN', a=6378249.2, b=6356515, f_=293.46602129, f=0.00340755, f2=0.0034192, n=0.00170668, e=0.08248326, e2=0.00680349, e22=0.00685009, e32=0.00341336, A=6367386.73667336, L=10001867.69724907, R1=6371004.46666667, R2=6371002.82383112, R3=6370996.22212395, Rbiaxial=6367391.37333829, Rtriaxial=6372822.59907505) @var Ellipsoids.Clarke1880Mod: Ellipsoid(name='Clarke1880Mod', a=6378249.145, b=6356514.96582849, 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, Rbiaxial=6367391.32873482, Rtriaxial=6372822.54926891) @var Ellipsoids.Delambre1810: Ellipsoid(name='Delambre1810', a=6376428, b=6355957.92616372, f_=311.5, f=0.00321027, f2=0.00322061, n=0.00160772, e=0.08006397, e2=0.00641024, e22=0.0064516, e32=0.00321543, A=6366197.07684334, L=9999998.98395793, R1=6369604.64205457, R2=6369603.18419749, R3=6369597.32739068, Rbiaxial=6366201.19059818, Rtriaxial=6371316.64722284) @var Ellipsoids.Engelis1985: Ellipsoid(name='Engelis1985', a=6378136.05, b=6356751.32272154, f_=298.2566, f=0.00335282, f2=0.0033641, n=0.00167922, e=0.08181928, e2=0.00669439, e22=0.00673951, e32=0.00335844, A=6367448.17507971, L=10001964.20447208, R1=6371007.80757385, R2=6371006.21707085, R3=6370999.82613573, Rbiaxial=6367452.66379074, Rtriaxial=6372796.59560563) @var Ellipsoids.Everest1969: Ellipsoid(name='Everest1969', a=6377295.664, b=6356094.667915, f_=300.8017, f=0.00332445, f2=0.00333554, n=0.00166499, e=0.08147298, e2=0.00663785, e22=0.0066822, e32=0.00332998, A=6366699.57839501, L=10000788.3115495, R1=6370228.665305, R2=6370227.10178537, R3=6370220.81951618, Rbiaxial=6366703.99082487, Rtriaxial=6372002.02812501) @var Ellipsoids.Fisher1968: Ellipsoid(name='Fisher1968', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367463.65604381, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995035, Rbiaxial=6367468.14345752, Rtriaxial=6372811.30979281) @var Ellipsoids.GEM10C: Ellipsoid(name='GEM10C', a=6378137, b=6356752.31424783, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14582474, L=10001965.7293148, R1=6371008.77141594, R2=6371007.18091936, R3=6371000.79001005, Rbiaxial=6367453.63451765, Rtriaxial=6372797.55596006) @var Ellipsoids.GRS67: Ellipsoid(name='GRS67', a=6378160, b=6356774.51609071, f_=298.24716743, f=0.00335292, f2=0.0033642, n=0.00167928, e=0.08182057, e2=0.00669461, e22=0.00673973, e32=0.00335854, A=6367471.74706533, L=10002001.2312605, R1=6371031.50536357, R2=6371029.91475409, R3=6371023.52339015, Rbiaxial=6367476.23607738, Rtriaxial=6372820.3568989) @var Ellipsoids.GRS80: Ellipsoid(name='GRS80', a=6378137, b=6356752.31414035, f_=298.2572221, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14577104, L=10001965.72923046, R1=6371008.77138012, R2=6371007.18088351, R3=6371000.78997414, Rbiaxial=6367453.634464, Rtriaxial=6372797.55593326) @var Ellipsoids.Helmert1906: Ellipsoid(name='Helmert1906', a=6378200, b=6356818.16962789, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367513.57227074, L=10002066.93013953, R1=6371072.7232093, R2=6371071.13315272, R3=6371064.74401563, Rbiaxial=6367518.05971963, Rtriaxial=6372861.26794141) @var Ellipsoids.IERS1989: Ellipsoid(name='IERS1989', a=6378136, b=6356751.30156878, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.13949125, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669088, R3=6370999.78577297, Rbiaxial=6367452.62819019, Rtriaxial=6372796.55279934) @var Ellipsoids.IERS1992TOPEX: Ellipsoid(name='IERS1992TOPEX', a=6378136.3, b=6356751.61659215, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.44699641, L=10001964.63159783, R1=6371008.07219738, R2=6371006.48170097, R3=6371000.09079236, Rbiaxial=6367452.93568883, Rtriaxial=6372796.85654541) @var Ellipsoids.IERS2003: Ellipsoid(name='IERS2003', a=6378136.6, b=6356751.85797165, f_=298.25642, f=0.00335282, f2=0.0033641, n=0.00167922, e=0.0818193, e2=0.0066944, e22=0.00673951, e32=0.00335844, A=6367448.71771058, L=10001965.05683465, R1=6371008.35265722, R2=6371006.76215217, R3=6371000.37120877, Rbiaxial=6367453.20642742, Rtriaxial=6372797.14192686) @var Ellipsoids.Intl1924: Ellipsoid(name='Intl1924', a=6378388, b=6356911.94612795, f_=297, f=0.003367, f2=0.00337838, n=0.00168634, e=0.08199189, e2=0.00672267, e22=0.00676817, e32=0.00337267, A=6367654.50005758, L=10002288.29898944, R1=6371229.31537598, R2=6371227.71133444, R3=6371221.26587487, Rbiaxial=6367659.02704315, Rtriaxial=6373025.77129687) @var Ellipsoids.Intl1967: Ellipsoid(name='Intl1967', a=6378157.5, b=6356772.2, f_=298.24961539, f=0.0033529, f2=0.00336418, n=0.00167926, e=0.08182023, e2=0.00669455, e22=0.00673967, e32=0.00335852, A=6367469.33894446, L=10001997.44859308, R1=6371029.06666667, R2=6371027.47608389, R3=6371021.08482752, Rbiaxial=6367473.827881, Rtriaxial=6372817.9027631) @var Ellipsoids.Krassovski1940: Ellipsoid(name='Krassovski1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367558.49687498, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439, Rbiaxial=6367562.98435553, Rtriaxial=6372906.23027515) @var