# Coverage for pygeodesy/sphericalTrigonometry.py : 95%

## 377 statements 358 run 19 missing 37 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 | # -*- coding: utf-8 -*- u'''Spherical, C{trigonometry}-based geodesy. Trigonometric classes geodetic (lat-/longitude) L{LatLon} and geocentric (ECEF) L{Cartesian} and functions L{areaOf}, L{intersection}, L{intersections2}, L{isPoleEnclosedBy}, L{meanOf}, L{nearestOn3} and L{perimeterOf}, I{all spherical}. Pure Python implementation of geodetic (lat-/longitude) methods using spherical trigonometry, transcoded from JavaScript originals by I{(C) Chris Veness 2011-2016} published under the same MIT Licence**, see U{Latitude/Longitude<https://www.Movable-Type.co.UK/scripts/latlong.html>}. ''' # make sure int/int division yields float quotient, see .basics from __future__ import division from pygeodesy.basics import copysign0, isnear0, isnon0, isscalar, map1, signOf from pygeodesy.datums import _ellipsoidal_datum, _mean_radius from pygeodesy.errors import _AssertionError, CrossError, crosserrors, \ _ValueError, IntersectionError, _xError, \ _xkwds, _xkwds_get from pygeodesy.fmath import favg, fdot, fmean, Fsum, fsum, fsum_, hypot from pygeodesy.formy import antipode_, bearing_, _bearingTo2, excessAbc, \ excessGirard, excessLHuilier, _radical2, vincentys_ from pygeodesy.interns import EPS, EPS1, PI, PI2, PI_2, PI_4, R_M, \ _EPS4, _coincident_, _colinear_, _convex_, \ _end_, _invalid_, _LatLon_, _near_concentric_, \ _not_, _points_, _SPACE_, _too_, _1_, _2_, \ _0_0, _0_5, _1_0, _2_0, _90_0 from pygeodesy.lazily import _ALL_LAZY, _ALL_OTHER from pygeodesy.named import notImplemented, _xnamed from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ NearestOn3Tuple, Triangle7Tuple, \ Triangle8Tuple from pygeodesy.nvectorBase import NvectorBase as _Nvector from pygeodesy.points import ispolar, nearestOn5 as _nearestOn5 from pygeodesy.props import deprecated_function, deprecated_method from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, \ LatLonSphericalBase, _rads3, _trilaterate5 from pygeodesy.streprs import Fmt as _Fmt # XXX shadowed from pygeodesy.units import Bearing_, Height, Lam_, Phi_, Radius, \ Radius_, Scalar from pygeodesy.utily import acos1, asin1, degrees90, degrees180, degrees2m, \ m2radians, radiansPI2, sincos2, tan_2, \ unrollPI, wrap180, wrapPI from pygeodesy.vector3d import sumOf, Vector3d from math import asin, atan2, cos, degrees, radians, sin __all__ = _ALL_LAZY.sphericalTrigonometry __version__ = '21.06.30' _infinite_ = 'infinite' _null_ = 'null' _parallel_ = 'parallel' _path_ = 'path' _PI_EPS4 = PI - _EPS4 if _PI_EPS4 >= PI: raise _AssertionError(EPS4=_EPS4, PI=PI, PI_EPS4=_PI_EPS4) def _destination2(a, b, r, t): '''(INTERNAL) Destination lat- and longitude in C{radians}. @arg a: Latitude (C{radians}). @arg b: Longitude (C{radians}). @arg r: Angular distance (C{radians}). @arg t: Bearing (compass C{radians}). @return: 2-Tuple (phi, lam) of (C{radians}, C{radiansPI}). ''' # see <https://www.EdWilliams.org/avform.htm#LL> sa, ca, sr, cr, st, ct = sincos2(a, r, t) a = asin1(ct * sr * ca + cr * sa) d = atan2(st * sr * ca, cr - sa * sin(a)) # note, in EdWilliams.org/avform.htm W is + and E is - return a, b + d def _r2m(r, radius): '''(INTERNAL) Angular distance in C{radians} to C{meter}. ''' if radius is not None: # not in (None, _0_0) r *= R_M if radius is R_M else Radius(radius) return r class Cartesian(CartesianSphericalBase): '''Extended to convert geocentric, L{Cartesian} points to spherical, geodetic L{LatLon}. ''' def toLatLon(self, **LatLon_datum_kwds): # PYCHOK LatLon=LatLon '''Convert this cartesian point to an C{Nvector}-based geodetic point. @kwarg LatLon_datum_kwds: Optional L{LatLon}, B{C{datum}} and other keyword arguments, ignored if C{B{LatLon}=None}. Use C{B{LatLon}=...} to override this L{LatLon} class or specify C{B{LatLon}=None}. @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with C{C} and C{M} if available. @raise TypeError: Invalid B{C{LatLon_datum_kwds}}. ''' kwds = _xkwds(LatLon_datum_kwds, LatLon=LatLon, datum=self.datum) return CartesianSphericalBase.toLatLon(self, **kwds) class LatLon(LatLonSphericalBase): '''New point on spherical model earth model. @example: >>> p = LatLon(52.205, 0.119) # height=0 ''' def alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False): '''Compute the (angular) distance (signed) from the start to the closest point on the great circle path defined by a start and an end point. That is, if a perpendicular is drawn from this point to the great circle path, the along-track distance is the distance from the start point to the point where the perpendicular crosses the path. @arg start: Start point of great circle path (L{LatLon}). @arg end: End point of great circle path (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Distance along the great circle path (C{meter}, same units as B{C{radius}}) or C{radians} if C{B{radius}=None}, positive if after the B{C{start}} toward the B{C{end}} point of the path or negative if before the B{C{start}} point. @raise TypeError: Invalid B{C{start}} or B{C{end}} point. @raise ValueError: Invalid B{C{radius}}. @example: >>> p = LatLon(53.2611, -0.7972) >>> s = LatLon(53.3206, -1.7297) >>> e = LatLon(53.1887, 0.1334) >>> d = p.alongTrackDistanceTo(s, e) # 62331.58 ''' r, x, b = self._trackDistanceTo3(start, end, radius, wrap) cx = cos(x) return _0_0 if isnear0(cx) else \ _r2m(copysign0(acos1(cos(r) / cx), cos(b)), radius) @deprecated_method def bearingTo(self, other, wrap=False, raiser=False): # PYCHOK no cover '''DEPRECATED, use method L{initialBearingTo}. ''' return self.initialBearingTo(other, wrap=wrap, raiser=raiser) def crossingParallels(self, other, lat, wrap=False): '''Return the pair of meridians at which a great circle defined by this and an other point crosses the given latitude. @arg other: The other point defining great circle (L{LatLon}). @arg lat: Latitude at the crossing (C{degrees}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: 2-Tuple C{(lon1, lon2)}, both in C{degrees180} or C{None} if the great circle doesn't reach B{C{lat}}. ''' self.others(other) a1, b1 = self.philam a2, b2 = other.philam a = radians(lat) db, b2 = unrollPI(b1, b2, wrap=wrap) sa, ca, sa1, ca1, \ sa2, ca2, sdb, cdb = sincos2(a, a1, a2, db) x = sa1 * ca2 * ca * sdb y = sa1 * ca2 * ca * cdb - ca1 * sa2 * ca z = ca1 * ca2 * sa * sdb h = hypot(x, y) if h < EPS or abs(z) > h: return None # great circle doesn't reach latitude m = atan2(-y, x) + b1 # longitude at max latitude d = acos1(z / h) # delta longitude to intersections return degrees180(m - d), degrees180(m + d) def crossTrackDistanceTo(self, start, end, radius=R_M, wrap=False): '''Compute the (angular) distance (signed) from this point to the great circle defined by a start and an end point. @arg start: Start point of great circle path (L{LatLon}). @arg end: End point of great circle path (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Distance to great circle (negative if to the left or positive if to the right of the path) (C{meter}, same units as B{C{radius}} or C{radians} if B{C{radius}} is C{None}). @raise TypeError: The B{C{start}} or B{C{end}} point is not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @example: >>> p = LatLon(53.2611, -0.7972) >>> s = LatLon(53.3206, -1.7297) >>> e = LatLon(53.1887, 0.1334) >>> d = p.crossTrackDistanceTo(s, e) # -307.5 ''' _, x, _ = self._trackDistanceTo3(start, end, radius, wrap) return _r2m(x, radius) def destination(self, distance, bearing, radius=R_M, height=None): '''Locate the destination from this point after having travelled the given distance on the given initial bearing. @arg distance: Distance travelled (C{meter}, same units as B{C{radius}}). @arg bearing: Bearing from this point (compass C{degrees360}). @kwarg radius: Mean earth radius (C{meter}). @kwarg height: Optional height at destination (C{meter}, same units a B{C{radius}}). @return: Destination point (L{LatLon}). @raise ValueError: Invalid B{C{distance}}, B{C{bearing}}, B{C{radius}} or B{C{height}}. @example: >>> p1 = LatLon(51.4778, -0.0015) >>> p2 = p1.destination(7794, 300.7) >>> p2.toStr() # '51.5135°N, 000.0983°W' @JSname: I{destinationPoint}. ''' a, b = self.philam r, t = _angular(distance, radius), Bearing_(bearing) a, b = _destination2(a, b, r, t) h = self.height if height is None else Height(height) return self.classof(degrees90(a), degrees180(b), height=h) def distanceTo(self, other, radius=R_M, wrap=False): '''Compute the (angular) distance from this to an other point. @arg other: The other point (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Distance between this and the B{C{other}} point (C{meter}, same units as B{C{radius}} or C{radians} if B{C{radius}} is C{None}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351); >>> d = p1.distanceTo(p2) # 404300 ''' self.others(other) a1, b1 = self.philam a2, b2 = other.philam db, _ = unrollPI(b1, b2, wrap=wrap) return _r2m(vincentys_(a2, a1, db), radius) # @Property_RO # def Ecef(self): # '''Get the ECEF I{class} (L{EcefVeness}), I{lazily}. # ''' # from pygeodesy.ecef import EcefKarney # return EcefKarney def greatCircle(self, bearing): '''Compute the vector normal to great circle obtained by heading on the given initial bearing from this point. Direction of vector is such that initial bearing vector b = c × n, where n is an n-vector representing this point. @arg bearing: Bearing from this point (compass C{degrees360}). @return: Vector representing great circle (L{Vector3d}). @raise ValueError: Invalid B{C{bearing}}. @example: >>> p = LatLon(53.3206, -1.7297) >>> g = p.greatCircle(96.0) >>> g.toStr() # (-0.794, 0.129, 0.594) ''' a, b = self.philam t = Bearing_(bearing) sa, ca, sb, cb, st, ct = sincos2(a, b, t) return Vector3d(sb * ct - cb * sa * st, -cb * ct - sb * sa * st, ca * st) # XXX .unit()? def initialBearingTo(self, other, wrap=False, raiser=False): '''Compute the initial bearing (forward azimuth) from this to an other point. @arg other: The other point (spherical L{LatLon}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @kwarg raiser: Optionally, raise L{CrossError} (C{bool}), use C{B{raiser}=True} for behavior like C{sphericalNvector.LatLon.initialBearingTo}. @return: Initial bearing (compass C{degrees360}). @raise CrossError: If this and the B{C{other}} point coincide, provided both B{C{raiser}} is C{True} and L{crosserrors} is C{True}. @raise TypeError: The B{C{other}} point is not L{LatLon}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351) >>> b = p1.initialBearingTo(p2) # 156.2 @JSname: I{bearingTo}. ''' self.others(other) a1, b1 = self.philam a2, b2 = other.philam # XXX behavior like