# Coverage for pygeodesy/geodesicx/gx.py : 93%

## 502 statements 465 run 37 missing 11 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 | # -*- coding: utf-8 -*- u'''A Python version of I{Karney}'s C++ class U{GeodesicExact <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>}. Classes L{GeodesicExact} and L{GeodesicLineExact} follow the naming, methods and return values from I{Karney}s' Python classes C{Geodesic} and C{GeodesicLine}, respectively. Copyright (C) Charles Karney (2012-2021) <Charles@Karney.com> and licensed under the MIT/X11 License. For more information, see U{GeographicLib<https://GeographicLib.SourceForge.io>}. ''' # make sure int/int division yields float quotient from __future__ import division as _; del _ # PYCHOK semicolon # A copy of comments from Karney's C{GeodesicExact.cpp}: # # This is a reformulation of the geodesic problem. The # notation is as follows: # - at a general point (no suffix or 1 or 2 as suffix) # - phi = latitude # - beta = latitude on auxiliary sphere # - omega = longitude on auxiliary sphere # - lambda = longitude # - alpha = azimuth of great circle # - sigma = arc length along great circle # - s = distance # - tau = scaled distance (= sigma at multiples of PI/2) # - at northwards equator crossing # - beta = phi = 0 # - omega = lambda = 0 # - alpha = alpha0 # - sigma = s = 0 # - a 12 suffix means a difference, e.g., s12 = s2 - s1. # - s and c prefixes mean sin and cos from pygeodesy.basics import copysign0, _xinstanceof, _xor, unsigned0 from pygeodesy.fmath import cbrt, fsum_, hypot, norm2 from pygeodesy.geodesicx.gxbases import _ALL_DOCS, Caps, _coSeries, \ _GeodesicBase, _polynomial, \ _sincos12, _TINY, _xnC4 from pygeodesy.geodesicx.gxline import _GeodesicLineExact, pairs, \ _update_glXs from pygeodesy.interns import EPS, EPS0, EPS02, MANT_DIG, NAN, NN, PI, PI_2, \ _COMMASPACE_, _convergence_, _EPSqrt, _no_, \ _0_0, _0_001, _0_01, _0_1, _0_5, _1_0, \ _N_1_0, _2_0, _3_0, _4_0, _6_0, _8_0, _16_0, \ _90_0, _180_0, _1000_0 from pygeodesy.karney import _around, _ellipsoid, _EWGS84, GDict, GeodesicError, \ _diff182, _fix90, _norm180, Property, Property_RO from pygeodesy.namedTuples import Destination3Tuple, Distance3Tuple # from pygeodesy.props import Property, Property_RO # from .karney # from pygeodesy.streprs import pairs # from .geodesicx.gxline from pygeodesy.utily import atan2d, sincos2, sincos2d, unroll180, wrap360 from math import atan2, cos, degrees, radians, sqrt __all__ = () __version__ = '21.11.30' _MAXIT1 = 20 _MAXIT2 = 10 + _MAXIT1 + MANT_DIG # MANT_DIG == C++ digits _SQRT2_2 = -0.7071 # negative sqrt(2) / 2 _1_75 = 1.75 # increased multiplier in defn of _TOL1 from 100 to 200 to fix Inverse # case 52.784459512564 0 -52.784459512563990912 179.634407464943777557 # which otherwise failed for Visual Studio 10 (Release and Debug) _TOL0 = EPS _TOL1 = _TOL0 * -200 # negative _TOL2 = _EPSqrt # == sqrt(_TOL0) _TOL3 = _TOL2 * _0_1 _TOLb = _TOL2 * _TOL0 # Check on bisection interval _THR1 = _TOL2 * _1000_0 + _1_0 _TINY3 = _TINY * _3_0 _TOL08 = _TOL0 * _8_0 _TOL016 = _TOL0 * _16_0 def _eTOL2(f): # Using the auxiliary sphere solution with dnm computed at # (bet1 + bet2) / 2, the relative error in the azimuth # consistency check is sig12^2 * abs(f) * min(1, 1-f/2) / 2. # (Error measured for 1/100 < b/a < 100 and abs(f) >= 1/1000. # For a given f and sig12, the max error occurs for lines # near the pole. If the old rule for computing dnm = (dn1 # + dn2)/2 is used, then the error increases by a factor of # 2.) Setting this equal to epsilon gives sig12 = etol2. # Here 0.1 is a safety factor (error decreased by 100) and # max(0.001, abs(f)) stops etol2 getting too large in the # nearly spherical case. t = min(_1_0, _1_0 - f * _0_5) * max(_0_001, abs(f)) * _0_5 return _TOL3 / (sqrt(t) if t > EPS02 else EPS0) class _PDict(GDict): '''(INTERNAL) Parameters passed around in C{._GDictInverse} and optionally returned when C{GeodesicExact.debug} is C{True}. ''' def setsigs(self, ssig1, csig1, ssig2, csig2): '''Update the C{sig1} and C{sig2} parameters. ''' self.set_(ssig1=ssig1, csig1=csig1, sncndn1=(ssig1, csig1, self.dn1), # PYCHOK dn1 ssig2=ssig2, csig2=csig2, sncndn2=(ssig2, csig2, self.dn2)) # PYCHOK dn2 def toGDict(self): '''Return as C{GDict} without attrs C{sncndn1} and C{sncndn2}. ''' def _rest(sncndn1=None, sncndn2=None, **rest): # PYCHOK unused return GDict(rest) return _rest(**self) class GeodesicExact(_GeodesicBase): '''A pure Python version of I{Karney}'s C++ class U{GeodesicExact <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>}, modeled after I{Karney}'s Python class U{Geodesic<https://GeographicLib.SourceForge.io/ html/python/code.html#module-geographiclib.geodesic>}. ''' _E = _EWGS84 _nC4 = 30 # default C4Order def __init__(self, a_ellipsoid=_EWGS84, f=None, name=NN, C4Order=None): '''New L{GeodesicExact} instance. @arg a_ellipsoid: An ellipsoid (L{Ellipsoid}) or datum (L{Datum}) or the equatorial radius of the ellipsoid (C{scalar}, conventionally in C{meter}), see B{C{f}}. @arg f: The flattening of the ellipsoid (C{scalar}) if B{C{a_ellipsoid}} is specified as C{scalar}. @kwarg name: Optional name (C{str}). @kwarg C4Order: Optional series expansion order (C{int}), see property L{C4Order}, default C{30}. @raise GeodesicError: Invalid B{C{C4Order}}. ''' if a_ellipsoid not in (GeodesicExact._E, None): self._E = _ellipsoid(a_ellipsoid, f, name=name) if name: self.name = name if C4Order: # XXX private copy, always? self.C4Order = C4Order @Property_RO def a(self): '''Get the I{equatorial} radius, semi-axis (C{meter}). ''' return self.ellipsoid.a def ArcDirect(self, lat1, lon1, azi1, a12, outmask=Caps.STANDARD): '''Solve the I{Direct} geodesic problem in terms of (spherical) arc length. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @arg a12: Arc length between the points (C{degrees}), can be negative. @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying the quantities to be returned. @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2, lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, C{lon1}, C{azi1} and arc length C{a12} always included. @see: C++ U{GeodesicExact.ArcDirect <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.ArcDirect<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GDictDirect(lat1, lon1, azi1, True, a12, outmask) def ArcDirectLine(self, lat1, lon1, azi1, a12, caps): '''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as arc length. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @arg a12: Arc length between the points (C{degrees}), can be negative. @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returned by calls to L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}. @return: A L{GeodesicLineExact} instance. @note: The third point of the L{GeodesicLineExact} is set to correspond to the second point of the I{Inverse} geodesic problem. @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}. @see: C++ U{GeodesicExact.ArcDirectLine <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.ArcDirectLine<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GenDirectLine(lat1, lon1, azi1, True, a12, caps) def Area(self, polyline=False, name=NN): '''Set up a L{GeodesicAreaExact} to compute area and perimeter of a polygon. @kwarg polyline: If C{True} perimeter only, otherwise area and perimeter (C{bool}). @kwarg name: Optional name (C{str}). @return: A L{GeodesicAreaExact} instance. @note: The B{C{debug}} setting is passed as C{verbose} to the returned L{GeodesicAreaExact} instance. ''' from pygeodesy.geodesicx.gxarea import GeodesicAreaExact gaX = GeodesicAreaExact(self, polyline=polyline, name=name or self.name) if self.debug: gaX.verbose = True return gaX Polygon = Area # for C{geographiclib} compatibility @Property_RO def b(self): '''Get the ellipsoid's I{polar} radius, semi-axis (C{meter}). ''' return self.ellipsoid.b @Property_RO def c2x(self): '''Get the ellipsoid's I{authalic} earth radius I{squared} (C{meter**2}). ''' # The Geodesic class substitutes atanh(sqrt(e2)) for asinh(sqrt(ep2)) # in the definition of _c2. The latter is more accurate for very # oblate ellipsoids (which the Geodesic class does not handle). Of # course, the area calculation in GeodesicExact is still based on a # series and only holds for moderately oblate (or prolate) ellipsoids. return self.ellipsoid.c2x c2 = c2x # in this particular case def C4f(self, eps): '''Evaluate the C{C4x} coefficients for B{C{eps}}. @arg eps: Polynomial factor (C{float}). @return: C{C4Order}-Tuple of C{C4x(B{eps})} coefficients. ''' def _c4(nC4, C4x, _p_eps, eps): i, e = 0, _1_0 for r in range(nC4, 0, -1): j = i + r yield _p_eps(eps, C4x, i, j) * e e *= eps i = j # assert i == (nC4 * (nC4 + 1)) // 2 return tuple(_c4(self._nC4, self._C4x, _polynomial, eps)) def _C4f_k2(self, k2): # in ._GDictInverse and gxline._GeodesicExactLine._C4a '''(INTERNAL) Compute C{eps} from B{C{k2}} and invoke C{C4f}. ''' return self.C4f(k2 / fsum_(_2_0, sqrt(k2 + _1_0) * _2_0, k2)) @Property def C4Order(self): '''Get the series expansion order (C{int}, 24, 27 or 30). ''' return self._nC4 @C4Order.setter # PYCHOK .setter def C4Order(self, order): '''Set the series expansion order. @arg order: New order (C{int}, 24, 27 or 30). @raise GeodesicError: Invalid B{C{order}}. ''' _xnC4(C4Order=order) if self._nC4 != order: GeodesicExact._C4x._update(self) _update_glXs(self) # zap cached/memoized _GeodesicLineExact attrs self._nC4 = order @Property_RO def _C4x(self): '''Get this ellipsoid's C{C4} coefficients, I{cached} tuple. @see: Property L{C4Order}. ''' # see C4coeff() in GeographicLib.src.GeodesicExactC4.cpp def _C4(nC4, coeffs, _p_n, n): i = 0 for r in range(nC4 + 1, 1, -1): for j in range(1, r): j = i + j # (j - i - 1) order of polynomial yield _p_n(n, coeffs, i, j) / coeffs[j] i = j + 1 # assert i == len(cs) == (nC4 * (nC4 + 1) * (nC4 + 5)) // 6 return tuple(_C4(self._nC4, self._coeffs(self._nC4), _polynomial, self.n)) # 3rd flattening def _coeffs(self, nC4): '''(INTERNAL) Get the series coefficients. ''' if nC4 == 30: from pygeodesy.geodesicx._C4_30 import _coeffs_30 as _coeffs elif nC4 == 27: from pygeodesy.geodesicx._C4_27 import _coeffs_27 as _coeffs elif nC4 == 24: from pygeodesy.geodesicx._C4_24 import _coeffs_24 as _coeffs else: # PYCHOK no cover _coeffs = () n = _xnC4(nC4=nC4) if len(_coeffs) != n: # double check raise GeodesicError(_coeffs=len(_coeffs), _xnC4=n, nC4=nC4) return _coeffs def Direct(self, lat1, lon1, azi1, s12, outmask=Caps.STANDARD): '''Solve the I{Direct} geodesic problem @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @arg s12: Distance between