# Coverage for pygeodesy/sphericalNvector.py : 96%

## 301 statements 290 run 11 missing 20 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 | # -*- coding: utf-8 -*- u'''N-vector-based classes geodetic (lat-/longitude) L{LatLon}, geocentric (ECEF) L{Cartesian} and L{Nvector} and functions L{areaOf}, L{intersection}, L{meanOf}, L{nearestOn2}, L{triangulate} and L{trilaterate}, I{all spherical}. Pure Python implementation of n-vector-based spherical geodetic (lat-/longitude) methods, transcribed from JavaScript originals by I{(C) Chris Veness 2011-2016}, published under the same MIT Licence**. See U{Vector-based geodesy <https://www.Movable-Type.co.UK/scripts/latlong-vectors.html>} and U{Module latlon-nvector-spherical <https://www.Movable-Type.co.UK/scripts/geodesy/docs/module-latlon-nvector-spherical.html>}. Tools for working with points and paths on (a spherical model of) the earth’s surface using using n-vectors rather than the more common spherical trigonometry. N-vectors make many calculations much simpler, and easier to follow, compared with the trigonometric equivalents. Based on Kenneth Gade’s U{‘Non-singular Horizontal Position Representation’ <https://www.NavLab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf>}, The Journal of Navigation (2010), vol 63, nr 3, pp 395-417. Note that the formulations below take x => 0°N,0°E, y => 0°N,90°E and z => 90°N while Gade uses x => 90°N, y => 0°N,90°E, z => 0°N,0°E. Also note that on a spherical earth model, an n-vector is equivalent to a normalised version of an (ECEF) cartesian coordinate. @newfield example: Example, Examples ''' from pygeodesy.basics import EPS, EPS_2, PI, PI2, PI_2, R_M, \ isscalar, map1, _xinstanceof, _xkwds from pygeodesy.datum import Datums from pygeodesy.ecef import EcefKarney from pygeodesy.errors import _ValueError from pygeodesy.fmath import fidw, fmean, fsum, fsum_ from pygeodesy.lazily import _ALL_LAZY from pygeodesy.named import NearestOn3Tuple from pygeodesy.nvectorBase import NvectorBase, NorthPole, LatLonNvectorBase, \ sumOf as _sumOf from pygeodesy.points import _imdex2, ispolar # PYCHOK exported from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, \ LatLonSphericalBase from pygeodesy.streprs import unstr from pygeodesy.units import Bearing, Bearing_, Height, Radius, Radius_, Scalar from pygeodesy.utily import degrees360, iterNumpy2, sincos2, sincos2d from math import atan2, fabs, sqrt # all public contants, classes and functions __all__ = _ALL_LAZY.sphericalNvector + ( 'Cartesian', 'LatLon', 'Nvector', # classes 'areaOf', # functions 'intersection', 'ispolar', 'meanOf', 'nearestOn2', 'perimeterOf', 'sumOf', 'triangulate', 'trilaterate') __version__ = '20.05.08' class Cartesian(CartesianSphericalBase): '''Extended to convert geocentric, L{Cartesian} points to L{Nvector} and n-vector-based, spherical L{LatLon}. ''' def toLatLon(self, **LatLon_datum_kwds): # PYCHOK LatLon=LatLon '''Convert this cartesian to an C{Nvector}-based geodetic point. @kwarg LatLon_datum_kwds: Optional L{LatLon}, B{C{datum}} and other keyword arguments, ignored if B{C{LatLon=None}}. Use B{C{LatLon=...}} to override the L{LatLon} class or specify B{C{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) def toNvector(self, **Nvector_datum_kwds): # PYCHOK Datums.WGS84 '''Convert this cartesian to L{Nvector} components, I{including height}. @kwarg Nvector_datum_kwds: Optional L{Nvector}, B{C{datum}} and other keyword arguments, ignored if B{C{Nvector=None}}. Use B{C{Nvector=...}} to override the L{Nvector} class or specify B{C{Nvector=None}}. @return: The C{n-vector} components (L{Nvector}) or a L{Vector4Tuple}C{(x, y, z, h)} if B{C{Nvector}} is C{None}. @raise TypeError: Invalid B{C{Nvector_datum_kwds}}. ''' # ll = CartesianBase.toLatLon(self, LatLon=LatLon, # datum=datum or self.datum) # kwds = _xkwds(kwds, Nvector=Nvector) # return ll.toNvector(**kwds) kwds = _xkwds(Nvector_datum_kwds, Nvector=Nvector, datum=self.datum) return CartesianSphericalBase.toNvector(self, **kwds) class LatLon(LatLonNvectorBase, LatLonSphericalBase): '''New n-vector based point on a spherical earth model. Tools for working with points and paths on (a spherical model of) the earth's surface using vector-based methods. @example: >>> from sphericalNvector import LatLon >>> p = LatLon(52.205, 0.119) ''' _Nv = None #: (INTERNAL) cache _toNvector L{Nvector}). def _gc3(self, start, end, namend, raiser='points'): '''(INTERNAL) Return