sphericalNvector.LatLon.initialBearingTo if raiser and crosserrors() and max(abs(a2 - a1), abs(b2 - b1)) < EPS: raise CrossError(_points_, self, txt=_coincident_) return degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)) def intermediateTo(self, other, fraction, height=None, wrap=False): '''Locate the point at given fraction between this and an other point. @arg other: The other point (L{LatLon}). @arg fraction: Fraction between both points (float, between 0.0 for this and 1.0 for the other point). @kwarg height: Optional height, overriding the fractional height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Intermediate point (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{fraction}} or B{C{height}}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351) >>> p = p1.intermediateTo(p2, 0.25) # 51.3721°N, 000.7073°E @JSname: I{intermediatePointTo}. ''' self.others(other) f = Scalar(fraction=fraction) # not high=_1_0 a1, b1 = self.philam a2, b2 = other.philam db, b2 = unrollPI(b1, b2, wrap=wrap) r = vincentys_(a2, a1, db) sr = sin(r) if isnon0(sr): sa1, ca1, sa2, ca2, \ sb1, cb1, sb2, cb2 = sincos2(a1, a2, b1, b2) A = sin((_1_0 - f) * r) / sr B = sin( f * r) / sr x = A * ca1 * cb1 + B * ca2 * cb2 y = A * ca1 * sb1 + B * ca2 * sb2 z = A * sa1 + B * sa2 a = atan2(z, hypot(x, y)) b = atan2(y, x) else: # points too close a = favg(a1, a2, f=f) b = favg(b1, b2, f=f) h = self._havg(other, f=f) if height is None else Height(height) return self.classof(degrees90(a), degrees180(b), height=h) def intersection(self, end1, other, end2, height=None, wrap=False): '''Compute the intersection point of two paths, each defined by two points or a start point and bearing from North. @arg end1: End point of this path (L{LatLon}) or the initial bearing at this point (compass C{degrees360}). @arg other: Start point of the other path (L{LatLon}). @arg end2: End point of the other path (L{LatLon}) or the initial bearing at the other start point (compass C{degrees360}). @kwarg height: Optional height for intersection point, overriding the mean height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: The intersection point (L{LatLon}). An alternate intersection point might be the L{antipode} to the returned result. @raise IntersectionError: Ambiguous or infinite intersection or colinear, parallel or otherwise non-intersecting paths. @raise TypeError: If B{C{other}} is not L{LatLon} or B{C{end1}} or B{C{end2}} not C{scalar} nor L{LatLon}. @raise ValueError: Invalid B{C{height}} or C{null} path. @example: >>> p = LatLon(51.8853, 0.2545) >>> s = LatLon(49.0034, 2.5735) >>> i = p.intersection(108.547, s, 32.435) # '50.9078°N, 004.5084°E' ''' try: s2 = self.others(other) return _intersect(self, end1, s2, end2, height=height, wrap=wrap, LatLon=self.classof) except (TypeError, ValueError) as x: raise _xError(x, start1=self, end1=end1, other=other, end2=end2) def intersections2(self, rad1, other, rad2, radius=R_M, eps=_0_0, height=None, wrap=True): '''Compute the intersection points of two circles, each defined by a center point and radius. @arg rad1: Radius of the this circle (C{meter} or C{radians}, see B{C{radius}}). @arg other: Center point of the other circle (L{LatLon}). @arg rad2: Radius of the other circle (C{meter} or C{radians}, see B{C{radius}}). @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}}, B{C{rad2}} and B{C{eps}} are given in C{radians}). @kwarg eps: Required overlap (C{meter} or C{radians}, see B{C{radius}}). @kwarg height: Optional height for the intersection points (C{meter}, conventionally) or C{None} for the I{"radical height"} at the I{radical line} between both centers. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: 2-Tuple of the intersection points, each a L{LatLon} instance. For abutting circles, both intersection points are the same instance. @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting circles. @raise TypeError: If B{C{other}} is not L{LatLon}. @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}}, B{C{eps}} or B{C{height}}. ''' try: c2 = self.others(other) return _intersects2(self, rad1, c2, rad2, radius=radius, eps=eps, height=height, wrap=wrap, LatLon=self.classof) except (TypeError, ValueError) as x: raise _xError(x, center=self, rad1=rad1, other=other, rad2=rad2) def isenclosedBy(self, points): '''Check whether a (convex) polygon encloses this point. @arg points: The polygon points (L{LatLon}[]). @return: C{True} if the polygon encloses this point, C{False} otherwise. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{points}}, non-convex polygon. @example: >>> b = LatLon(45,1), LatLon(45,2), LatLon(46,2), LatLon(46,1) >>> p = LatLon(45,1, 1.1) >>> inside = p.isEnclosedBy(b) # True ''' Ps = self.PointsIter(points, loop=2, dedup=True) n0 = self._N_vector v2 = Ps[0]._N_vector v1 = Ps[1]._N_vector gc1 = v2.cross(v1) # check whether this point on same side of all # polygon edges (to the left or right depending # on anti-/clockwise polygon direction) t0 = gc1.angleTo(n0) > PI_2 s0 = None # get great-circle vector for each edge for i, p in Ps.enumerate(closed=True): v2 = p._N_vector gc = v1.cross(v2) v1 = v2 t = gc.angleTo(n0) > PI_2 if t != t0: # different sides of edge i return False # outside # check for convex polygon: angle between # gc vectors, signed by direction of n0 # (otherwise the test above