Ellipsoids.Krassowsky1940: Ellipsoid(name='Krassowsky1940', a=6378245, b=6356863.01877305, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367558.49687498, L=10002137.49754285, R1=6371117.67292435, R2=6371116.08285656, R3=6371109.69367439, Rbiaxial=6367562.98435553, Rtriaxial=6372906.23027515) @var Ellipsoids.Maupertuis1738: Ellipsoid(name='Maupertuis1738', a=6397300, b=6363806.28272251, f_=191, f=0.0052356, f2=0.00526316, n=0.00262467, e=0.10219488, e2=0.01044379, e22=0.01055402, e32=0.00524931, A=6380564.13011837, L=10022566.69846922, R1=6386135.42757417, R2=6386131.54144847, R3=6386115.8862823, Rbiaxial=6380575.11882818, Rtriaxial=6388943.03218495) @var Ellipsoids.Mercury1960: Ellipsoid(name='Mercury1960', a=6378166, b=6356784.28360711, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367479.62923643, L=10002013.61254591, R1=6371038.76120237, R2=6371037.17115427, R3=6371030.78205124, Rbiaxial=6367484.1166614, Rtriaxial=6372827.29640037) @var Ellipsoids.Mercury1968Mod: Ellipsoid(name='Mercury1968Mod', a=6378150, b=6356768.33724438, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367463.65604381, L=10001988.52191361, R1=6371022.77908146, R2=6371021.18903735, R3=6371014.79995035, Rbiaxial=6367468.14345752, Rtriaxial=6372811.30979281) @var Ellipsoids.NWL1965: Ellipsoid(name='NWL1965', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802667, Rbiaxial=6367461.36258457, Rtriaxial=6372805.42010473) @var Ellipsoids.OSU86F: Ellipsoid(name='OSU86F', a=6378136.2, b=6356751.51693008, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.3471653, L=10001964.47478349, R1=6371007.97231003, R2=6371006.38181364, R3=6370999.99090513, Rbiaxial=6367452.83585765, Rtriaxial=6372796.75662978) @var Ellipsoids.OSU91A: Ellipsoid(name='OSU91A', a=6378136.3, b=6356751.6165948, f_=298.2572236, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.44699773, L=10001964.6315999, R1=6371008.07219827, R2=6371006.48170186, R3=6371000.09079324, Rbiaxial=6367452.93569015, Rtriaxial=6372796.85654607) @var Ellipsoids.Plessis1817: Ellipsoid(name='Plessis1817', a=6376523, b=6355862.93325557, f_=308.64, f=0.00324002, f2=0.00325055, n=0.00162264, e=0.08043347, e2=0.00646954, e22=0.00651167, e32=0.00324527, A=6366197.15710739, L=9999999.11003639, R1=6369636.31108519, R2=6369634.82608583, R3=6369628.85999668, Rbiaxial=6366201.34758009, Rtriaxial=6371364.26393357) @var Ellipsoids.SGS85: Ellipsoid(name='SGS85', a=6378136, b=6356751.30156878, f_=298.257, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181922, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367448.13949125, L=10001964.14856985, R1=6371007.76718959, R2=6371006.17669087, R3=6370999.78577297, Rbiaxial=6367452.62819019, Rtriaxial=6372796.55279934) @var Ellipsoids.SoAmerican1969: Ellipsoid(name='SoAmerican1969', a=6378160, b=6356774.71919531, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367471.84853228, L=10002001.39064442, R1=6371031.5730651, R2=6371029.98248581, R3=6371023.59124344, Rbiaxial=6367476.337459, Rtriaxial=6372820.40754721) @var Ellipsoids.Sphere: Ellipsoid(name='Sphere', a=6371008.771415, b=6371008.771415, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6371008.771415, L=10007557.17611675, R1=6371008.771415, R2=6371008.771415, R3=6371008.771415, Rbiaxial=6371008.771415, Rtriaxial=6371008.771415) @var Ellipsoids.SphereAuthalic: Ellipsoid(name='SphereAuthalic', a=6371000, b=6371000, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6371000, L=10007543.39801029, R1=6371000, R2=6371000, R3=6371000, Rbiaxial=6371000, Rtriaxial=6371000) @var Ellipsoids.SpherePopular: Ellipsoid(name='SpherePopular', a=6378137, b=6378137, f_=0, f=0, f2=0, n=0, e=0, e2=0, e22=0, e32=0, A=6378137, L=10018754.17139462, R1=6378137, R2=6378137, R3=6378137, Rbiaxial=6378137, Rtriaxial=6378137) @var Ellipsoids.Struve1860: Ellipsoid(name='Struve1860', a=6378298.3, b=6356657.14266956, f_=294.73, f=0.00339294, f2=0.00340449, n=0.00169935, e=0.0823065, e2=0.00677436, e22=0.00682056, e32=0.00339869, A=6367482.31832549, L=10002017.83655714, R1=6371084.58088985, R2=6371082.95208988, R3=6371076.40691418, Rbiaxial=6367486.91530791, Rtriaxial=6372894.90029454) @var Ellipsoids.WGS60: Ellipsoid(name='WGS60', a=6378165, b=6356783.28695944, f_=298.3, f=0.00335233, f2=0.00336361, n=0.00167898, e=0.08181333, e2=0.00669342, e22=0.00673853, e32=0.00335795, A=6367478.63091189, L=10002012.0443814, R1=6371037.76231981, R2=6371036.17227197, R3=6371029.78316994, Rbiaxial=6367483.11833616, Rtriaxial=6372826.29723739) @var Ellipsoids.WGS66: Ellipsoid(name='WGS66', a=6378145, b=6356759.76948868, f_=298.25, f=0.00335289, f2=0.00336417, n=0.00167926, e=0.08182018, e2=0.00669454, e22=0.00673966, e32=0.00335851, A=6367456.87366841, L=10001977.86818326, R1=6371016.58982956, R2=6371014.999254, R3=6371008.60802667, Rbiaxial=6367461.36258457, Rtriaxial=6372805.42010473) @var