the points (C{meter}), can be negative. @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying the quantities to be returned. @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2, lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, C{lon1}, C{azi1} and distance C{s12} always included. @see: C++ U{GeodesicExact.Direct <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.Direct<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GDictDirect(lat1, lon1, azi1, False, s12, outmask) def Direct3(self, lat1, lon1, azi1, s12): # PYCHOK outmask '''Return the destination lat, lon and reverse azimuth (final bearing) in C{degrees}. @return: L{Destination3Tuple}C{(lat, lon, final)}. ''' r = self._GDictDirect(lat1, lon1, azi1, False, s12, Caps._AZIMUTH_LATITUDE_LONGITUDE) return Destination3Tuple(r.lat2, r.lon2, r.azi2) def DirectLine(self, lat1, lon1, azi1, s12, caps=Caps.STANDARD): '''Define a L{GeodesicLineExact} in terms of the I{direct} geodesic problem and as distance. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @arg s12: Distance between the points (C{meter}), can be negative. @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returned by calls to L{GeodesicLineExact.Position}. @return: A L{GeodesicLineExact} instance. @note: The third point of the L{GeodesicLineExact} is set to correspond to the second point of the I{Inverse} geodesic problem. @note: Latitude B{C{lat1}} should in the range C{[-90, +90]}. @see: C++ U{GeodesicExact.DirectLine <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.DirectLine<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GenDirectLine(lat1, lon1, azi1, False, s12, caps) def _dn(self, sbet, cbet): # in gxline._GeodesicLineExact.__init__ '''(INTERNAL) Helper. ''' return sqrt(_1_0 + self.ep2 * sbet**2) if self.f >= 0 else ( sqrt(_1_0 - self.e2 * cbet**2) / self.f1) @Property_RO def e2(self): '''Get the ellipsoid's I{(1st) eccentricity squared} (C{float}), M{f * (2 - f)}. ''' return self.ellipsoid.e2 @Property_RO def _e2a2(self): '''(INTERNAL) Cache M{E.e2 * E.a2}. ''' return self.e2 * self.ellipsoid.a2 @Property_RO def _e2_f1(self): '''(INTERNAL) Cache M{E.e2 * E.f1}. ''' return self.e2 / self.f1 @Property_RO def _eF(self): '''(INTERNAL) Get the elliptic function, aka C{.E}. ''' from pygeodesy.elliptic import Elliptic return Elliptic(k2=-self.ep2) def _eF_reset_e2_f1_cH(self, x, y): '''(INTERNAL) Reset elliptic function and return M{e2 / f1 * cH * ...}. ''' self._eF_reset_k2(x) return self._e2_f1 * y * self._eF.cH def _eF_reset_k2(self, x): '''(INTERNAL) Reset elliptic function and return C{k2}. ''' ep2 = self.ep2 k2 = x**2 * ep2 self._eF.reset(k2=-k2, alpha2=-ep2, kp2=k2 + _1_0, alphap2=ep2 + _1_0) # defaults _update_glXs(self) # zap cached/memoized _GeodesicLineExact attrs return k2 @Property_RO def ellipsoid(self): '''Get the ellipsoid (C{Ellipsoid}). ''' return self._E @Property_RO def ep2(self): # aka .e22 '''Get the ellipsoid's I{2nd eccentricity squared} (C{float}), M{e2 / (1 - e2)}. ''' return self.ellipsoid.e22 # == self.e2 / self.f1**2 @Property_RO def _eTOL2(self): '''(INTERNAL) The si12 threshold for "really short". ''' return _eTOL2(self.f) @Property_RO def f(self): '''Get the ellipsoid's I{flattening} (C{float}), M{(a - b) / a}, C{0} for spherical, negative for prolate. ''' return self.ellipsoid.f flattening = f @Property_RO def f1(self): # aka .b_a '''Get the ellipsoid's ratio I{polar} over I{equatorial} radius (C{float}), M{b / a} == M{1 - f}. ''' return self.ellipsoid.b_a # _1_0 - self.f @Property_RO def _f_180(self): '''(INTERNAL) Cached/memoized. ''' return self.f * _180_0 def _GDictDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask=Caps.STANDARD): '''(INTERNAL) As C{_GenDirect}, but returning a L{GDict}. @return: A L{GDict} ... ''' if not arcmode: # supply DISTANCE_IN outmask |= Caps.DISTANCE_IN glX = self._LineTemp(lat1, lon1, azi1, outmask) return glX._GDictPosition(arcmode, s12_a12, outmask) def _GDictInverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): # MCCABE 33, 41 vars '''(INTERNAL) As C{_GenInverse}, but returning a L{GDict}. @return: A L{GDict} ... ''' if self.debug: # PYCHOK no cover outmask |= Caps._DEBUG_INVERSE & self._debug outmask &= Caps._OUTMASK # incl. _SALPs_CALPs and _DEBUG_ # compute longitude difference carefully (with _diff182): # result is in [-180, +180] but -180 is only for west-going # geodesics, +180 is for east-going and meridional geodesics lon12, lon12s = _diff182(lon1, lon2) # see C{result} from geographiclib.geodesic.Inverse if (outmask & Caps.LONG_UNROLL): # == (lon1 + lon12) + lon12s r = GDict(lon1=lon1, lon2=fsum_(lon1, lon12, lon12s)) else: r = GDict(lon1=_norm180(lon1), lon2=_norm180(lon2)) # make longitude difference positive and if very close to # being on the same half-meridian, then make it so. if lon12 < 0: lon_, lon12 = True, -_around(lon12) lon12s = _around((_180_0 - lon12) + lon12s) else: lon_, lon12 = False, _around(lon12) lon12s = _around((_180_0 - lon12) - lon12s) lam12 = radians(lon12) if lon12 > _90_0: slam12, clam12 = sincos2d(lon12s) clam12 = -clam12 else: slam12, clam12 = sincos2d(lon12) # If really close to the equator, treat as on equator. lat1 = _around(_fix90(lat1)) lat2 = _around(_fix90(lat2)) r.set_(lat1=lat1, lat2=lat2) # Swap points so that point with higher (abs) latitude