great circle, start and end Nvectors. ''' s = start.toNvector() if isscalar(end): # bearing gc = s.greatCircle(end) e = None else: self.others(end, name=namend) e = end.toNvector() gc = s.cross(e, raiser=raiser) # XXX .unit()? return gc, s, e def _update(self, updated, *attrs): # PYCHOK args '''(INTERNAL) Zap cached attributes if updated. ''' if updated: # reset caches LatLonNvectorBase._update(self, updated, _Nv=self._Nv) # special case LatLonSphericalBase._update(self, updated, *attrs) def alongTrackDistanceTo(self, start, end, radius=R_M): '''Compute the (signed) distance 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}) or initial bearing from start point (compass C{degrees360}). @kwarg radius: Mean earth radius (C{meter}). @return: Distance along the great circle path (positive if after the start toward the end point of the path or negative if before the start point). @raise TypeError: If B{C{start}} or B{C{end}} point is not L{LatLon}. @raise Valuerror: Some points coincide. @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 ''' self.others(start, name='start') gc, _, _ = self._gc3(start, end, 'end') p = self.toNvector() a = gc.cross(p).cross(gc) # along-track point gc × p × gc return start.toNvector().angleTo(a, vSign=gc) * radius def bearingTo(self, other, **unused): # PYCHOK no cover '''DEPRECATED, use method C{initialBearingTo}. ''' return self.initialBearingTo(other) def crossTrackDistanceTo(self, start, end, radius=R_M): '''Compute the (signed) distance from this point to great circle defined by a start and end point. @arg start: Start point of great circle path (L{LatLon}). @arg end: End point of great circle path (L{LatLon}) or initial bearing from start point (compass C{degrees360}). @kwarg radius: Mean earth radius (C{meter}). @return: Distance to great circle (negative if to the left or positive if to the right of the path). @raise TypeError: If B{C{start}} or B{C{end}} point is not L{LatLon}. @raise Valuerror: Some points coincide. @example: >>> p = LatLon(53.2611, -0.7972) >>> s = LatLon(53.3206, -1.7297) >>> d = p.crossTrackDistanceTo(s, 96) # -305.7 >>> e = LatLon(53.1887, 0.1334) >>> d = p.crossTrackDistanceTo(s, e) # -307.5 ''' self.others(start, name='start') gc, _, _ = self._gc3(start, end, 'end') p = self.toNvector() return (gc.angleTo(p) - PI_2) * 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 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, overriding the default height (C{meter}, same units as B{C{radius}}). @return: Destination point (L{LatLon}). @raise Valuerror: Polar coincidence ior invalid B{C{distance}}, B{C{bearing}}, B{C{radius}} or B{C{height}}. @example: >>> p = LatLon(51.4778, -0.0015) >>> q = p.destination(7794, 300.7) >>> q.toStr() # 51.513546°N, 000.098345°W @JSname: I{destinationPoint}. ''' a = _angular(distance, radius) sa, ca, sb, cb = sincos2(a, Bearing_(bearing)) p = self.toNvector() e = NorthPole.cross(p, raiser='pole').unit() # east vector at p n = p.cross(e) # north vector at p q = n.times(cb).plus(e.times(sb)) # direction vector @ p n = p.times(ca).plus(q.times(sa)) return n.toLatLon(height=height, LatLon=self.classof) # Nvector(n.x, n.y, n.z).toLatLon(...) def distanceTo(self, other, radius=R_M): '''Compute the distance from this to an other point. @arg other: The other point (L{LatLon}). @kwarg radius: Mean earth radius (C{meter}). @return: Distance between this and the B{C{other}} point (C{meter}, same units as B{C{radius}}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @example: >>> p = LatLon(52.205, 0.119) >>> q = LatLon(48.857, 2.351); >>> d = p.distanceTo(q) # 404.3 km ''' self.others(other) return self.toNvector().angleTo(other.toNvector()) * Radius(radius) def greatCircle(self, bearing): '''Compute the vector normal to great circle obtained by heading on the given 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: N-vector representing the great circle (L{Nvector}). ''' a, b = self.philam t = Bearing_(bearing) sa, ca, sb, cb, st, ct = sincos2(a, b, t) return Nvector(sb * ct - sa * cb * st, -cb * ct - sa * sb * st, ca * st, name=self.name) # XXX .unit() def greatCircleTo(self, other): '''Compute the vector normal to great circle obtained by heading from this to an other point or on a given bearing. Direction of vector is such that initial bearing vector b = c × n, where n is an n-vector representing this point. @arg other: The other point (L{LatLon}) or the bearing from this point (compass C{degrees360}). @return: N-vector representing the great circle (L{Nvector}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise Valuerror: Points coincide. @example: >>> p = LatLon(53.3206, -1.7297) >>> gc = p.greatCircle(96.0) >>> gc.toStr() # (-0.79408, 0.12856, 0.59406) >>> q = LatLon(53.1887, 0.1334) >>> g = p.greatCircleTo(q) >>> g.toStr() # (-0.79408, 0.12859, 0.59406) ''' gc, _, _ = self._gc3(self, other, 'other') return gc.unit() def initialBearingTo(self, other, **unused): '''Compute the initial bearing (forward azimuth) from this to an other point. @arg other: The other point (L{LatLon}). @arg unused: Optional keyword argument B{C{wrap}} ignored. @return: Initial bearing (compass C{degrees360}). @raise Crosserror: This point coincides with the B{C{other}} point or the C{NorthPole}, provided 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, name='other') # see <https://MathForum.org/library/drmath/view/55417.html> n = self.toNvector() # gc1 = self.greatCircleTo(other) gc1 = n.cross(other.toNvector(), raiser='points') # .unit() # gc2 = self.greatCircleTo(NorthPole) gc2 = n.cross(NorthPole, raiser='pole') # .unit() return degrees360(gc1.angleTo(gc2, vSign=n)) def intermediateChordTo(self, other, fraction, height=None): '''Locate the point projected from the point at given fraction on a straight line (chord) 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 at the intermediate point, overriding the fractional height (C{meter}). @return: Intermediate point (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @example: >>> p = LatLon(52.205, 0.119) >>> q = LatLon(48.857, 2.351) >>> i = p.intermediateChordTo(q, 0.25) # 51.3723°N, 000.7072°E @JSname: I{intermediatePointOnChordTo}, I{intermediatePointDirectlyTo}. ''' self.others(other) f = Scalar(fraction, name='fraction') i = other.toNvector().times(f).plus( self.toNvector().times(1 - f)) # i = other.toNvector() * f + \ # self.toNvector() * (1 - f)) h = self._havg(other, f=f) if height is None else Height(height) return i.toLatLon(height=h, LatLon=self.classof) # Nvector(i.x, i.y, i.z).toLatLon(...) def intermediateTo(self, other, fraction, height=None): '''Locate the point at a 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 at the intermediate point, overriding the fractional height (C{meter}). @return: Intermediate point (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise Valuerror: Points coincide or invalid B{C{height}}. @example: >>> p = LatLon(52.205, 0.119) >>> q = LatLon(48.857, 2.351) >>> i = p.intermediateTo(q, 0.25) # 51.3721°N, 000.7074°E @JSname: I{intermediatePointTo}. ''' self.others(other) f = Scalar(fraction, name='fraction') p = self.toNvector() q = other.toNvector() x = p.cross(q, raiser='points') d = x.unit().cross(p) # unit(p × q) × p # angular distance α, tan(α) = |p × q| / p ⋅ q s, c = sincos2(atan2(x.length, p.dot(q)) * f) # interpolated i = p.times(c).plus(d.times(s)) # p * cosα + d * sinα h = self._havg(other, f=f) if height is None else Height(height) return i.toLatLon(height=h, LatLon=self.classof) # Nvector(i.x, i.y, i.z).toLatLon(...) def intersection(self, end1, start2, end2, height=None): '''Locate the intersection point of two paths each defined by two points or a start point and bearing from North. @arg end1: End point of the first path (L{LatLon}) or the initial bearing at this 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 point (compass C{degrees}). @kwarg height: Optional height at the intersection point, overriding the mean height (C{meter}). @return: The intersection point (L{LatLon}) or C{None} if no unique intersection exists. @raise TypeError: If B{C{start2}}, B{C{end1}} or B{C{end2}} point is not L{LatLon}. @raise ValueError: Intersection is ambiguous or infinite or the paths are parallel, coincident or null. @example: >>> s = LatLon(51.8853, 0.2545) >>> e = LatLon(49.0034, 2.5735) >>> i = s.intersection(108.55, e, 32.44) # 50.9076°N, 004.5086°E ''' return intersection(self, end1, start2, end2, height=height, LatLon=self.classof) def isenclosedBy(self, points): '''Check whether this point is enclosed by a (convex) polygon. @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}. @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 @JSname: I{enclosedBy}. ''' n, points = self.points2(points, closed=True) # use normal vector to this point for sign of α n0 = self.toNvector() if