Ellipsoids.WGS72: Ellipsoid(name='WGS72', a=6378135, b=6356750.52001609, f_=298.26, f=0.00335278, f2=0.00336406, n=0.0016792, e=0.08181881, e2=0.00669432, e22=0.00673943, e32=0.0033584, A=6367447.24862383, L=10001962.74919858, R1=6371006.84000536, R2=6371005.24953886, R3=6370998.8587507, Rbiaxial=6367451.7372317, Rtriaxial=6372795.60727472) @var Ellipsoids.WGS84: Ellipsoid(name='WGS84', a=6378137, b=6356752.31424518, f_=298.25722356, f=0.00335281, f2=0.00336409, n=0.00167922, e=0.08181919, e2=0.00669438, e22=0.0067395, e32=0.00335843, A=6367449.14582341, L=10001965.72931272, R1=6371008.77141506, R2=6371007.18091847, R3=6371000.79000916, Rbiaxial=6367453.63451633, Rtriaxial=6372797.5559594) ''' # make sure int/int division yields float quotient, see .basics from __future__ import division as _; del _ # PYCHOK semicolon from pygeodesy.basics import copysign0, isfinite, isint, _xinstanceof from pygeodesy.errors import _AssertionError, _ValueError from pygeodesy.fmath import cbrt, cbrt2, fdot, fhorner, fpowers, hypot, hypot_, \ hypot1, hypot2, sqrt3 from pygeodesy.fsums import Fsum, fsum_ from pygeodesy.interns import EPS, EPS0, EPS02, EPS1, INF, NN, PI4, PI_2, R_M, _a_, \ _Airy1830_, _AiryModified_, _Bessel1841_, _Clarke1866_, \ _Clarke1880IGN_, _DOT_, _1_EPS, _EPStol as _TOL, _f_, \ _finite_, _float as _F, _floatuple as _T, _GRS80_, _height_, \ _Intl1924_, _Krassovski1940_, _Krassowsky1940_, _lat_, \ _meridional_, _negative_, _not_, _null_, _prime_vertical_, \ _radius_, _Sphere_, _SPACE_, _vs_, _WGS72_, _WGS84_, \ _0_0, _0_5, _1_0, _2_0, _4_0, _90_0 from pygeodesy.interns import _0_25, _3_0 # PYCHOK used! from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS from pygeodesy.named import _lazyNamedEnumItem as _lazy, _NamedEnum, \ _NamedEnumItem, _NamedTuple, _Pass from pygeodesy.namedTuples import Distance2Tuple, Vector3Tuple, Vector4Tuple from pygeodesy.props import deprecated_Property_RO, Property_RO, property_doc_ 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, 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__ = '22.01.17' _f_0_0 = Float(f =_0_0) _f__0_0 = Float(f_=_0_0) # like U{WGS84_f()<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Constants.html>} _f_WGS84 = Float(f = 1 / (1000000000 / 298257223563)) # 10000000000 / 2982572235629 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. ''' ecef = _MODS.ecef if Ecef is None: Ecef = ecef.EcefKarney else: _xinstanceof(ecef.EcefKarney, ecef.EcefVeness, ecef.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_) # name 'f' not 'f_' _Units_ = (_Pass, _Pass) def __new__(cls, a, f, **name): '''New L{a_f2Tuple} ellipsoid specification. @arg a: Equatorial radius (C{scalar} > 0). @arg f: Flattening (C{scalar} < 1, negative for I{prolate}). @return: An L{a_f2Tuple}C{(a, f)} instance. @raise UnitError: Invalid B{C{a}} or B{C{f}}. @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. Negative C{B{f}} produces a I{prolate} ellipsoid. ''' a = Radius_(a=a) # low=EPS, high=None f = Float_( f=f, low=None, high=EPS1) if abs(f) < EPS: # force spherical f = _f_0_0 return _NamedTuple.__new__(cls, a, f, **name) @Property_RO def b(self): '''Get the I{polar} radius (C{meter}), M{a * (1 - f)}. ''' return a_f2b(self.a, self.f) # PYCHOK .a and .f @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} radii, I{flattening}, I{inverse flattening} and other, often used, I{cached} attributes, supporting I{oblate} and I{prolate} ellipsoidal and I{spherical} earth models. ''' _a = 0 # equatorial radius, semi-axis (C{meter}) _b = 0 # polar radius, semi-axis (C{meter}): a * (f - 1) / f _f = 0 # (1st) flattening: (a - b) / a _f_ = 0 # inverse flattening: 1 / f = a / (a - b) _KsOrder = 8 # Krüger series order (4, 6 or 8) def __init__(self, a, b=None, f_=None, 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_: f_ = Float(f_=f_) b = a_f_2b(a, f_) # a * (f_ - 1) / f_ f = a_b2f(a, b) 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{meter**2}), M{a**2}. ''' return Meter2(a2=self.a**2) @Property_RO def a2_(self): '''Get the inverse of the I{equatorial} radius I{squared} (C{meter**2}), 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>}, see L{c2} and L{R2} or L{Rauthalic}. ''' return Radius(a2_b=self.a2 / self.b if self.f else self.a) # = rocPolar @Property_RO def a2_b2(self): '''Get the ratio I{equatorial} over I{polar} radius I{squared} (C{float}), M{(a / b)**2} == M{1 / (1 - e**2)} == M{1 / (1 - e2)} == M{1 / 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}. ''' return _MODS.albers.AlbersEqualAreaCylindrical(datum=self, name=self.name) @Property_RO def AlphaKs(self): '''Get the I{Krüger} U{Alpha series coefficients<https://GeographicLib.SourceForge.io/html/tmseries30.html>} (C{KsOrder}C{-tuple}). ''' return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 _T(1/2, -2/3, 5/16, 41/180, -127/288, 7891/37800, 72161/387072, -18975107/50803200), _T(13/48, -3/5, 557/1440, 