is # point 1. If one latitude is a NAN, then it becomes lat1. swap_ = abs(lat1) < abs(lat2) if swap_: lat1, lat2 = lat2, lat1 lon_ = not lon_ if lat1 < 0: lat_ = False else: # make lat1 <= 0 lat_ = True lat1, lat2 = -lat1, -lat2 # Now 0 <= lon12 <= 180, -90 <= lat1 <= 0 and lat1 <= lat2 <= -lat1 # and lat_, lon_, swap_ register the transformation to bring the # coordinates to this canonical form, where False means no change # made. We make these transformations so that there are few cases # to check, e.g., on verifying quadrants in atan2. In addition, # this enforces some symmetries in the results returned. # Initialize for the meridian. No longitude calculation is # done in this case to let the parameter default to 0. sbet1, cbet1 = self._sincos2f1(lat1) sbet2, cbet2 = self._sincos2f1(lat2) # If cbet1 < -sbet1, then cbet2 - cbet1 is a sensitive measure # of the |bet1| - |bet2|. Alternatively (cbet1 >= -sbet1), # abs(sbet2) + sbet1 is a better measure. This logic is used # in assigning calp2 in _Lambda10. Sometimes these quantities # vanish and in that case we force bet2 = +/- bet1 exactly. An # example where is is necessary is the inverse problem # 48.522876735459 0 -48.52287673545898293 179.599720456223079643 # which failed with Visual Studio 10 (Release and Debug) if cbet1 < -sbet1: if cbet2 == cbet1: sbet2 = sbet1 if sbet2 < 0 else -sbet1 elif abs(sbet2) == -sbet1: cbet2 = cbet1 p = _PDict(sbet1=sbet1, cbet1=cbet1, dn1=self._dn(sbet1, cbet1), sbet2=sbet2, cbet2=cbet2, dn2=self._dn(sbet2, cbet2)) _meridian = True # i.e. meridian = False if lat1 == -90 or slam12 == 0: # Endpoints are on a single full meridian, # so the geodesic might lie on a meridian. salp1, calp1 = slam12, clam12 # head to target lon salp2, calp2 = _0_0, _1_0 # then head north # tan(bet) = tan(sig) * cos(alp) p.setsigs(sbet1, calp1 * cbet1, sbet2, calp2 * cbet2) # sig12 = sig2 - sig1 sig12 = atan2(*_sincos12(sbet1, p.csig1, sbet2, p.csig2, noneg=True)) # PYCHOK csig* s12x, m12x, _, \ M12, M21 = self._Lengths5(sig12, outmask | Caps.REDUCEDLENGTH, p) # Add the check for sig12 since zero length geodesics # might yield m12 < 0. Test case was # echo 20.001 0 20.001 0 | GeodSolve -i # In fact, we will have sig12 > PI/2 for meridional # geodesic which is not a shortest path. if m12x >= 0 or sig12 < _1_0: # Need at least 2 to handle 90 0 90 180 # Prevent negative s12 or m12 from geographiclib 1.52 if sig12 < _TINY3 or (sig12 < _TOL0 and (s12x < 0 or m12x < 0)): sig12 = m12x = s12x = a12 = _0_0 else: m12x *= self.b s12x *= self.b a12 = degrees(sig12) _meridian = False C = 1 # elif m12 < 0: # prolate and too close to anti-podal # _meridian = True if _meridian: if sbet1 == 0 and ( # sbet2 == 0 and self.f <= 0 or lon12s >= self._f_180): # Geodesic runs along equator calp1 = calp2 = _0_0 salp1 = salp2 = _1_0 sig12 = lam12 / self.f1 # == omg12 somg12, comg12 = sincos2(sig12) m12x = self.b * somg12 s12x = self.a * lam12 if (outmask & Caps.GEODESICSCALE): r.set_(M12=comg12, M21=comg12) a12 = lon12 / self.f1 C = 2 else: # Now point1 and point2 belong within a hemisphere bounded by a # meridian and geodesic is neither meridional or equatorial. p.set_(slam12=slam12, clam12=clam12) # Figure a starting point for Newton's method sig12, salp1, calp1, \ salp2, calp2, dnm = self._InverseStart6(lam12, p) if sig12 is None: # use Newton's method # pre-compute the constant _Lambda10 term, once p.set_(bet12=None if cbet2 == cbet1 and abs(sbet2) == -sbet1 else ( ((cbet1 + cbet2) * (cbet2 - cbet1)) if cbet1 < -sbet1 else ((sbet1 + sbet2) * (sbet1 - sbet2)))) sig12, salp1, calp1, salp2, calp2, domg12, \ s12x, m12x, M12, M21 = self._Newton10(salp1, calp1, outmask, p) if (outmask & Caps.AREA): # omg12 = lam12 - domg12 s, c = sincos2(domg12) somg12, comg12 = _sincos12(s, c, slam12, clam12) if (outmask & Caps.GEODESICSCALE): if swap_: M12, M21 = M21, M12 r.set_(M12=unsigned0(M12), M21=unsigned0(M21)) C = 3 # Newton else: C = 4 # Short lines, _Inverse6 set salp2, calp2, dnm s, c = sincos2(sig12 / dnm) m12x = s * dnm**2 s12x = sig12 * dnm if (outmask & Caps.GEODESICSCALE): r.set_(M12=c, M21=c) if (outmask & Caps.AREA): somg12, comg12 = sincos2(lam12 / (self.f1 * dnm)) a12 = degrees(sig12) m12x *= self.b s12x *= self.b else: # _meridian is False somg12 = comg12 = NAN r.set_(a12=a12) # in [0, 180] if outmask == Caps._ANGLE_ONLY: return r # for .Inverse1 if (outmask & Caps.DISTANCE): r.set_(s12=unsigned0(s12x)) if (outmask & Caps.REDUCEDLENGTH): r.set_(m12=unsigned0(m12x)) if (outmask & Caps.AREA): S12 = self._InverseArea(_meridian, salp1, calp1, salp2, calp2, somg12, comg12, p) if _xor(swap_, lat_, lon_): S12 = -S12 r.set_(S12=unsigned0(S12)) if swap_: salp1, salp2 = salp2, salp1 calp1, calp2 = calp2, calp1 if _xor(swap_, lon_): salp1, salp2 = -salp1, -salp2 if _xor(swap_, lat_): calp1, calp2 = -calp1, -calp2 if (outmask & Caps.AZIMUTH): r.set_(azi1=atan2d(salp1, calp1), azi2=atan2d(salp2, calp2, reverse=outmask & Caps.REVERSE2)) if (outmask & Caps._SALPs_CALPs): r.set_(salp1=salp1, calp1=calp1, salp2=salp2, calp2=calp2) if (outmask & Caps._DEBUG_INVERSE): # PYCHOK no cover eF = self._eF p.set_(C=C, a=self.a, f=self.f, f1=self.f1, e=self.ellipsoid.e, e2=self.e2, ep2=self.ep2, c2x=self.c2x, c2=self.ellipsoid.c2, eFcD=eF.cD, eFcE=eF.cE, eFcH=eF.cH, eFk2=eF.k2, eFa2=eF.alpha2) p.update(r) # r