iterNumpy2(points): def _subtangles(n, points, n0): # iterate vs = n0.minus(points[n-1].toNvector()) for i in range(n): vs1 = n0.minus(points[i].toNvector()) yield vs.angleTo(vs1, vSign=n0) # PYCHOK false vs = vs1 # sum subtended angles s = fsum(_subtangles(n, points, n0)) else: # get vectors from this to each point vs = [n0.minus(points[i].toNvector()) for i in range(n)] # close the polygon vs.append(vs[0]) # sum subtended angles of each edge (using n0 to determine sign) s = fsum(vs[i].angleTo(vs[i+1], vSign=n0) for i in range(n)) # Note, this method uses angle summation test: on a plane, # angles for an enclosed point will sum to 360°, angles for # an exterior point will sum to 0°. On a sphere, enclosed # point angles will sum to less than 360° (due to spherical # excess), exterior point angles will be small but non-zero. # XXX are winding number optimisations equally applicable to # spherical surface? return abs(s) > PI def isEnclosedBy(self, points): # PYCHOK no cover '''DEPRECATED, use method C{isenclosedBy}. ''' return self.isenclosedBy(points) def iswithin(self, point1, point2): '''Check whether this point is between two other points. If this point is not on the great circle arc defined by both points, return whether it is within the area bound by perpendiculars to the great circle at each point (in the same hemispere). @arg point1: Start point of the arc (L{LatLon}). @arg point2: End point of the arc (L{LatLon}). @return: C{True} if this point is within the arc, C{False} otherwise. @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}. @JSname: I{isBetween}. ''' self.others(point1, name='point1') self.others(point2, name='point2') n0 = self.toNvector() n1 = point1.toNvector() n2 = point2.toNvector() # corner case, null arc if n1.isequalTo(n2): return n0.isequalTo(n1) or n0.isequalTo(n2) # PYCHOK returns if n0.dot(n1) < 0 or n0.dot(n2) < 0: # different hemisphere return False # PYCHOK returns # get vectors representing d0=p0->p1 and d2=p2->p1 # and dot product d0⋅d2 tells us if p0 is on the # p2 side of p1 or on the other side (similarly # for d0=p0->p2 and d1=p1->p2 and dot product # d0⋅d1 and p0 on the p1 side of p2 or not) return n0.minus(n1).dot(n2.minus(n1)) >= 0 and \ n0.minus(n2).dot(n1.minus(n2)) >= 0 def isWithin(self, point1, point2): # PYCHOK no cover '''DEPRECATED, use method C{iswithin}. ''' return self.iswithin(point1, point2) def midpointTo(self, other, height=None): '''Find the midpoint between this and an other point. @arg other: The other point (L{LatLon}). @kwarg height: Optional height at the midpoint, overriding the mean height (C{meter}). @return: Midpoint (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @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) m = self.toNvector().plus(other.toNvector()) h = self._havg(other) if height is None else height return m.toLatLon(height=h, LatLon=self.classof) def nearestOn(self, point1, point2, height=None): '''Locate the point on the great circle arc between two points closest to this point. If this point is within the extent of the arc between both end points, return the closest point on the arc. Otherwise, return the closest of the arc's end points. @arg point1: Start point of the arc (L{LatLon}). @arg point2: End point of the arc (L{LatLon}). @kwarg height: Optional height, overriding the mean height for the point within the arc (C{meter}). @return: Closest point on the arc (L{LatLon}). @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}. @example: >>> s1 = LatLon(51.0, 1.0) >>> s2 = LatLon(51.0, 2.0) >>> s = LatLon(51.0, 1.9) >>> p = s.nearestOn(s1, s2) # 51.0004°N, 001.9000°E >>> d = p.distanceTo(s) # 42.71 m >>> s = LatLon(51.0, 2.1) >>> p = s.nearestOn(s1, s2) # 51.0000°N, 002.0000°E @JSname: I{nearestPointOnSegment}. ''' if self.isWithin(point1, point2) and not point1.isequalTo(point2, EPS): # closer to arc than to its endpoints, # find the closest point on the arc gc1 = point1.toNvector().cross(point2.toNvector()) gc2 = self.toNvector().cross(gc1) p = gc1.cross(gc2).toLatLon(height=height or 0, LatLon=self.classof) if height is None: # interpolate height d = point1.distanceTo(point2) f = 0.5 if d < EPS else point1.distanceTo(p) / d p.height = point1._havg(point2, f=f) # beyond arc extent, take closer endpoint elif self.distanceTo(point1) < self.distanceTo(point2): p = point1 else: p = point2 return p def nearestOn2(self, points, **closed_radius_height): # PYCHOK no cover '''DEPRECATED, use method L{sphericalNvector.LatLon.nearestOn3}. @return: ... 