281/630, -1983433/1935360, 13769/28800, 148003883/174182400), # PYCHOK unaligned _T(61/240, -103/140, 15061/26880, 167603/181440, -67102379/29030400, 79682431/79833600), # PYCHOK unaligned _T(49561/161280, -179/168, 6601661/7257600, 97445/49896, -40176129013/7664025600), # PYCHOK unaligned _T(34729/80640, -3418889/1995840, 14644087/9123840, 2605413599/622702080), # PYCHOK unaligned _T(212378941/319334400, -30705481/10378368, 175214326799/58118860800), # PYCHOK unaligned _T(1522256789/1383782400, -16759934899/3113510400), # PYCHOK unaligned _T(1424729850961/743921418240)) # PYCHOK unaligned @Property_RO def area(self): '''Get the ellipsoid's surface area (C{meter**2}), M{4 * PI * c2}. @see: Properties L{areax}, L{c2} and L{R2}. ''' return Meter2(area=self.c2 * PI4) @Property_RO def areax(self): '''Get the ellipsoid's surface area (C{meter**2}), M{4 * PI * c2x}, more accurate for very I{oblate} ellipsoids. @see: Properties L{area}, L{c2x}, L{R2x} and class L{GeodesicExact}. ''' return Meter2(areax=self.c2x * 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 C{rocMeridional}C{(0)}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>}. ''' return Radius(b2_a=self.b2 / self.a if self.f else self.b) @Property_RO def b2_a2(self): '''Get the ratio I{polar} over I{equatorial} radius I{squared} (C{float}), M{(b / a)**2} == M{(1 - f)**2} == M{1 - e**2} == C{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) s *= self.a_b if inverse else self.b_a return atan2d(s, c) # == atand(s / c) if c else copysign0(_90_0, lat) @Property_RO def BetaKs(self): '''Get the I{Krüger} U{Beta series coefficients<https://GeographicLib.SourceForge.io/html/tmseries30.html>} (C{KsOrder}C{-tuple}). ''' return self._Kseries( # XXX int/int quotients may require from __future__ import division as _; del _ # PYCHOK semicolon # n n**2 n**3 n**4 n**5 n**6 n**7 n**8 _T(1/2, -2/3, 37/96, -1/360, -81/512, 96199/604800, -5406467/38707200, 7944359/67737600), _T(1/48, 1/15, -437/1440, 46/105, -1118711/3870720, 51841/1209600, 24749483/348364800), # PYCHOK unaligned _T(17/480, -37/840, -209/4480, 5569/90720, 9261899/58060800, -6457463/17740800), # PYCHOK unaligned _T(4397/161280, -11/504, -830251/7257600, 466511/2494800, 324154477/7664025600), # PYCHOK unaligned _T(4583/161280, -108847/3991680, -8005831/63866880, 22894433/124540416), # PYCHOK unaligned _T(20648693/638668800, -16363163/518918400, -2204645983/12915302400), # PYCHOK unaligne _T(219941297/5535129600, -497323811/12454041600), # PYCHOK unaligned _T(191773887257/3719607091200)) # PYCHOK unaligned @deprecated_Property_RO def c(self): '''DEPRECATED, use property C{R2} or C{Rauthalic}.''' return self.R2 @Property_RO def c2(self): '''Get the I{authalic} earth radius I{squared} (C{meter**2}). @see: Properties L{c2x}, L{area}, L{R2}, L{Rauthalic}, I{Karney's} U{equation 60 <https://Link.Springer.com/article/10.1007%2Fs00190-012-0578-z>} and C++ U{Ellipsoid.Area() <https://GeographicLib.SourceForge.io/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>}. ''' return self._c2f(False) @Property_RO def c2x(self): '''Get the I{authalic} earth radius I{squared} (C{meter**2}), more accurate for very I{oblate} ellipsoids. @see: Properties L{c2}, L{areax}, L{R2x}, L{Rauthalicx}, class L{GeodesicExact} and I{Karney}'s comments at C++ attribute U{GeodesicExact._c2<https://GeographicLib.SourceForge.io/html/GeodesicExact_8cpp_source.html>}. ''' return self._c2f(True) def _c2f(self, c2x): '''(INTERNAL) Helper for C{.c2} and C{.c2x}. ''' f = self.f if f: c2, e = self.b2, self.e if e > EPS0: if f > 0: # .isOblate c2 *= (asinh(sqrt(self.e22abs)) if c2x else atanh(e)) / e elif f < 0: # .isProlate c2 *= atan(e) / e # XXX asin? c2 = Meter2(c2=(self.a2 + c2) * _0_5) 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{pygeodesy.rangerrors} set to C{True}. @raise TypeError: Invalid B{C{lat}}. @raise ValueError: Invalid B{C{lat}}. @see: Definition of U{I{p} and I{z} under B{Parametric (or reduced) latitude}<https://WikiPedia.org/wiki/Latitude>} 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, copysign0(self.b, lat) else: # equator r = self.a z = lat = b = _0_0 return Circle4Tuple(r, z, lat, b) def degrees2m(self, deg, lat=0): '''Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. @arg deg: The angle (C{degrees}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Distance (C{meter}, same units as the equatorial and polar radii) or C{0} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{deg}} or B{C{lat}}. ''' return self.radians2m(radians(deg), lat=lat) def distance2(self, lat0, lon0, lat1, lon1): '''I{Approximate} the distance and (initial) bearing between two points based on the U{local, flat earth approximation <https://www.EdWilliams.org/avform.htm#flat>} aka U{Hubeny <https://www.OVG.AT/de/vgi/files/pdf/3781/>} formula. I{Suitable only for distances of several hundred Km or Miles and only between points not near-polar}. @arg lat0: From