overrides r = p.toGDict() return r def _GenDirect(self, lat1, lon1, azi1, arcmode, s12_a12, outmask): '''(INTERNAL) The general I{Inverse} geodesic calculation. @return: L{Direct9Tuple}C{(a12, lat2, lon2, azi2, s12, m12, M12, M21, S12)}. ''' r = self._GDictDirect(lat1, lon1, azi1, arcmode, s12_a12, outmask) return r.toDirect9Tuple() def _GenDirectLine(self, lat1, lon1, azi1, arcmode, s12_a12, caps): '''(INTERNAL) Helper for C{ArcDirectLine} and C{DirectLine}. @return: A L{GeodesicLineExact} instance. ''' azi1 = _norm180(azi1) # guard against underflow in salp0. Also -0 is converted to +0. s, c = sincos2d(_around(azi1)) if not arcmode: # supply DISTANCE_IN caps |= Caps.DISTANCE_IN return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug, s, c)._GenSet(arcmode, s12_a12) def _GenInverse(self, lat1, lon1, lat2, lon2, outmask): '''(INTERNAL) The general I{Inverse} geodesic calculation. @return: L{Inverse10Tuple}C{(a12, s12, salp1, calp1, salp2, calp2, m12, M12, M21, S12)}. ''' r = self._GDictInverse(lat1, lon1, lat2, lon2, outmask | Caps._SALPs_CALPs) return r.toInverse10Tuple() def Inverse(self, lat1, lon1, lat2, lon2, outmask=Caps.STANDARD): '''Perform the I{Inverse} geodesic calculation. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg lat2: Latitude of the second point (C{degrees}). @arg lon2: Longitude of the second point (C{degrees}). @kwarg outmask: Bit-or'ed combination of L{Caps} values specifying the quantities to be returned. @return: A L{GDict} with up to 12 items C{lat1, lon1, azi1, lat2, lon2, azi2, m12, a12, s12, M12, M21, S12} with C{lat1}, C{lon1}, C{azi1} and distance C{s12} always included. @note: The third point of the L{GeodesicLineExact} is set to correspond to the second point of the I{Inverse} geodesic problem. @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}. @see: C++ U{GeodesicExact.InverseLine <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return self._GDictInverse(lat1, lon1, lat2, lon2, outmask) def Inverse1(self, lat1, lon1, lat2, lon2, wrap=False): '''Return the non-negative, I{angular} distance in C{degrees}. ''' # see .FrechetKarney.distance, .HausdorffKarney._distance # and .HeightIDWkarney._distances _, lon2 = unroll180(lon1, lon2, wrap=wrap) # self.LONG_UNROLL return abs(self._GDictInverse(lat1, lon1, lat2, lon2, Caps._ANGLE_ONLY).a12) def Inverse3(self, lat1, lon1, lat2, lon2): # PYCHOK outmask '''Return the distance in C{meter} and the forward and reverse azimuths (initial and final bearing) in C{degrees}. @return: L{Distance3Tuple}C{(distance, initial, final)}. ''' r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps._AZIMUTH_DISTANCE) return Distance3Tuple(r.s12, wrap360(r.azi1), wrap360(r.azi2)) def InverseLine(self, lat1, lon1, lat2, lon2, caps=Caps.STANDARD): '''Define a L{GeodesicLineExact} in terms of the I{Inverse} geodesic problem. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg lat2: Latitude of the second point (C{degrees}). @arg lon2: Longitude of the second point (C{degrees}). @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returned by calls to L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}. @return: A L{GeodesicLineExact} instance. @note: The third point of the L{GeodesicLineExact} is set to correspond to the second point of the I{Inverse} geodesic problem. @note: Both B{C{lat1}} and B{C{lat2}} should in the range C{[-90, +90]}. @see: C++ U{GeodesicExact.InverseLine <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.InverseLine<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' r = self._GDictInverse(lat1, lon1, lat2, lon2, Caps._SALPs_CALPs) # No need for AZIMUTH azi1 = atan2d(r.salp1, r.calp1) if (caps & (Caps._OUTMASK & Caps.DISTANCE_IN)): caps |= Caps.DISTANCE # ensure that a12 is a distance return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug, r.salp1, r.calp1)._GenSet(True, r.a12) def _InverseArea(self, _meridian, salp1, calp1, # PYCHOK 9 args salp2, calp2, somg12, comg12, p): '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length. @return: Area C{S12}. ''' # from _Lambda10: sin(alp1) * cos(bet1) = sin(alp0) salp0 = salp1 * p.cbet1 calp0 = hypot(calp1, salp1 * p.sbet1) # calp0 > 0 if calp0 and salp0: # from _Lambda10: tan(bet) = tan(sig) * cos(alp) k2 = calp0**2 * self.ep2 C4a = self._C4f_k2(k2) B41 = _coSeries(C4a, *norm2(p.sbet1, calp1 * p.cbet1)) B42 = _coSeries(C4a, *norm2(p.sbet2, calp2 * p.cbet2)) # multiplier = a^2 * e^2 * cos(alpha0) * sin(alpha0) A4 = self._e2a2 * calp0 * salp0 S12 = A4 * (B42 - B41) p.set_(A4=A4, B41=B41, B42=B42, k2=k2) else: # avoid problems with indeterminate sig1, sig2 on equator S12 = _0_0 if (_meridian and # omg12 < 3/4 * PI comg12 > _SQRT2_2 and # lon diff not too big (p.sbet2 - p.sbet1) < _1_75): # lat diff not too big # use tan(Gamma/2) = tan(omg12/2) * # (tan(bet1/2) + tan(bet2/2)) / # (tan(bet1/2) * tan(bet2/2) + 1)) # with tan(x/2) = sin(x) / (1 + cos(x)) cbet1 = _1_0 + p.cbet1 cbet2 = _1_0 + p.cbet2 salp12 = (p.sbet1 * cbet2 + cbet1 * p.sbet2) * somg12 calp12 = (p.sbet1 * p.sbet2 + cbet1 * cbet2) * (comg12 + _1_0) alp12 = _2_0 * atan2(salp12, calp12) else: # alp12 = alp2 - alp1, used in atan2, no need to normalize salp12, calp12 = _sincos12(salp1, calp1, salp2, calp2) # The right thing appears to happen if alp1 = +/-180 and # alp2 = 0, viz