2-Tuple C{(closest, distance)} of the C{closest} point (L{LatLon}) on the polygon and the C{distance} to that point from this point ... ''' r = self.nearestOn3(points, **closed_radius_height) return r.closest, r.distance def nearestOn3(self, points, closed=False, radius=R_M, height=None): '''Locate the point on a polygon (with great circle arcs joining consecutive points) closest to this point. If this point is within the extent of any great circle arc, the closest point is on that arc. Otherwise, the closest is the nearest of the arc's end points. @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @kwarg height: Optional height, overriding the mean height for a point within the arc (C{meter}). @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of the C{closest} point (L{LatLon}), the C{distance} between this and the C{closest} point in C{meter}, same units as B{C{radius}} and the C{angle} from this to the C{closest} point in compass C{degrees360}. @raise TypeError: Some B{C{points}} are not C{LatLon}. @raise ValueError: No B{C{points}}. ''' n, points = self.points2(points, closed=closed) i, m = _imdex2(closed, n) c = p2 = points[i] r = self.distanceTo(c, radius=1) # force radians for i in range(m, n): p1, p2 = p2, points[i] p = self.nearestOn(p1, p2, height=height) t = self.distanceTo(p, radius=1) # force radians if t < r: c, r = p, t return NearestOn3Tuple(c, r * Radius(radius), degrees360(r)) def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None '''Convert this point to C{Nvector}-based cartesian (ECEF) coordinates. @kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} or other keyword arguments, ignored if B{C{Cartesian=None}}. Use B{C{Cartesian=...}} to override this L{Cartesian} class or specify B{C{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 toNvector(self, **Nvector_kwds): # PYCHOK signature '''Convert this point to L{Nvector} components, I{including height}. @kwarg Nvector_kwds: Optional L{Nvector} keyword arguments, ignored if B{C{Nvector=None}}. Use B{C{Nvector=...}} to override this L{Nvector} class or specify B{C{Nvector=None}} @return: The C{n-vector} components (L{Nvector}) or a L{Vector4Tuple}C{(x, y, z, h)} if B{C{Nvector}} is C{None}. @raise TypeError: Invalid B{C{Nvector_kwds}}. @example: >>> p = LatLon(45, 45) >>> n = p.toNvector() >>> n.toStr() # [0.50, 0.50, 0.70710] @JSname: I{toVector}. ''' kwds = _xkwds(Nvector_kwds, Nvector=Nvector) return LatLonNvectorBase.toNvector(self, **kwds) def triangulate(self, bearing1, other, bearing2, height=None): '''Locate a point given this and an other point and a bearing at this and the other point. @arg bearing1: Bearing at this point (compass C{degrees360}). @arg other: The other point (L{LatLon}). @arg bearing2: Bearing at the other point (compass C{degrees360}). @kwarg height: Optional height at the triangulated point, overriding the mean height (C{meter}). @return: Triangulated point (L{LatLon}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @raise Valuerror: Points coincide. @example: >>> p = LatLon("47°18.228'N","002°34.326'W") # Basse Castouillet >>> q = LatLon("47°18.664'N","002°31.717'W") # Basse Hergo >>> t = p.triangulate(7, q, 295) # 47.323667°N, 002.568501°W' ''' return triangulate(self, bearing1, other, bearing2, height=height, LatLon=self.classof) def trilaterate(self, distance1, point2, distance2, point3, distance3, radius=R_M, height=None, useZ=False): '''Locate a point at given distances from this and two other points. See also U{Trilateration<https://WikiPedia.org/wiki/Trilateration>}. @arg distance1: Distance to this point (C{meter}, same units as B{C{radius}}). @arg point2: Second reference point (L{LatLon}). @arg distance2: Distance to point2 (C{meter}, same units as B{C{radius}}). @arg point3: Third reference point (L{LatLon}). @arg distance3: Distance to point3 (C{meter}, same units as B{C{radius}}). @kwarg radius: Mean earth radius (C{meter}). @kwarg height: Optional height at trilaterated point, overriding the mean height (C{meter}, same units as B{C{radius}}). @kwarg useZ: Include Z component iff non-NaN, non-zero (C{bool}). @return: Trilaterated point (L{LatLon}). @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Distance(s) exceeds trilateration or some B{C{points}} coincide. ''' return trilaterate(self, distance1, point2, distance2, point3, distance3, radius=radius, height=height, LatLon=self.classof, useZ=useZ) class