latitude (C{degrees}). @arg lon0: From longitude (C{degrees}). @arg lat1: To latitude (C{degrees}). @arg lon1: To longitude (C{degrees}). @return: A L{Distance2Tuple}C{(distance, initial)} with C{distance} in same units as this ellipsoid's axes. @note: The meridional and prime_vertical radii of curvature are taken and scaled I{at the initial latitude}, see C{roc2}. @see: Function L{pygeodesy.flatLocal}/L{pygeodesy.hubeny}. ''' phi0 = Phi_(lat0=lat0) m, n = self.roc2_(phi0, scaled=True) m *= Phi_(lat1=lat1) - phi0 n *= Lam_(lon1=lon1) - Lam_(lon0=lon0) return Distance2Tuple(hypot(m, n), atan2b(n, m)) @Property_RO def e(self): '''Get the I{(1st) eccentricity} (C{float}), M{sqrt(1 - (b / a)**2))}, see C{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 C{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 == (1 - f)**2} == M{b**2 / a**2}, see C{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) == e2 / (1 - f)**2 == (a / b)**2 - 1}, see C{a_b2e22}. ''' return self._assert(a_b2e22(self.a, self.b), e22=f2e22(self.f)) 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 C{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}. ''' return _MODS.elliptic.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=copysign0(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=(exp(self.es_atanh(_1_0)) * self.b_a) if self.f else _1_0) def es_tauf(self, taup): '''Compute I{Karney}'s U{equations (19), (20) and (21) <https://ArXiv.org/abs/1002.1417>}. @see: 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 C{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 C{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) return _MODS.karney._wrapped.Geodesic(self) @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}. ''' return self.geodesic, _MODS.karney._wrapped.Math @Property_RO def geodesicx(self): '''Get this ellipsoid's L{GeodesicExact}. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.geodesicx.GeodesicExact(self, name=self.name) @Property_RO def geodsolve(self): '''Get this ellipsoid's L{GeodesicSolve}, the I{wrapper} around utility U{GeodSolve<https://GeographicLib.SourceForge.io/html/GeodSolve.1.html>}, provided the path to the C{GeodSolve} executable is specified with env variable C{PYGEODESY_GEODSOLVE}. ''' # if not self.isEllipsoidal: # raise _IsnotError(_ellipsoidal_, ellipsoid=self) return _MODS.geodsolve.GeodesicSolve(self, name=self.name) def height4(self, xyz, normal=True): '''Compute the height of a cartesian above or below and the projection on this ellipsoid's surface. @arg xyz: The cartesian (C{Cartesian}, L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). @kwarg normal: If C{True} the projection is the nearest point on the ellipsoid's surface, otherwise the intersection of the radial line to the center and the ellipsoid's surface. @return: L{Vector4Tuple}C{(x, y, z, h)} with the intersection C{x}, C{y} and C{z} coordinates and height C{h} in C{meter}, conventionally. @raise ValueError: Null B{C{xyz}}. @raise TypeError: Non-cartesian B{C{xyz}}. @see: U{Distance to<https://StackOverflow.com/questions/22959698/distance-from-given-point-to-given-ellipse>} and U{intersection with<https://MathWorld.wolfram.com/Ellipse-LineIntersection.html>} an ellipse. ''' v = _MODS.vector3d._otherV3d(xyz=xyz) r = v.length if r < EPS0: # EPS raise _ValueError(xyz=xyz, txt=_null_) a, b = self.a, self.b if self.isSpherical: v = v.times(a / r) h = r - a elif normal: # perpendicular to ellipsoid x, y = hypot(v.x, v.y), abs(v.z) if x < EPS0: # polar z = copysign0(b, v.z) v = Vector3Tuple(v.x, v.y, z) h = y - b elif y < EPS0: # equatorial t = a / r v = v.times_(t, t, 0) # force z=0 h = x - a else: # normal in 1st quadrant x, y = _normal2(x, y, self) t, v = v, v.times_(x, x, y) h = t.minus(v).length else: # radial to ellipsoid center t = hypot_(a * v.z, b * v.x, b * v.y) if t < EPS0: # EPS raise _ValueError(xyz=xyz, txt=_null_) t = a * b / t v = v.times(t) h = r * (_1_0 - t) return Vector4Tuple(v.x, v.y, v.z, h, name=self.height4.__name__) def _hubeny2_(self, phi2, phi1, lam21): '''(INTERNAL) like function L{pygeodesy.flatLocal_}/L{pygeodesy.hubeny_} but returning the I{angular} distance in C{radians squared}. ''' m, n = self.roc2_((phi2 + phi1) * _0_5, scaled=True) return hypot2(m * (phi2 - phi1), n * lam21) * self.a2_ @Property_RO def isEllipsoidal(self): '''Is this model I{ellipsoidal} (C{bool})? ''' return self.f != 0 @Property_RO def isOblate(self): '''Is this ellipsoid I{oblate} (C{bool})? I{Prolate} or spherical otherwise. ''' return self.f > 0 @Property_RO def isProlate(self): '''Is this ellipsoid I{prolate} (C{bool})? I{Oblate} or spherical otherwise. ''' return self.f < 0 @Property_RO def isSpherical(self): '''Is this model I{spherical} (C{bool})? ''' return self.f == 0 def _Kseries(self, *AB8Ks): '''(INTERNAL) Compute the 4-, 6- or 8-th order I{Krüger} Alpha or Beta series coefficients per I{Karney}'s U{equations 35 