is not reliable) s = signOf(gc1.angleTo(gc, vSign=n0)) if s != s0: if s0 is None: s0 = s else: t = _Fmt.SQUARE(points=i) raise _ValueError(t, p, txt=_not_(_convex_)) gc1 = gc return True # inside @deprecated_method def isEnclosedBy(self, points): # PYCHOK no cover '''DEPRECATED, use method C{isenclosedBy}.''' return self.isenclosedBy(points) def midpointTo(self, other, height=None, wrap=False): '''Find the midpoint between this and an other point. @arg other: The other point (L{LatLon}). @kwarg height: Optional height for midpoint, overriding the mean height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Midpoint (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise ValueError: Invalid B{C{height}}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351) >>> m = p1.midpointTo(p2) # '50.5363°N, 001.2746°E' ''' self.others(other) # see <https://MathForum.org/library/drmath/view/51822.html> a1, b1 = self.philam a2, b2 = other.philam db, b2 = unrollPI(b1, b2, wrap=wrap) sa1, ca1, sa2, ca2, sdb, cdb = sincos2(a1, a2, db) x = ca2 * cdb + ca1 y = ca2 * sdb a = atan2(sa1 + sa2, hypot(x, y)) b = atan2(y, x) + b1 h = self._havg(other) if height is None else Height(height) return self.classof(degrees90(a), degrees180(b), height=h) def nearestOn(self, point1, point2, radius=R_M, **options): '''Locate the point between two points closest to this point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg point1: Start point (L{LatLon}). @arg point2: End point (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: Closest point on the great circle path (L{LatLon}). @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, see function L{equirectangular_}. @raise NotImplementedError: Keyword argument C{B{within}=False} is not (yet) supported. @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. @raise ValueError: Invalid B{C{radius}} or B{C{options}}. @see: Functions L{equirectangular_} and L{nearestOn5} and method L{sphericalTrigonometry.LatLon.nearestOn3}. ''' try: # remove kwarg B{C{within}} if present within = options.pop('within') if not within: notImplemented(self, within=within) # # UNTESTED - handle C{B{within}=False} and C{B{within}=True} # wrap = options.get('wrap', False) # a = self.alongTrackDistanceTo(point1, point2, radius=radius, wrap=wrap) # if abs(a) < EPS or (within and a < EPS): # return point1 # d = point1.distanceTo(point2, radius=radius, wrap=wrap) # if isnear0(d): # return point1 # or point2 # elif abs(d - a) < EPS or (a + EPS) > d: # return point2 # f = a / d # if within: # if f > EPS1: # return point2 # elif f < EPS: # return point1 # return point1.intermediateTo(point2, f, wrap=wrap) except KeyError: pass # without kwarg B{C{within}}, use backward compatible .nearestOn3 return self.nearestOn3([point1, point2], closed=False, radius=radius, **options)[0] @deprecated_method def nearestOn2(self, points, closed=False, radius=R_M, **options): # PYCHOK no cover '''DEPRECATED, use method L{sphericalTrigonometry.LatLon.nearestOn3}. @return: ... 2-Tuple C{(closest, distance)} of the closest point (L{LatLon}) on the polygon and the distance to that point from this point in C{meter}, same units of B{C{radius}}. ''' r = self.nearestOn3(points, closed=closed, radius=radius, **options) return r.closest, r.distance def nearestOn3(self, points, closed=False, radius=R_M, **options): '''Locate the point on a polygon closest to this point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of the C{closest} point (L{LatLon}), the L{equirectangular_} C{distance} between this and the C{closest} point in C{meter}, same units as B{C{radius}}. The C{angle} from this to the C{closest} point is in compass C{degrees360}, like function L{compassAngle}. @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, see function L{equirectangular_}. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not C{LatLon}. @raise ValueError: Invalid B{C{radius}} or B{C{options}}. @see: Functions L{compassAngle}, L{equirectangular_} and L{nearestOn5}. ''' if _LatLon_ in options: raise _ValueError(**options) lat, lon, d, c, h = _nearestOn5(self, points, closed=closed, **options) return NearestOn3Tuple(self.classof(lat, lon, height=h), degrees2m(d, radius=radius), c) def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None '''Convert this point to C{Karney}-based cartesian (ECEF) coordinates. @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} and other keyword arguments, ignored if C{B{Cartesian}=None}. Use C{B{Cartesian}=...} to override this L{Cartesian} class or specify C{B{Cartesian}=None}. @return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}} is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} with C{C} and C{M} if available. @raise TypeError: Invalid B{C{Cartesian_datum_kwds}}. ''' kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, datum=self.datum) return LatLonSphericalBase.toCartesian(self, **kwds) def _trackDistanceTo3(self, start, end, radius, wrap): '''(INTERNAL) Helper for .along-/crossTrackDistanceTo. ''' self.others(start=start) self.others(end=end) r = Radius_(radius) r = start.distanceTo(self, r, wrap=wrap) / r b = radians(start.initialBearingTo(self, wrap=wrap)) e = radians(start.initialBearingTo(end, wrap=wrap)) x = asin(sin(r) * sin(b - e)) return r, x, (e - b) def triangle7(self, otherB, otherC, radius=R_M, wrap=False): '''Compute