Nvector(NvectorBase): '''An n-vector is a position representation using a (unit) vector normal to the earth's surface. Unlike lat-/longitude points, n-vectors have no singularities or discontinuities. For many applications, n-vectors are more convenient to work with than other position representations like lat-/longitude, earth-centred earth-fixed (ECEF) vectors, UTM coordinates, etc. On a spherical model earth, an n-vector is equivalent to an earth-centred earth-fixed (ECEF) vector. Note commonality with L{ellipsoidalNvector.Nvector}. ''' _datum = Datums.Sphere #: (INTERNAL) Default datum (L{Datum}). _Ecef = EcefKarney #: (INTERNAL) Preferred C{Ecef...} class. def toCartesian(self, **Cartesian_h_kwds): # PYCHOK Cartesian=Cartesian '''Convert this n-vector to C{Nvector}-based cartesian (ECEF) coordinates. @kwarg Cartesian_h_kwds: Optional L{Cartesian}, B{C{h}} and other keyword arguments, ignored if B{C{Cartesian=None}}. Use B{C{Cartesian=...}} to override this L{Cartesian} class or specify B{C{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}}, B{C{h}} or other B{C{Cartesian_h_kwds}}. ''' kwds = _xkwds(Cartesian_h_kwds, h=self.h, Cartesian=Cartesian) return NvectorBase.toCartesian(self, **kwds) # class or .classof def toLatLon(self, **LatLon_height_kwds): # PYCHOK height=None, LatLon=LatLon '''Convert this n-vector to an C{Nvector}-based geodetic point. @kwarg LatLon_height_kwds: Optional L{LatLon}, B{C{height}} and other keyword arguments, ignored if B{C{LatLon=None}}. Use B{C{LatLon=...}} to override this L{LatLon} class or specify B{C{LatLon=None}}. @return: The geodetic point (L{LatLon}) or if B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, lon, height)}. @raise TypeError: Invalid B{C{LatLon}}, B{C{height}} or other B{C{LatLon_height_kwds}}. @raise ValueError: Invalid B{C{height}}. ''' kwds = _xkwds(LatLon_height_kwds, height=self.h, LatLon=LatLon) return NvectorBase.toLatLon(self, **kwds) # class or .classof def greatCircle(self, bearing): '''Compute the n-vector normal to great circle obtained by heading on given compass bearing from this point as its n-vector. Direction of vector is such that initial bearing vector b = c × p. @arg bearing: Initial compass bearing (C{degrees}). @return: N-vector representing great circle (L{Nvector}). @raise Valuerror: Polar coincidence. @example: >>> n = LatLon(53.3206, -1.7297).toNvector() >>> gc = n.greatCircle(96.0) # [-0.794, 0.129, 0.594] ''' s, c = sincos2d(Bearing(bearing)) e = NorthPole.cross(self, raiser='pole') # easting n = self.cross(e, raiser='point') # northing e = e.times(c / e.length) n = n.times(s / n.length) return n.minus(e) _Nvll = LatLon(0, 0, name='Nv00') #: (INTERNAL) Reference instance (L{LatLon}). def areaOf(points, radius=R_M): '''Calculate the area of a (spherical) polygon (with great circle arcs joining consecutive points). @arg points: The polygon points (L{LatLon}[]). @kwarg radius: Mean earth radius (C{meter}). @return: Polygon area (C{meter}, same units as B{C{radius}}, squared). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @see: L{pygeodesy.areaOf}, L{sphericalTrigonometry.areaOf} and L{ellipsoidalKarney.areaOf}. @example: >>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1) >>> areaOf(b) # 8666058750.718977 ''' n, points = _Nvll.points2(points, closed=True) # use vector to 1st point as plane normal for sign of α n0 = points[0].toNvector() if iterNumpy2(points): def _interangles(n, points, n0): # iterate v2 = points[n-2]._N_vector v1 = points[n-1]._N_vector gc = v2.cross(v1) for i in range(n): v2 = points[i]._N_vector gc1 = v1.cross(v2) v1 = v2 yield gc.angleTo(gc1, vSign=n0) gc = gc1 # sum interior angles s = fsum(_interangles(n, points, n0)) else: # get great-circle vector for each edge gc, v1 = [], points[n-1]._N_vector for i in range(n): v2 = points[i]._N_vector gc.append(v1.cross(v2)) # PYCHOK false, does have .cross v1 = v2 gc.append(gc[0]) # XXX needed? # sum interior angles: depending on whether polygon is cw or ccw, # angle between edges is π−α or π+α, where α is angle between # great-circle vectors; so sum α, then take n·π − |Σα| (cannot # use Σ(π−|α|) as concave polygons would fail) s = fsum(gc[i].angleTo(gc[i+1], vSign=n0) for i in range(n)) # using Girard’s theorem: A = [Σθᵢ − (n−2)·π]·R² # (PI2 - abs(s) == (n*PI - abs(s)) - (n-2)*PI) return abs(PI2 - abs(s)) * Radius(radius)**2 