and 36<https://Arxiv.org/pdf/1002.1417v3.pdf>}. @arg AB8Ks: 8-Tuple of 8-th order I{Krüger} Alpha or Beta series coefficient tuples. @return: I{Krüger} series coefficients (L{KsOrder}C{-tuple}). @see: I{Karney}s 30-th order U{TMseries30 <https://GeographicLib.SourceForge.io/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 properties C{AlphaKs}, C{BetaKs}.''') def KsOrder(self): '''Get the I{Krüger} series' order (C{int} 4, 6 or 8). ''' return self._KsOrder @KsOrder.setter # PYCHOK setter! def KsOrder(self, order): '''Set the I{Krüger} series' order. @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 C{a} or C{Requatorial}.''' return self.a def m2degrees(self, distance, lat=0): '''Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. @arg distance: Distance (C{meter}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Angle (C{degrees}) or C{INF} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{distance}} or B{C{lat}}. ''' return degrees(self.m2radians(distance, lat=lat)) def m2radians(self, distance, lat=0): '''Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. @arg distance: Distance (C{meter}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Angle (C{radians}) or C{INF} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{distance}} or B{C{lat}}. ''' r = self.circle4(lat).radius if lat else self.a return m2radians(distance, radius=r, lat=0) @deprecated_Property_RO def minoradius(self): # PYCHOK no cover '''DEPRECATED, use property C{b} or C{Rpolar}.''' return self.b @Property_RO def n(self): '''Get the I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}, see C{a_b2n}. ''' return self._assert(a_b2n(self.a, self.b), n=f2n(self.f)) flattening3rd = n @deprecated_Property_RO def quarteradius(self): # PYCHOK no cover '''DEPRECATED, use property C{L} or method C{Llat}.''' return self.L @Property_RO def R1(self): '''Get the I{mean} earth radius per I{IUGG} (C{meter}), M{(2 * a + b) / 3}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and method C{Rgeometric}. ''' r = (fsum_(self.a, self.a, self.b) / _3_0) if self.f else self.a return Radius(R1=r) Rmean = R1 @Property_RO def R2(self): '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2)}. @see: C{R2x}, C{c2}, C{area} and U{Earth radius <https://WikiPedia.org/wiki/Earth_radius>}. ''' return Radius(R2=sqrt(self.c2) if self.f else self.a) Rauthalic = R2 @Property_RO def R2x(self): '''Get the I{authalic} earth radius (C{meter}), M{sqrt(c2x)}. @see: C{R2}, C{c2x} and C{areax}. ''' return Radius(R2x=sqrt(self.c2x) if self.f else self.a) Rauthalicx = R2x @Property_RO def R3(self): '''Get the I{volumetric} earth radius (C{meter}), M{(a * a * b)**(1/3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>} and C{volume}. ''' r = (cbrt(self.b_a) * self.a) if self.f else self.a return Radius(R3=r) Rvolumetric = R3 def radians2m(self, rad, lat=0): '''Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. @arg rad: The angle (C{radians}). @kwarg lat: Parallel latitude (C{degrees90}, C{str}). @return: Distance (C{meter}, same units as the equatorial and polar radii) or C{0} for near-polar B{C{lat}}. @raise RangeError: Latitude B{C{lat}} outside valid range and L{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{rad}} or B{C{lat}}. ''' r = self.circle4(lat).radius if lat else self.a return radians2m(rad, radius=r, lat=0) @Property_RO def Rbiaxial(self): '''Get the I{biaxial, quadratic} mean earth radius (C{meter}), M{sqrt((a**2 + b**2) / 2)}. @see: C{Rtriaxial} ''' b = (sqrt((_1_0 + self.b2_a2) * _0_5) * self.a) 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: C{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{pygeodesy.rangerrors} set to C{True}. @raise TypeError: Invalid B{C{lat}}. @raise ValueError: Invalid B{C{lat}}. @note: C{Rlat(B{90})} equals C{Rpolar}. @see: Method C{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 @deprecated_Property_RO def Rquadratic(self): # PYCHOK no cover '''DEPRECATED, use property C{Rbiaxial} or C{Rtriaxial}.''' return self.Rbiaxial @deprecated_Property_RO def Rr(self): # PYCHOK no cover '''DEPRECATED, use property C{Rrectifying}.''' return self.Rrectifying @Property_RO def Rrectifying(self): '''Get the I{rectifying} earth radius (C{meter}), M{((a**(3/2) + b**(3/2)) / 2)**(2/3)}. @see: U{Earth radius<https://WikiPedia.org/wiki/Earth_radius>}. ''' r = (cbrt2((_1_0 + sqrt3(self.b_a)) * _0_5) * self.a) if self.f else self.a return Radius(Rrectifying=r) def roc1_(self, sa, ca=None): '''Compute the I{prime-vertical}, I{normal} radius of curvature of (geodetic) latitude, I{unscaled}. @arg sa: Sine of the latitude (C{float}, [-1.0..+1.0]). @kwarg ca: Optional cosine of the latitude (C{float}, [-1.0..