salp12 = -0 and alp12 = -180. However, # this depends on the sign being attached to 0 correctly. # Following ensures the correct behavior. if salp12 == 0 and calp12 < 0: alp12 = copysign0(PI, calp1) else: alp12 = atan2(salp12, calp12) p.set_(alp12=alp12) return S12 + self.c2x * alp12 def _InverseStart6(self, lam12, p): '''(INTERNAL) Return a starting point for Newton's method in C{salp1} and C{calp1} indicated by C{sig12=None}. If Newton's method doesn't need to be used, return also C{salp2}, C{calp2}, C{dnm} and C{sig12} non-C{None}. @return: 6-Tuple C{(sig12, salp1, calp1, salp2, calp2, dnm)} and updated C{p.setsigs} if C{sig12==None}. ''' sig12 = None # use Newton salp1 = calp1 = salp2 = calp2 = dnm = NAN # bet12 = bet2 - bet1 in [0, PI) sbet12, cbet12 = _sincos12(p.sbet1, p.cbet1, p.sbet2, p.cbet2) shortline = cbet12 >= 0 and sbet12 < _0_5 and (p.cbet2 * lam12) < _0_5 if shortline: # sin((bet1 + bet2)/2)^2 = (sbet1 + sbet2)^2 / ( # (cbet1 + cbet2)^2 + (sbet1 + sbet2)^2) t = (p.sbet1 + p.sbet2)**2 s = t / ((p.cbet1 + p.cbet2)**2 + t) dnm = sqrt(_1_0 + self.ep2 * s) somg12, comg12 = sincos2(lam12 / (self.f1 * dnm)) else: somg12, comg12 = p.slam12, p.clam12 # bet12a = bet2 + bet1 in (-PI, 0], note -sbet1 sbet12a, cbet12a = _sincos12(-p.sbet1, p.cbet1, p.sbet2, p.cbet2) s = somg12**2 / (abs(comg12) + _1_0) t = p.cbet2 * p.sbet1 * s salp1 = p.cbet2 * somg12 calp1 = (sbet12a - t) if comg12 < 0 else (sbet12 + t) ssig12 = hypot(salp1, calp1) csig12 = p.sbet1 * p.sbet2 + p.cbet1 * p.cbet2 * comg12 if shortline and ssig12 < self._eTOL2: # really short lines if comg12 < 0: s = _1_0 - comg12 salp2, calp2 = norm2(somg12 * p.cbet1, sbet12 - p.cbet1 * p.sbet2 * s) sig12 = atan2(ssig12, csig12) # return value elif (self._n_0_1 or # Skip astroid calc if too eccentric csig12 >= 0 or ssig12 >= (self._n_6PI * p.cbet1**2)): pass # nothing to do, 0th order spherical approximation OK else: # Scale lam12 and bet2 to x, y coordinate system where antipodal # point is at origin and singular point is at y = 0, x = -1 lam12x = atan2(-p.slam12, -p.clam12) # lam12 - PI f = self.f if f < 0: # x = dlat, y = dlon # ssig1=sbet1, csig1=-cbet1, ssig2=sbet2, csig2=cbet2 p.setsigs(p.sbet1, -p.cbet1, p.sbet2, p.cbet2) # if lon12 = 180, this repeats a calculation made in Inverse _, m12b, m0, _, _ = self._Lengths5(atan2(sbet12a, cbet12a) + PI, Caps.REDUCEDLENGTH, p) t = p.cbet1 * PI x = m12b / (t * p.cbet2 * m0) - _1_0 sca = (sbet12a / (x * p.cbet1)) if x < -_0_01 else (-f * t) y = lam12x / sca else: # _f >= 0, in fact f == 0 does not get here sca = self._eF_reset_e2_f1_cH(p.sbet1, p.cbet1 * _2_0) x = lam12x / sca # dlon y = sbet12a / (sca * p.cbet1) # dlat if y > _TOL1 and x > -_THR1: # strip near cut if f < 0: calp1 = max((_0_0 if x > _TOL1 else _N_1_0), x) salp1 = sqrt(_1_0 - calp1**2) else: salp1 = min(_1_0, -x) calp1 = -sqrt(_1_0 - salp1**2) else: # Estimate alp1, by solving the astroid problem. # # Could estimate alpha1 = theta + PI/2, directly, i.e., # calp1 = y/k; salp1 = -x/(1+k); for _f >= 0 # calp1 = x/(1+k); salp1 = -y/k; for _f < 0 (need to check) # # However, it's better to estimate omg12 from astroid and use # spherical formula to compute alp1. This reduces the mean # number of Newton iterations for astroid cases from 2.24 # (min 0, max 6) to 2.12 (min 0 max 5). The changes in the # number of iterations are as follows: # # change percent # 1 5 # 0 78 # -1 16 # -2 0.6 # -3 0.04 # -4 0.002 # # The histogram of iterations is (m = number of iterations # estimating alp1 directly, n = number of iterations # estimating via omg12, total number of trials = 148605): # # iter m n # 0 148 186 # 1 13046 13845 # 2 93315 102225 # 3 36189 32341 # 4 5396 7 # 5 455 1 # 6 56 0 # # omg12 is near PI, estimate work with omg12a = PI - omg12 k = _Astroid(x, y) k1 = k + _1_0 sca *= (y * k1 / k) if f < 0 else (x * k / k1) s, c = sincos2(-sca) # update spherical estimate of alp1 using omg12 instead of lam12 salp1 = p.cbet2 * s calp1 = sbet12a - p.cbet2 * p.sbet1 * s**2 / (_1_0 + c) # sanity check on starting guess. Backwards check allows NaN through. salp1, calp1 = norm2(salp1, calp1) if salp1 > 0 else (_1_0, _0_0) return sig12, salp1, calp1, salp2, calp2, dnm def _Lambda5(self, salp1, calp1, diffp, p): '''(INTERNAL) Helper. @return: 5-Tuple C{(lam12, sig12, salp2, calp2, dlam12)} and updated C{p.setsigs} and C{p.domg12}. ''' eF = self._eF f1 = self.f1 if p.sbet1 == 0 and calp1 == 0: # Break degeneracy of equatorial line calp1 = -_TINY # sin(alp1) * cos(bet1) = sin(alp0) salp0 = salp1 * p.cbet1 calp0 = hypot(calp1, salp1 * p.sbet1) # calp0 > 0 # tan(bet1) = tan(sig1) * cos(alp1) # tan(omg1) = sin(alp0) * tan(sig1) # = sin(bet1) * tan(alp1) somg1 = salp0 * p.sbet1 comg1 = calp1 * p.cbet1 ssig1, csig1 = norm2(p.sbet1, comg1) # Without normalization we have schi1 = somg1 cchi1 = f1 * p.dn1 * comg1 # Enforce symmetries in the case abs(bet2) = -bet1. # Need to be careful about this case, since this can # yield singularities in the Newton iteration. # sin(alp2) * cos(bet2) = sin(alp0) salp2 = (salp0 / p.cbet2) if p.cbet2 != p.cbet1 else salp1 # calp2 = sqrt(1 - sq(salp2)) # = sqrt(sq(calp0) - sq(sbet2)) / cbet2 # and subst for calp0 and rearrange to give (choose # positive sqrt to give alp2 in [0, PI/2]). calp2 = abs(calp1) if p.bet12 is None else ( sqrt(p.bet12 + (calp1 * p.cbet1)**2) / p.cbet2) # tan(bet2) = tan(sig2) * cos(alp2) # tan(omg2) = sin(alp0) * tan(sig2). somg2 = salp0 * p.sbet2 comg2 = calp2 * p.cbet2 ssig2, csig2 = norm2(p.sbet2, comg2) # without normalization we have schi2 = somg2 cchi2 = f1 * p.dn2 * comg2 # omg12 = omg2 - omg1, limit to [0, PI] somg12, comg12 = _sincos12(somg1, comg1, somg2, comg2, noneg=True) # chi12 = chi2 - chi1, limit to [0, PI] schi12, cchi12 = _sincos12(somg1, cchi1, somg2, cchi2, noneg=True) p.setsigs(ssig1, csig1, ssig2, csig2) # sig12 = sig2 - sig1, limit to [0, PI] sig12 = atan2(*_sincos12(ssig1, csig1, ssig2, csig2, noneg=True)) eta12 = self._eF_reset_e2_f1_cH(calp0, salp0) / PI_2 # then ... eta12 *= fsum_(eF.deltaH(*p.sncndn2), -eF.deltaH(*p.sncndn1), sig12) # eta = chi12 - lam12 lam12 = atan2(*_sincos12(p.slam12, p.clam12, schi12, cchi12)) - eta12 # domg12 = chi12 - omg12 - deta12 domg12 = atan2(*_sincos12( somg12, comg12, schi12, cchi12)) - eta12 if not diffp: dlam12 = NAN # dv > 0 in ._Newton9 elif calp2: _, dlam12, _, _, _ = self._Lengths5(sig12, Caps.REDUCEDLENGTH, p) dlam12 *= f1 / (calp2 * p.cbet2) else: dlam12 = -f1 * _2_0 * p.dn1 / p.sbet1 # p.set_(deta12=-eta12, lam12=lam12) return lam12, sig12, salp2, calp2, domg12, dlam12 def _Lengths5(self, sig12, outmask, p): '''(INTERNAL) Return M{m12b = (reduced length) / self.b} and calculate M{s12b = distance / self.b} and M{m0}, the coefficient of secular term in expression for reduced length. @return: 5-Tuple C{(s12b, m12b, m0, M12, M21)}. ''' s12b = m12b = m0 = M12 = M21 = NAN eF = self._eF # outmask &= Caps._OUTMASK if (outmask & Caps.DISTANCE): # Missing a factor of self.b s12b = eF.cE / PI_2 * fsum_(eF.deltaE(*p.sncndn2), -eF.deltaE(*p.sncndn1), sig12) if (outmask & Caps._REDUCEDLENGTH_GEODESICSCALE): m0x = -eF.k2 * eF.cD / PI_2 J12 = -m0x * fsum_(eF.deltaD(*p.sncndn2), -eF.deltaD(*p.sncndn1), sig12) c12 = p.csig1 * p.csig2 if (outmask & Caps.REDUCEDLENGTH): m0 = m0x # Missing a factor of self.b. Add parens around # (csig1 * ssig2) and (ssig1 * csig2) to ensure # accurate cancellation for coincident points. m12b = fsum_(p.dn2 * (p.csig1 * p.ssig2), -p.dn1 * (p.ssig1 * p.csig2), J12 * c12) if (outmask & Caps.GEODESICSCALE): M12 = M21 = p.ssig1 * p.ssig2 + c12 t = (p.cbet1 - p.cbet2) * self.ep2 * \ (p.cbet1 + p.cbet2) / (p.dn1 + p.dn2) M12 += (p.ssig2 * t + p.csig2 * J12) * p.ssig1 / p.dn1 M21 -= (p.ssig1 * t + p.csig1 * J12) * p.ssig2 / p.dn2 return s12b, m12b, m0, M12, M21 def Line(self, lat1, lon1, azi1, caps=Caps.ALL): '''Set up a L{GeodesicLineExact} to compute several points on a single geodesic. @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first point (compass C{degrees}). @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returnedby calls to L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}. @return: A L{GeodesicLineExact} instance. @note: If the point is at a pole, the azimuth is defined by keeping B{C{lon1}} fixed, writing C{B{lat1} = ±(90 − ε)}, and taking the limit C{ε → 0+}. @see: C++ U{GeodesicExact.Line <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicExact.html>} and Python U{Geodesic.Line<https://GeographicLib.SourceForge.io/html/python/code.html>}. ''' return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug) def _LineTemp(self, lat1, lon1, azi1, caps): # for ._GDictDirect and in .azimuthal._GnomonicBase.reverse '''(INTERNAL) Get a temporary L{GeodesicLineExact} instance. ''' return _GeodesicLineExact(self, lat1, lon1, azi1, caps, self._debug, append=False) @Property_RO def n(self): '''Get the ellipsoid's I{3rd flattening} (C{float}), M{f / (2 - f) == (a - b) / (a + b)}. ''' return self.ellipsoid.n @Property_RO def _n_0_1(self): '''(INTERNAL) Cached once. ''' return abs(self.n) > _0_1 @Property_RO def _n_6PI(self): '''(INTERNAL) Cached once. ''' return abs(self.n) * _6_0 * PI def _Newton10(self, salp1, calp1, outmask, p): '''(INTERNAL) Split off from C{_GDictInverse} to reduce complexity/length. @return: 10-Tuple C{(sig12, salp1, calp1, salp2, calp2, domg12, s12x, m12x, M12, M21)}. ''' # This is a straightforward solution of f(alp1) = lambda12(alp1) - # lam12 = 0 with one wrinkle. f(alp) has exactly one root in the # interval (0, PI) and its derivative is positive at the root. # Thus f(alp) is positive for alp > alp1 and negative for alp < alp1. # During the course of the iteration, a range (alp1a, alp1b) is # maintained which brackets the root and with each evaluation of # f(alp) the range is shrunk, if possible. Newton's method is # restarted whenever the derivative of f is negative (because the # new value of alp1 is then further from the solution) or if the # new estimate of alp1 lies outside (0,PI); in this case, the new # starting guess is taken to be (alp1a + alp1b) / 2. salp1a = salp1b = _TINY calp1a, calp1b = _1_0, _N_1_0 tripb, TOLv = False, _TOL0 for it in range(_MAXIT2): # 1/4 meridian = 10e6 meter and random input, # estimated max error in nm (nano meter, by # checking Inverse problem by Direct). # # max iterations # log2(b/a) error mean sd # -7 387 5.33 3.68 # -6 345 5.19 3.43 # -5 269 5.00 3.05 # -4 210 4.76 2.44 # -3 115 4.55 1.87 # -2 69 4.35 1.38 # -1 36 4.05 1.03 # 0 15 0.01 0.13 # 1 25 5.10 1.53 # 2 96 5.61 2.09 # 3 318 6.02 2.74 # 4 985 6.24 3.22 # 5 2352 6.32 3.44 # 6 6008 6.30 3.45 # 7 19024 6.19 3.30 