def intersection(start1, end1, start2, end2, height=None, LatLon=LatLon, **LatLon_kwds): '''Locate the intersection of two paths each defined by two points or by a start point and an initial bearing. @arg start1: Start point of the first path (L{LatLon}). @arg end1: End point of the 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 at the intersection point, overriding the mean height (C{meter}). @kwarg LatLon: Optional class to return the intersection point (L{LatLon}). @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if B{C{LatLon=None}}. @return: The intersection point (B{C{LatLon}}) or 3-tuple (C{degrees90}, C{degrees180}, height) if B{C{LatLon}} is C{None} or C{None} if no unique intersection exists. @raise TypeError: If B{C{start*}} or B{C{end*}} is not L{LatLon}. @raise ValueError: Intersection is ambiguous or infinite or the paths are parallel, coincident or null. @example: >>> p = LatLon(51.8853, 0.2545) >>> q = LatLon(49.0034, 2.5735) >>> i = intersection(p, 108.55, q, 32.44) # 50.9076°N, 004.5086°E ''' _Nvll.others(start1, name='start1') _Nvll.others(start2, name='start2') # If gc1 and gc2 are great circles through start and end points # (or defined by start point and bearing), then the candidate # intersections are simply gc1 × gc2 and gc2 × gc1. Most of the # work is deciding the correct intersection point to select! If # bearing is given, that determines the intersection, but if both # paths are defined by start/end points, take closer intersection. gc1, s1, e1 = _Nvll._gc3(start1, end1, 'end1') gc2, s2, e2 = _Nvll._gc3(start2, end2, 'end2') hs = start1.height, start2.height # there are two (antipodal) candidate intersection # points ... we have to choose the one to return i1 = gc1.cross(gc2, raiser='paths') # postpone computing i2 until needed # i2 = gc2.cross(gc1, raiser='paths') # selection of intersection point depends on how # paths are defined (by bearings or endpoints) if e1 and e2: # endpoint+endpoint d = sumOf((s1, s2, e1, e2)).dot(i1) hs += end1.height, end2.height elif e1 and not e2: # endpoint+bearing # gc2 x v2 . i1 +ve means v2 bearing points to i1 d = gc2.cross(s2).dot(i1) hs += end1.height, elif e2 and not e1: # bearing+endpoint # gc1 x v1 . i1 +ve means v1 bearing points to i1 d = gc1.cross(s1).dot(i1) hs += end2.height, else: # bearing+bearing # if gc x v . i1 is +ve, initial bearing is # towards i1, otherwise towards antipodal i2 d1 = gc1.cross(s1).dot(i1) # +ve means p1 bearing points to i1 d2 = gc2.cross(s2).dot(i1) # +ve means p2 bearing points to i1 if d1 > 0 and d2 > 0: d = 1 # both point to i1 elif d1 < 0 and d2 < 0: d = -1 # both point to i2 else: # d1, d2 opposite signs # intersection is at further-away intersection # point, take opposite intersection from mid- # point of v1 and v2 [is this always true?] d = -s1.plus(s2).dot(i1) i = i1 if d > 0 else gc2.cross(gc1, raiser='paths') h = fmean(hs) if height is None else height kwds = _xkwds(LatLon_kwds, height=h, LatLon=LatLon) return i.toLatLon(**kwds) # Nvector(i.x, i.y, i.z).toLatLon(...) def meanOf(points, height=None, LatLon=LatLon, **LatLon_kwds): '''Compute the geographic mean of the supplied points. @arg points: Array of points to be averaged (L{LatLon}[]). @kwarg height: Optional height, overriding the mean height (C{meter}). @kwarg LatLon: Optional class to return the mean point (L{LatLon}). @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if B{C{LatLon=None}}. @return: Point at geographic mean and mean height (B{C{LatLon}}). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not C{LatLon}. ''' n, points = _Nvll.points2(points, closed=False) # geographic mean m = sumOf(points[i]._N_vector for i in range(n)) kwds = _xkwds(LatLon_kwds, height=height, LatLon=LatLon) return m.toLatLon(**kwds) def nearestOn2(point, points, **closed_radius_height): # PYCHOK no cover '''DEPRECATED, use method L{sphericalNvector.