+1.0]) to use an alternate formula. @return: The prime-vertical radius of curvature (C{float}). @note: The delta between both formulae with C{Ellipsoids.WGS84} is less than 2 nanometer over the entire latitude range. @see: Method L{roc2_} and class L{EcefYou}. ''' if not self.f: # .isSpherical n = self.a elif ca is None: r = self.e2s2(sa) # see .roc2_ and _EcefBase._forward n = (self.a / sqrt(r)) if r > EPS02 else _0_0 elif ca: # derived from EcefYou.forward h = hypot(ca, self.b_a * sa) if sa else abs(ca) n = self.a / h elif sa: n = self.a2_b / abs(sa) else: n = self.a return n def roc2(self, lat, scaled=False): '''Compute the I{meridional} and I{prime-vertical}, I{normal} radii of curvature of (geodetic) latitude. @arg lat: Latitude (C{degrees90}). @kwarg scaled: Scale prime_vertical by C{cos(radians(B{lat}))} (C{bool}). @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the radii of curvature. @raise ValueError: Invalid B{C{lat}}. @see: Methods L{roc2_} and L{roc1_} 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 C{cos(B{phi})} (C{bool}). @return: An L{Curvature2Tuple}C{(meridional, prime_vertical)} with the radii of curvature. @raise ValueError: Invalid B{C{phi}}. @see: Methods L{roc2} and L{roc1_} 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 > EPS02: 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{pygeodesy.rangerrors} set to C{True}. @raise ValueError: Invalid B{C{lat}} or B{C{bearing}}. @see: U{Radii of Curvature <https://WikiPedia.org/wiki/Earth_radius#Radii_of_curvature>} ''' if self.f: s2, c2 = _s2_c2(Bearing_(bearing)) m, n = self.roc2_(Phi_(lat)) if n < m: # == n / (c2 * n / m + s2) c2 *= n / m elif m < n: # == m / (c2 + s2 * m / n) s2 *= m / n n = m b = n / (c2 + s2) # == 1 / (c2 / m + s2 / n) else: b = self.b # == self.a return Radius(rocBearing=b) 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 @deprecated_Property_RO def Rs(self): # PYCHOK no cover '''DEPRECATED, use property C{Rgeometric}.''' return self.Rgeometric @Property_RO def Rtriaxial(self): '''Get the I{triaxial, quadratic} mean earth radius (C{meter}), M{sqrt((3 * a**2 + b**2) / 4)}. @see: C{Rbiaxial} ''' t = (sqrt((_3_0 + self.b2_a2) * _0_25) * self.a) if self.f else self.a return Radius(Rtriaxial=t) 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}). ''' E = Ellipsoid return self._instr(prec, _a_, E.b.name, E.f_.name, _f_, E.f2.name, E.n.name, E.e.name, E.e2.name, E.e22.name, E.e32.name, E.A.name, E.L.name, E.R1.name, E.R2.name, E.R3.name, E.Rbiaxial.name, E.Rtriaxial.name) @Property_RO def volume(self): '''Get the ellipsoid's I{volume} (C{meter**3}), M{4 / 3 * PI * R3**3}. @see: C{R3}. ''' return Meter3(volume=self.a2 * self.b * (PI4 / _3_0)) class Ellipsoid2(Ellipsoid): '''An L{Ellipsoid} specified by I{equatorial} radius and I{flattening}. ''' def __init__(self, a, f, name=NN): '''New L{Ellipsoid2}. @arg a: Equatorial radius, semi-axis (C{meter}). @arg f: Flattening: (C{float} < 1.0, negative for I{prolate}). @kwarg name: Optional, unique name (C{str}). @raise NameError: Ellipsoid with that B{C{name}} already exists. @raise ValueError: Invalid B{C{a}} or B{C{f}}. @note: C{abs(B{f}) < EPS} is forced to C{B{f}=0}, I{spherical}. Negative C{B{f}} produces a I{prolate} ellipsoid. ''' try: t = a_f2Tuple(a, f) except (TypeError, ValueError) as x: t = instr(self, a=a, f=f, name=name) raise _ValueError(t, txt=str(x)) Ellipsoid.__init__(self, t.a, t.b, 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 _normal2(px, py, E): '''(INTERNAL) Nearest point on a 2-D ellipse. ''' a, b = E.a, E.b if min(px, py, a, b) < EPS0: raise _AssertionError(px=px, py=py, a=a, b=b, E=E) a2 = a - b * E.b_a b2 = b - a * E.a_b tx = ty = 0.70710678118 for _ in range(9): # 4..5 trips max ex = a2 * tx**3 ey = b2 * ty**3 qx = px - ex qy = py - ey q = hypot(qx, qy) if q < EPS0: break r = hypot(ex - tx * a, ey - ty * b) / q sx, tx = tx, min(_1_0, max(0, (ex + qx * r) / a)) sy, ty = ty, min(_1_0, max(0, (ey + qy * r) / b)) t = hypot(ty, tx) if t < EPS0: break tx = tx / t # /= t chokes PyChecker ty = ty / t if max(abs(sx - tx), abs(sy - ty)) < EPS: break tx *= a / px ty *= b / py return tx, ty # x and y as fractions 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}. ''' return Ellipsoid(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_, *_T(6377563.396, _0_0, 299.3249646)), # b=6356256.909 AiryModified = _lazy(_AiryModified_, *_T(6377340.189, _0_0, 299.3249646)), # b=6356034.448 # ANS = _lazy('ANS', *_T(6378160.0, _0_0, 298.25)), # Australian Nat. Spheroid Australia1966 = _lazy('Australia1966', *_T(6378160.0, _0_0, 298.25)), # b=6356774.7192 # Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.963, 299.152815351)), Bessel1841 = _lazy(_Bessel1841_, *_T(6377397.155, 6356078.962818, 299.152812797)), Clarke1866 = _lazy(_Clarke1866_, *_T(6378206.4, 6356583.8, 294.978698214)), Clarke1880 = _lazy('Clarke1880', *_T(6378249.145, 6356514.86954978, 293.465)), Clarke1880IGN = _lazy(_Clarke1880IGN_, *_T(6378249.2, 6356515.0, 293.466021294)), Clarke1880Mod = _lazy('Clarke1880Mod', *_T(6378249.145, 6356514.96582849, 293.4663)), CPM1799 = _lazy('CPM1799', *_T(6375738.7, 6356671.92557493, 