the angles, sides and area of a spherical triangle. @arg otherB: Second triangle point (C{LatLon}). @arg otherC: Third triangle point (C{LatLon}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}. @kwarg wrap: Wrap/unroll angular distances (C{bool}). @return: L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} or if B{C{radius}} is C{None}, a L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)}. @see: Function L{triangle7} and U{Spherical trigonometry <https://WikiPedia.org/wiki/Spherical_trigonometry>}. ''' self.others(otherB=otherB) self.others(otherC=otherC) r = self.philam + otherB.philam + otherC.philam t = triangle8_(*r, wrap=wrap) return self._xnamed(_t7Tuple(t, radius)) def trilaterate5(self, distance1, point2, distance2, point3, distance3, area=True, eps=EPS1, radius=R_M, wrap=False): '''Trilaterate three points by area overlap or perimeter intersection of three corresponding circles. @arg distance1: Distance to this point (C{meter}, same units as B{C{radius}}). @arg point2: Second center point (C{LatLon}). @arg distance2: Distance to point2 (C{meter}, same units as B{C{radius}}). @arg point3: Third center point (C{LatLon}). @arg distance3: Distance to point3 (C{meter}, same units as B{C{radius}}). @kwarg area: If C{True} compute the area overlap, otherwise the perimeter intersection of the circles (C{bool}). @kwarg eps: The required I{minimal overlap} for C{B{area}=True} or the I{intersection margin} for C{B{area}=False} (C{meter}, same units as B{C{radius}}). @kwarg radius: Mean earth radius (C{meter}, conventionally). @kwarg wrap: Wrap/unroll angular distances (C{bool}). @return: A L{Trilaterate5Tuple}C{(min, minPoint, max, maxPoint, n)} with C{min} and C{max} in C{meter}, same units as B{C{eps}}, the corresponding trilaterated points C{minPoint} and C{maxPoint} as I{spherical} C{LatLon} and C{n}, the number of trilatered points found for the given B{C{eps}}. If only a single trilaterated point is found, C{min I{is} max}, C{minPoint I{is} maxPoint} and C{n = 1}. For C{B{area}=True}, C{min} and C{max} are the smallest respectively largest I{radial} overlap found. For C{B{area}=False}, C{min} and C{max} represent the nearest respectively farthest intersection margin. If C{B{area}=True} and all 3 circles are concentric, C{n = 0} and C{minPoint} and C{maxPoint} are both the B{C{point#}} with the smallest B{C{distance#}} C{min} and C{max} the largest B{C{distance#}}. @raise IntersectionError: Trilateration failed for the given B{C{eps}}, insufficient overlap for C{B{area}=True} or no intersection or all (near-)concentric for C{B{area}=False}. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @raise ValueError: Coincident B{C{points}} or invalid B{C{distance1}}, B{C{distance2}}, B{C{distance3}} or B{C{radius}}. ''' return _trilaterate5(self, distance1, self.others(point2=point2), distance2, self.others(point3=point3), distance3, area=area, radius=radius, eps=eps, wrap=wrap) _T00 = LatLon(0, 0, name='T00') # reference instance (L{LatLon}) def areaOf(points, radius=R_M, wrap=True): '''Calculate the area of a (spherical) polygon (with the points joined by great circle arcs). @arg points: The polygon points (L{LatLon}[]). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Polygon area (C{meter} I{quared}, same units as B{C{radius}} or C{radians} if B{C{radius}} is C{None}). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{radius}} or semi-circular polygon edge. @note: The area is based on I{Karney}'s U{'Area of a spherical polygon'<https://OSGeo-org.1560.x6.nabble.com/ Area-of-a-spherical-polygon-td3841625.html>}. @see: L{pygeodesy.areaOf}, L{sphericalNvector.areaOf}, L{ellipsoidalKarney.areaOf} and L{excessKarney}. @example: >>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1) >>> areaOf(b) # 8666058750.718977 >>> c = LatLon(0, 0), LatLon(1, 0), LatLon(0, 1) >>> areaOf(c) # 6.18e9 ''' Ps = _T00.PointsIter(points, loop=1) p1 = p2 = Ps[0] A = Fsum() # mean phi E = Fsum() # see L{excessKarney_} a1, b1 = p1.philam ta1 = tan_2(a1) # ispolar: Summation of course deltas around pole is 0° rather than normally ±360° # <https://blog.Element84.com/determining-if-a-spherical-polygon-contains-a-pole.html> # XXX duplicate of function C{points.ispolar} to avoid copying all iterated points D = Fsum() z1 = _0_0 # z1, _ = _bearingTo2(p2, p1, wrap=wrap) for i, p2 in Ps.enumerate(closed=True): a2, b2 = p2.philam db, b2 = unrollPI(b1, b2, wrap=wrap if i else False) ta2 = tan_2(a2) E += atan2(tan_2(db, points=i) * (ta1 + ta2), _1_0 + ta1 * ta2) ta1, b1 = ta2, b2 if not p2.isequalTo(p1, EPS): z, z2 = _bearingTo2(p1, p2, wrap=wrap) D.fadd_(wrap180(z - z1), # (z - z1 + 540) % 360 - 180 wrap180(z2 - z)) # (z2 - z + 540) % 360 - 180 p1, z1 = p2, z2 A += a2 r = abs(E.fsum()) * _2_0 if abs(D.fsum()) < _90_0: # ispolar(points) r = abs(r - PI2) if radius: a = degrees(A.fsum() / len(A)) # mean lat r *= _mean_radius(radius, a)**2 return r def _ib(a1, b1, end, a, b, wrap): # difference between the bearing to (a, b) and the given # bearing is negative if both are in opposite directions r = bearing_(a1, b1, a, b, wrap=wrap) return abs(wrapPI(r - radians(end))) > PI_2 def _i3d2(start, end, wrap, _i_, hs): # see <https://www.EdWilliams.org/intersect.htm> (5) ff a1, b1 = start.philam if isscalar(end): # bearing, make a point a2, b2 = _destination2(a1, b1, PI_4, radians(end)) else: # must be a point start.others(end, name=_end_ + _i_) hs.append(end.height) a2, b2 = end.philam db, b2 = unrollPI(b1, b2, wrap=wrap) if max(abs(db), abs(a2 - a1)) < EPS: raise _ValueError(_SPACE_(_path_ + _i_, _null_)) # note, in EdWilliams.org/avform.htm W is + and E is - b21, b12 = db * _0_5, -(b1 + b2) * _0_5 sb21, cb21, sb12, cb12, \ sa21, _, sa12, _ = sincos2(b21, b12, a1 - a2, a1 + a2) x = _Nvector(sa21 * sb12 * cb21 - sa12 * cb12 * sb21, sa21 * cb12 * cb21 + sa12 * sb12 * sb21, cos(a1) * cos(a2) * sin(db)) # ll=start return x.unit(), (db, (a2 - a1)) # negated d def _idot(ds, a1, b1, a, b, wrap): # compute dot product d . (-b + b1, a - a1) db, _ = unrollPI(b1, b, wrap=wrap) return fdot(ds, db, a - a1) def intersection(start1, end1, start2, end2, height=None, wrap=False, LatLon=LatLon, **LatLon_kwds): '''Compute the intersection point of two paths, each defined by two points or a start point and bearing from North. @arg start1: Start point of the first path (L{LatLon}). @arg end1: End point ofthe first path (L{LatLon}) or the initial bearing at the first start point (compass C{degrees360}). @arg start2: Start point of the second path (L{LatLon}). @arg end2: End point of the second path (L{LatLon}) or the initial bearing at the second start point (compass C{degrees360}). @kwarg height: Optional height for the intersection point, overriding the mean height (C{meter}). @kwarg wrap: Wrap and unroll longitudes (C{bool}). @kwarg LatLon: Optional class to return the intersection point (L{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}=None}. @return: The intersection point as a (B{C{LatLon}}) or a if C{B{LatLon}=None} a L{LatLon3Tuple}C{(lat, lon, height)}. An alternate intersection point might be the L{antipode} to the returned result. @raise IntersectionError: Ambiguous or infinite intersection or colinear, parallel or otherwise non-intersecting paths. @raise TypeError: A B{C{start}} or B{C{end}} point not L{LatLon}. @raise ValueError: Invalid B{C{height}} or C{null} path. @example: >>> p = LatLon(51.8853, 0.2545) >>> s = LatLon(49.0034, 2.5735) >>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E' ''' _T00.others(start1=start1) _T00.others(start2=start2) try: return _intersect(start1, end1, start2, end2, height=height, wrap=wrap, LatLon=LatLon, **LatLon_kwds) except (TypeError, ValueError) as x: raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2) def _intersect(start1, end1, start2, end2, height=None, wrap=False, # in.ellipsoidalBaseDI._intersect3 LatLon=None, **LatLon_kwds): # (INTERNAL) Intersect two (spherical) path, see L{intersection} # above, separated to allow callers to embellish any exceptions hs = [start1.height, start2.height] a1, b1 = start1.philam a2, b2 = start2.philam db, b2 = unrollPI(b1, b2, wrap=wrap) r12 = vincentys_(a2, a1, db) if abs(r12) < EPS: # [nearly] coincident points a, b = favg(a1, a2), favg(b1, b2) # see <https://www.EdWilliams.org/avform.htm#Intersection> elif isscalar(end1) and isscalar(end2): # both bearings sa1, ca1, sa2, ca2, sr12, cr12 = sincos2(a1, a2, r12) x1, x2 = (sr12 * ca1), (sr12 * ca2) if isnear0(x1) or isnear0(x2): raise IntersectionError(_parallel_) # handle domain error for equivalent longitudes, # see also functions asin_safe and acos_safe at # <https://www.EdWilliams.org/avform.htm#Math> t1, t2 = map1(acos1, (sa2 - sa1 * cr12) / x1, (sa1 - sa2 * cr12) / x2) if sin(db) > 0: t12, t21 = t1, PI2 - t2 else: t12, t21 = PI2 - t1, t2 t13, t23 = map1(radiansPI2, end1, end2) x1, x2 = map1(wrapPI, t13 - t12, # angle 2-1-3 t21 - t23) # angle 1-2-3 sx1, cx1, sx2, cx2 = sincos2(x1, x2) if sx1 == 0 and sx2 == 0: # max(abs(sx1), abs(sx2)) < EPS raise IntersectionError(_infinite_) sx3 = sx1 * sx2 # XXX if sx3 < 0: # XXX raise ValueError(_ambiguous_) x3 = acos1(cr12 * sx3 - cx2 * cx1) r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3)) a, b = _destination2(a1, b1, r13, t13) # choose antipode for opposing bearings if _ib(a1, b1, end1, a, b, wrap) or \ _ib(a2, b2, end2, a, b, wrap): a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple else: # end point(s) or bearing(s) x1, d1 = _i3d2(start1, end1, wrap, _1_, hs) x2, d2 = _i3d2(start2, end2, wrap, _2_, hs) x = x1.cross(x2) if x.length < EPS: # [nearly] colinear or parallel paths raise IntersectionError(_colinear_) a, b = x.philam # choose intersection similar to sphericalNvector d1 = _idot(d1, a1, b1, a, b, wrap) if d1: d2 = _idot(d2, a2, b2, a, b, wrap) if (d2 < 0 and d1 > 0) or (d2 > 0 and d1 < 0): a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple h = fmean(hs) if height is None else Height(height) return _LL3Tuple(degrees90(a), degrees180(b), h, intersection, LatLon, LatLon_kwds) def intersections2(center1, rad1, center2, rad2, radius=R_M, eps=_0_0, height=None, wrap=True, LatLon=LatLon, **LatLon_kwds): '''Compute the intersection points of two circles each defined by a center point and a radius. @arg center1: Center of the first circle (L{LatLon}). @arg rad1: Radius of the first circle (C{meter} or C{radians}, see B{C{radius}}). @arg center2: Center of the second circle (L{LatLon}). @arg rad2: Radius of the second circle (C{meter} or C{radians}, see B{C{radius}}). @kwarg radius: Mean earth radius (C{meter} or C{None} if B{C{rad1}}, B{C{rad2}} and B{C{eps}} are given in C{radians}). @kwarg eps: Required overlap (C{meter} or C{radians}, see B{C{radius}}). @kwarg height: Optional height for the intersection points (C{meter}, conventionally) or C{None} for the I{"radical height"} at the I{radical line} between both centers. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @kwarg LatLon: Optional class to return the intersection points (L{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}=None}. @return: 2-Tuple of the intersection points, each a B{C{LatLon}} instance or if C{B{LatLon}=None} a L{LatLon3Tuple}C{(lat, lon, height)}. For abutting circles, both intersection points are the same instance. @raise IntersectionError: Concentric, antipodal, invalid or non-intersecting circles. @raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}. @raise ValueError: Invalid B{C{rad1}}, B{C{rad2}}, B{C{radius}}, B{C{eps}} or B{C{height}}. @note: Courtesy U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}. @see: This U{Answer<https://StackOverflow.com/questions/53324667/ find-intersection-coordinates-of-two-circles-on-earth/53331953>}. ''' c1 = _T00.others(center1=center1) c2 = _T00.others(center2=center2) try: return _intersects2(c1, rad1, c2, rad2, radius=radius, eps=eps, height=height, wrap=wrap, LatLon=LatLon, **LatLon_kwds) except (TypeError, ValueError) as x: raise _xError(x, center1=center1, rad1=rad1, center2=center2, rad2=rad2) def _intersects2(c1, rad1, c2, rad2, radius=R_M, eps=_0_0, # in .ellipsoidalBaseDI._intersects2 height=None, too_d=None, wrap=True, LatLon=LatLon, **LatLon_kwds): # (INTERNAL) Intersect two spherical circles, see L{intersections2} # above, separated to allow callers to embellish any exceptions def _dest3(bearing, h): a, b = _destination2(a1, b1, r1, bearing) return _LL3Tuple(degrees90(a), degrees180(b), h, intersections2, LatLon, LatLon_kwds) r1, r2, f = _rads3(rad1, rad2, radius) if f: # swapped c1, c2 = c2, c1 # PYCHOK swap a1, b1 = c1.philam a2, b2 = c2.philam db, b2 = unrollPI(b1, b2, wrap=wrap) d = vincentys_(a2, a1, db) # radians if d < max(r1 - r2, EPS): raise IntersectionError(_near_concentric_) r = eps if radius is None else (m2radians( eps, radius=radius) if eps else _0_0) if r < _0_0: raise _ValueError(eps=r) x = fsum_(r1, r2, -d) # overlap if x > max(r, EPS): sd, cd, sr1, cr1, _, cr2 = sincos2(d, r1, r2) x = sd * sr1 if isnear0(x): raise _ValueError(_invalid_) x = acos1((cr2 - cd * cr1) / x) # 0 <= x <= PI elif x < r: t = (d * radius) if too_d is None else too_d raise IntersectionError(_too_(_Fmt.distant(t))) if height is None: # "radical height" f = _radical2(d, r1, r2).ratio h = Height(favg(c1.height, c2.height, f=f)) else: h = Height(height) b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap) if x < _EPS4: # externally ... r = _dest3(b, h) elif x > _PI_EPS4: # internally ... r = _dest3(b + PI, h) else: return _dest3(b + x, h), _dest3(b - x, h) return r, r # ... abutting circles @deprecated_function def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover '''DEPRECATED, use function L{ispolar}. ''' return ispolar(points, wrap=wrap) def _LL3Tuple(lat, lon, height, func, LatLon, LatLon_kwds): '''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanOf}. ''' n = func.__name__ if LatLon is None: r = LatLon3Tuple(lat, lon, height, name=n) else: kwds = _xkwds(LatLon_kwds, height=height, name=n) r = LatLon(lat, lon, **kwds) return r def meanOf(points, height=None, LatLon=LatLon, **LatLon_kwds): '''Compute the geographic mean of several points. @arg points: Points to be averaged (L{LatLon}[]). @kwarg height: Optional height at mean point, overriding the mean height (C{meter}). @kwarg LatLon: Optional class to return the mean point (L{LatLon}) or C{None}. @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}=None}. @return: Point at geographic mean and height (B{C{LatLon}}) or a L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: No B{C{points}} or invalid B{C{height}}. ''' # geographic mean m = sumOf(p._N_vector for p in _T00.PointsIter(points).iterate(closed=False)) lat, lon = m._N_vector.latlon if height is None: h = fmean(p.height for p in _T00.PointsIter(points).iterate(closed=False)) else: h = Height(height) return _LL3Tuple(lat, lon, h, meanOf, LatLon, LatLon_kwds) @deprecated_function def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}. @return: ... 2-tuple C{(closest, distance)} of the C{closest} point (L{LatLon}) on the polygon and the C{distance} between the C{closest} and the given B{C{point}}. The C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat, lon)} if B{C{LatLon}} is C{None} ... ''' ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None: ll = LatLon2Tuple(ll.lat, ll.lon) return ll, d def nearestOn3(point, points, closed=False, radius=R_M, LatLon=LatLon, **options): '''Locate the point on a polygon closest to an other, reference point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @arg point: The other, reference point (L{LatLon}). @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg LatLon: Optional class to return the closest point (L{LatLon}) or C{None}. @kwarg options: Optional keyword arguments for function L{equirectangular_}. @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the C{closest} point as B{C{LatLon}} or L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. The C{distance} is the L{equirectangular_} distance between the C{closest} and the given B{C{point}} in C{meter}, same units as B{C{radius}}. The C{angle} from the given B{C{point}} to the C{closest} is in compass C{degrees360}, like function L{compassAngle}. The C{height} is the (interpolated) height at the C{closest} point. @raise LimitError: Lat- and/or longitudinal delta exceeds the B{C{limit}}, see function L{equirectangular_}. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not C{LatLon}. @raise ValueError: Invalid B{C{radius}}. @see: Functions L{equirectangular_} and L{nearestOn5}. ''' lat, lon, d, c, h = _nearestOn5(point, points, closed=closed, LatLon=None, **options) r = LatLon3Tuple(lat, lon, h) if LatLon is None else \ LatLon(lat, lon, height=h) r = NearestOn3Tuple(r, degrees2m(d, radius=radius), c) return _xnamed(r, nearestOn3.__name__) def perimeterOf(points, closed=False, radius=R_M, wrap=True): '''Compute the perimeter of a (spherical) polygon (with great circle arcs joining the points). @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}) or C{None}. @kwarg wrap: Wrap and unroll longitudes (C{bool}). @return: Polygon perimeter (C{meter}, same units as B{C{radius}} or C{radians} if B{C{radius}} is C{None}). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Invalid B{C{radius}}. @note: This perimeter is based on the L{haversine} formula. @see: L{pygeodesy.perimeterOf}, L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}. ''' def _rads(Ps, closed, wrap): # angular edge lengths in radians a1, b1 = Ps[0].philam for i, p in Ps.enumerate(closed=closed): a2, b2 = p.philam db, b2 = unrollPI(b1, b2, wrap=wrap if i else False) yield vincentys_(a2, a1, db) a1, b1 = a2, b2 r = fsum(_rads(_T00.PointsIter(points, loop=1), closed, wrap)) return r if radius is None else (Radius(radius) * r) def triangle7(latA, lonA, latB, lonB, latC, lonC, radius=R_M, excess=excessAbc, wrap=False): '''Compute the angles, sides, and area of a (spherical) triangle. @arg latA: First corner latitude (C{degrees}). @arg lonA: First corner longitude (C{degrees}). @arg latB: Second corner latitude (C{degrees}). @arg lonB: Second corner longitude (C{degrees}). @arg latC: Third corner latitude (C{degrees}). @arg lonC: Third corner longitude (C{degrees}). @kwarg radius: Mean earth radius, ellipsoid or datum (C{meter}, L{Ellipsoid}, L{Ellipsoid2}, L{Datum} or L{a_f2Tuple}) or C{None}. @kwarg excess: I{Spherical excess} callable (L{excessAbc}, L{excessGirard} or L{excessLHuilier}). @kwarg wrap: Wrap and L{unroll180} longitudes (C{bool}). @return: A L{Triangle7Tuple}C{(A, a, B, b, C, c, area)} with spherical angles C{A}, C{B} and C{C}, angular sides C{a}, C{b} and C{c} all in C{degrees} and C{area} in I{square} C{meter} or units of B{C{radius}} I{squared} or if B{C{radius}} is C{None}, a L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} all in C{radians} with I{spherical excess} C{E} as the C{unit area} in C{radians}. ''' t = triangle8_(Phi_(latA=latA), Lam_(lonA=lonA), Phi_(latB=latB), Lam_(lonB=lonB), Phi_(latC=latC), Lam_(lonC=lonC), excess=excess, wrap=wrap) return _t7Tuple(t, radius) def triangle8_(phiA, lamA, phiB, lamB, phiC, lamC, excess=excessAbc, wrap=False): '''Compute the angles, sides, I{spherical deficit} and I{spherical excess} of a (spherical) triangle. @arg phiA: First corner latitude (C{radians}). @arg lamA: First corner longitude (C{radians}). @arg phiB: Second corner latitude (C{radians}). @arg lamB: Second corner longitude (C{radians}). @arg phiC: Third corner latitude (C{radians}). @arg lamC: Third corner longitude (C{radians}). @kwarg excess: I{Spherical excess} callable (L{excessAbc}, L{excessGirard} or L{excessLHuilier}). @kwarg wrap: Wrap and L{unrollPI} longitudes (C{bool}). @return: A L{Triangle8Tuple}C{(A, a, B, b, C, c, D, E)} with spherical angles C{A}, C{B} and C{C}, angular sides C{a}, C{b} and C{c}, I{spherical deficit} C{D} and I{spherical excess} C{E}, all in C{radians}. ''' def _a_r(w, phiA, lamA, phiB, lamB, phiC, lamC): d, _ = unrollPI(lamB, lamC, wrap=w) a = vincentys_(phiC, phiB, d) return a, (phiB, lamB, phiC, lamC, phiA, lamA) def _A_r(a, sa, ca, sb, cb, sc, cc): s = sb * sc A = acos1((ca - cb * cc) / s) if isnon0(s) else a return A, (sb, cb, sc, cc, sa, ca) # rotate sincos2's # notation: side C{a} is oposite to corner C{A}, etc. a, r = _a_r(wrap, phiA, lamA, phiB, lamB, phiC, lamC) b, r = _a_r(wrap, *r) c, _ = _a_r(wrap, *r) A, r = _A_r(a, *sincos2(a, b, c)) B, r = _A_r(b, *r) C, _ = _A_r(c, *r) D = fsum_(PI2, -a, -b, -c) # deficit aka defect E = excessGirard(A, B, C) if excess in (excessGirard, True) else ( excessLHuilier(a, b, c) if excess in (excessLHuilier, False) else excessAbc(*max((A, b, c), (B, c, a), (C, a, b)))) return Triangle8Tuple(A, a, B, b, C, c, D, E) def _t7Tuple(t, radius): '''(INTERNAL) Convert a L{Triangle8Tuple} to L{Triangle7Tuple}. ''' if radius: # not in (None, _0_0) r = radius if isscalar(radius) else \ _ellipsoidal_datum(radius).ellipsoid.Rmean A, B, C = map1(degrees, t.A, t.B, t.C) t = Triangle7Tuple(A, (r * t.a), B, (r * t.b), C, (r * t.c), t.E * r**2) return t __all__ += _ALL_OTHER(Cartesian, LatLon, # classes areaOf, # functions intersection, intersections2, ispolar, isPoleEnclosedBy, # DEPRECATED, use ispolar meanOf, nearestOn2, nearestOn3, perimeterOf, sumOf, # == vector3d.sumOf triangle7, triangle8_) # **) MIT License # # Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.