v, sig12, salp2, calp2, \ domg12, dv = self._Lambda5(salp1, calp1, it < _MAXIT1, p) # 2 * _TOL0 is approximately 1 ulp [0, PI] # reversed test to allow escape with NaNs if tripb or abs(v) < TOLv: break # update bracketing values if v > 0 and (it > _MAXIT1 or (calp1 / salp1) > (calp1b / salp1b)): salp1b, calp1b = salp1, calp1 elif v < 0 and (it > _MAXIT1 or (calp1 / salp1) < (calp1a / salp1a)): salp1a, calp1a = salp1, calp1 if it < _MAXIT1 and dv > 0: dalp1 = -v / dv if abs(dalp1) < PI: s, c = sincos2(dalp1) # nalp1 = alp1 + dalp1 s, c = _sincos12(-s, c, salp1, calp1) if s > 0: salp1, calp1 = norm2(s, c) # in some regimes we don't get quadratic convergence # because slope -> 0. So use convergence conditions # based on epsilon instead of sqrt(epsilon) TOLv = _TOL0 if abs(v) > _TOL016 else _TOL08 continue # Either dv was not positive or updated value was outside # legal range. Use the midpoint of the bracket as the next # estimate. This mechanism is not needed for the WGS84 # ellipsoid, but it does catch problems with more eccentric # ellipsoids. Its efficacy is such for the WGS84 test set # with the starting guess set to alp1 = 90deg: the WGS84 # test set: mean = 5.21, sd = 3.93, max = 24 WGS84 and # random input: mean = 4.74, sd = 0.99 salp1, calp1 = norm2((salp1a + salp1b) * _0_5, (calp1a + calp1b) * _0_5) tripb = fsum_(calp1a, -calp1, abs(salp1a - salp1)) < _TOLb or \ fsum_(calp1b, -calp1, abs(salp1b - salp1)) < _TOLb TOLv = _TOL0 else: raise GeodesicError(_no_(_convergence_), _MAXIT2, txt=repr(self)) # self.toRepr() s12x, m12x, _, M12, M21 = self._Lengths5(sig12, outmask, p) p.set_(iter=it, trip=tripb) return (sig12, salp1, calp1, salp2, calp2, domg12, s12x, m12x, M12, M21) def _sincos2f1(self, lat): '''(INTERNAL) Helper. ''' sbet, cbet = sincos2d(lat) # ensure cbet1 = +epsilon at poles; doing the fix on beta means # that sig12 will be <= 2*tiny for two points at the same pole sbet, cbet = norm2(sbet * self.f1, cbet) return sbet, max(_TINY, cbet) def toStr(self, prec=6, sep=_COMMASPACE_, **unused): # PYCHOK signature '''Return this C{GeodesicExact} as string. @kwarg prec: The C{float} precision, number of decimal digits (0..9). Trailing zero decimals are stripped for B{C{prec}} values of 1 and above, but kept for negative B{C{prec}} values. @kwarg sep: Optional separator to join (C{str}). @return: Tuple items (C{str}). ''' d = dict(ellipsoid=self.ellipsoid, C4Order=self.C4Order) return sep.join(pairs(d, prec=prec)) class GeodesicLineExact(_GeodesicLineExact): '''A pure Python version of I{Karney}'s C++ class U{GeodesicLineExact <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1GeodesicLineExact.html>}, modeled after I{Karney}'s Python class U{GeodesicLine<https://GeographicLib.SourceForge.io/ html/python/code.html#module-geographiclib.geodesicline>}. ''' def __init__(self, geodesic, lat1, lon1, azi1, caps=Caps.STANDARD, name=NN): '''New L{GeodesicLineExact} instance, allowing points to be found along a geodesic starting at C{(B{lat1}, B{lon1})} with azimuth B{C{azi1}}. @arg geodesic: The geodesic to use (L{GeodesicExact}). @arg lat1: Latitude of the first point (C{degrees}). @arg lon1: Longitude of the first point (C{degrees}). @arg azi1: Azimuth at the first points (compass C{degrees}). @kwarg caps: Bit-or'ed combination of L{Caps} values specifying the capabilities the L{GeodesicLineExact} instance should possess, i.e., which quantities can be returned by calls to L{GeodesicLineExact.Position} and L{GeodesicLineExact.ArcPosition}. @kwarg name: Optional name (C{str}). @raise TypeError: Invalid B{C{geodesic}}. ''' _xinstanceof(GeodesicExact, geodesic=geodesic) _GeodesicLineExact.__init__(self, geodesic, lat1, lon1, azi1, caps, 0, name=name) def _Astroid(x, y): '''(INTERNAL) Solve M{k^4 + 2 * k^3 - (x^2 + y^2 - 1 ) * k^2 - (2 * k + 1) * y^2 = 0} for positive root k. ''' p = x**2 q = y**2 r = q + p - _1_0 if q or r > 0: r = r / _6_0 # /= chokes PyChecker # avoid possible division by zero when r = 0 # by multiplying s and t by r^3 and r, resp. S = p * q / _4_0 # S = r^3 * s r3 = r**3 T3 = r3 + S # discriminant of the quadratic equation for T3 is # zero on the evolute curve p^(1/3)+q^(1/3) = 1 d = S * (S + _2_0 * r3) if d < 0: # T is complex, but u is defined for a real result a = atan2(sqrt(-d), -T3) # There are three possible cube roots, choose the one which # avoids cancellation. Note d < 0 implies that r < 0. u = r * (_1_0 + _2_0 * cos(a / _3_0)) else: # pick the sign on the sqrt to maximize abs(T3) to # minimize loss of precision due to cancellation. if d: T3 += copysign0(sqrt(d), T3) # T3 = (r * t)^3 # cbrt always returns the real root, cbrt(-8) = -2 u = cbrt(T3) # T = r * t if u: # T can be zero; but then r2 / T -> 0 u += r**2 / u u += r v = sqrt(u**2 + q) # avoid loss of accuracy when u < 0 uv = (q / (v - u)) if u < 0 else (u + v) w = (uv - q) / (_2_0 * v) # positive? # rearrange expression for k to avoid loss of accuracy due to # subtraction, division by 0 impossible because uv > 0, w >= 0 k = uv / (sqrt(w**2 + uv) + w) # guaranteed positive else: # q == 0 && r <= 0 # y = 0 with |x| <= 1. Handle this case directly, for # y small, positive root is k = abs(y) / sqrt(1 - x^2) k = _0_0 return k __all__ += _ALL_DOCS(GeodesicExact, GeodesicLineExact) # **) 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.