334.39)), # Comm. des Poids et Mesures Delambre1810 = _lazy('Delambre1810', *_T(6376428.0, 6355957.92616372, 311.5)), # Belgium Engelis1985 = _lazy('Engelis1985', *_T(6378136.05, 6356751.32272154, 298.2566)), Everest1969 = _lazy('Everest1969', *_T(6377295.664, 6356094.667915, 300.801699997)), Fisher1968 = _lazy('Fisher1968', *_T(6378150.0, 6356768.33724438, 298.3)), GEM10C = _lazy('GEM10C', *_T(R_MA, 6356752.31424783, 298.2572236)), # GPES = _lazy('GPES', *_T(6378135.0, 6356750.0, _0_0)), # "Gen. Purpose Earth Spheroid" GRS67 = _lazy('GRS67', *_T(6378160.0, _0_0, 298.247167427)), # Lucerne b=6356774.516 GRS80 = _lazy(_GRS80_, *_T(R_MA, 6356752.314140347, 298.257222101)), # ITRS, ETRS89 # Hayford1924 = _lazy('Hayford1924', *_T(6378388.0, 6356911.94612795, _0_0)), # aka Intl1924 f_=297 Helmert1906 = _lazy('Helmert1906', *_T(6378200.0, 6356818.16962789, 298.3)), IERS1989 = _lazy('IERS1989', *_T(6378136.0, _0_0, 298.257)), # b=6356751.302 IERS1992TOPEX = _lazy('IERS1992TOPEX', *_T(6378136.3, 6356751.61659215, 298.257223563)), # IERS/TOPEX/Poseidon/McCarthy IERS2003 = _lazy('IERS2003', *_T(6378136.6, 6356751.85797165, 298.25642)), Intl1924 = _lazy(_Intl1924_, *_T(6378388.0, _0_0, 297.0)), # aka Hayford b=6356911.9462795 Intl1967 = _lazy('Intl1967', *_T(6378157.5, 6356772.2, 298.24961539)), # New Int'l Krassovski1940 = _lazy(_Krassovski1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling Krassowsky1940 = _lazy(_Krassowsky1940_, *_T(6378245.0, 6356863.01877305, 298.3)), # spelling Maupertuis1738 = _lazy('Maupertuis1738', *_T(6397300.0, 6363806.28272251, 191.0)), # France Mercury1960 = _lazy('Mercury1960', *_T(6378166.0, 6356784.28360711, 298.3)), Mercury1968Mod = _lazy('Mercury1968Mod', *_T(6378150.0, 6356768.33724438, 298.3)), NWL1965 = _lazy('NWL1965', *_T(6378145.0, 6356759.76948868, 298.25)), # Naval Weapons Lab. OSU86F = _lazy('OSU86F', *_T(6378136.2, 6356751.51693008, 298.2572236)), OSU91A = _lazy('OSU91A', *_T(6378136.3, 6356751.6165948, 298.2572236)), # Plessis1817 = _lazy('Plessis1817', *_T(6397523.0, 6355863.0, 153.56512242)), # XXX incorrect? Plessis1817 = _lazy('Plessis1817', *_T(6376523.0, 6355862.93325557, 308.64)), # XXX IGN France 1972 SGS85 = _lazy('SGS85', *_T(6378136.0, 6356751.30156878, 298.257)), # Soviet Geodetic System SoAmerican1969 = _lazy('SoAmerican1969', *_T(6378160.0, 6356774.71919531, 298.25)), # South American Struve1860 = _lazy('Struve1860', *_T(6378298.3, 6356657.14266956, 294.73)), WGS60 = _lazy('WGS60', *_T(6378165.0, 6356783.28695944, 298.3)), WGS66 = _lazy('WGS66', *_T(6378145.0, 6356759.76948868, 298.25)), WGS72 = _lazy(_WGS72_, *_T(6378135.0, _0_0, 298.26)), # b=6356750.52 WGS84 = _lazy(_WGS84_, *_T(R_MA, _0_0, _f_WGS84)), # GPS b=6356752.3142451793 # Prolate = _lazy('Prolate', *_T(6356752.3, R_MA, _0_0)), Sphere = _lazy(_Sphere_, *_T(R_M, R_M, _0_0)), # pseudo SphereAuthalic = _lazy('SphereAuthalic', *_T(R_FM, R_FM, _0_0)), # pseudo SpherePopular = _lazy('SpherePopular', *_T(R_MA, R_MA, _0_0)) # EPSG:3857 Spheroid ) if __name__ == '__main__': from pygeodesy.interns import _COMMA_, _NL_, _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) 2016-2022 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE. # % python3 -m pygeodesy.ellipsoids # 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=6371000.7900091587, Rbiaxial=6367453.6345163295, Rtriaxial=6372797.5559594007, # e=8.1819190842622e-02, f_=2.98257223563e+02, f=3.3528106647475e-03, n=1.6792203863837e-03 (0.0e+00), # AlphaKs=(0.00083773182062447786, 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=6371000.7899741363, Rbiaxial=6367453.6344640013, Rtriaxial=6372797.5559332585, # e=8.1819191042833e-02, f_=2.98257222101e+02, f=3.3528106811837e-03, n=1.6792203946295e-03 (0.0e+00), # AlphaKs=(0.00083773182472890429, 0.00000076085278481561, 0.00000000119764552086, 0.00000000000242917073, 0.00000000000000571182, 0.0000000000000000148, 0.00000000000000000004, 0.0), # BetaKs= (0.0008377321681623882, 0.00000005905870210374, 0.000000000167348269, 0.00000000000021647982, 0.00000000000000037879, 0.00000000000000000072, 0.0, 0.0), # KsOrder= 8, # 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, Rbiaxial=6371008.7714149999, Rtriaxial=6371008.7714149999, # e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00), # AlphaKs=(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # BetaKs= (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # KsOrder= 8, # 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, Rbiaxial=6378137, Rtriaxial=6378137, # e=0.0e+00, f_=0.0e+00, f=0.0e+00, n=0.0e+00 (0.0e+00), # AlphaKs=(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # BetaKs= (0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0), # KsOrder= 8, # 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=6363872.5644020075, Rbiaxial=6367453.6345163304, Rtriaxial=6362105.2243882548, # 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)