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}} ... ''' r = nearestOn3(point, points, **closed_radius_height) return r.closest, r.distance def nearestOn3(point, points, closed=False, radius=R_M, height=None): '''Locate the point on a polygon (with great circle arcs joining consecutive points) closest to an other point. If the given point is within the extent of any great circle arc, the closest point is on that arc. Otherwise, the closest is the nearest of the arc's end points. @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 height: Optional height, overriding the mean height for a point within the arc (C{meter}). @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of the C{closest} point (L{LatLon}) on the polygon, the C{distance} and the C{angle} between the C{closest} and the given B{C{point}}. The C{distance} is in C{meter}, same units as B{C{radius}}, the C{angle} is in compass C{degrees360}. @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} or B{C{point}} not C{LatLon}. ''' _xinstanceof(LatLon, point=point) return point.nearestOn3(points, closed=closed, radius=radius, height=height) def perimeterOf(points, closed=False, radius=R_M): '''Compute the perimeter of a (spherical) polygon (with great circle arcs joining consecutive points). @arg points: The polygon points (L{LatLon}[]). @kwarg closed: Optionally, close the polygon (C{bool}). @kwarg radius: Mean earth radius (C{meter}). @return: Polygon perimeter (C{meter}, same units as B{C{radius}}). @raise PointsError: Insufficient number of B{C{points}}. @raise TypeError: Some B{C{points}} are not L{LatLon}. @see: L{pygeodesy.perimeterOf}, L{sphericalTrigonometry.perimeterOf} and L{ellipsoidalKarney.perimeterOf}. ''' n, points = _Nvll.points2(points, closed=closed) def _rads(n, points, closed): # angular edge lengths in radians i, m = _imdex2(closed, n) v1 = points[i]._N_vector for i in range(m, n): v2 = points[i]._N_vector yield v1.angleTo(v2) v1 = v2 r = fsum(_rads(n, points, closed)) return r * Radius(radius) def sumOf(nvectors, Vector=Nvector, h=None, **Vector_kwds): '''Return the vectorial sum of two or more n-vectors. @arg nvectors: Vectors to be added (L{Nvector}[]). @kwarg Vector: Optional class for the vectorial sum (L{Nvector}). @kwarg h: Optional height, overriding the mean height (C{meter}). @kwarg kwds: Optional, additional B{C{Vector}} keyword arguments. @return: Vectorial sum (B{C{Vector}}). @raise VectorError: No B{C{nvectors}}. ''' return _sumOf(nvectors, Vector=Vector, h=h, **Vector_kwds) def triangulate(point1, bearing1, point2, bearing2, height=None, LatLon=LatLon, **LatLon_kwds): '''Locate a point given two known points and initial bearings from those points. @arg point1: First reference point (L{LatLon}). @arg bearing1: Bearing at the first point (compass C{degrees360}). @arg point2: Second reference point (L{LatLon}). @arg bearing2: Bearing at the second point (compass C{degrees360}). @kwarg height: Optional height at the triangulated point, overriding the mean height (C{meter}). @kwarg LatLon: Optional class to return the triangulated point (L{LatLon}). @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if B{C{LatLon=None}}. @return: Triangulated point (B{C{LatLon}}). @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}. @raise Valuerror: Points coincide. @example: >>> p = LatLon("47°18.228'N","002°34.326'W") # Basse Castouillet >>> q = LatLon("47°18.664'N","002°31.717'W") # Basse Hergo >>> t = triangulate(p, 7, q, 295) # 47.323667°N, 002.568501°W' ''' _Nvll.others(point1, name='point1') _Nvll.others(point2, name='point2') if point1.isequalTo(point2, EPS): raise _ValueError(points=point2, txt='coincident') def _gc(p, b, i): n = p.toNvector() de = NorthPole.cross(n, raiser='pole').unit() # east vector @ n dn = n.cross(de) # north vector @ n s, c = sincos2d(Bearing(b, name='bearing' + i)) dest = de.times(s) dnct = dn.times(c) d = dnct.plus(dest) # direction vector @ n return n.cross(d) # great circle point + bearing gc1 = _gc(point1, bearing1, '1') # great circle p1 + b1 gc2 = _gc(point2, bearing2, '2') # great circle p2 + b2 n = gc1.cross(gc2, raiser='points') # n-vector of intersection point h = point1._havg(point2) if height is None else Height(height) kwds = _xkwds(LatLon_kwds, height=h, LatLon=LatLon) return n.toLatLon(**kwds) # Nvector(n.x, n.y, n.z).toLatLon(...) def trilaterate(point1, distance1, point2, distance2, point3, distance3, radius=R_M, height=None, useZ=False, LatLon=LatLon, **LatLon_kwds): '''Locate a point at given distances from three other points. See also U{Trilateration<https://WikiPedia.org/wiki/Trilateration>}. @arg point1: First point (L{LatLon}). @arg distance1: Distance to the first point (C{meter}, same units as B{C{radius}}). @arg point2: Second point (L{LatLon}). @arg distance2: Distance to the second point (C{meter}, same units as B{C{radius}}). @arg point3: Third point (L{LatLon}). @arg distance3: Distance to the third point (C{meter}, same units as B{C{radius}}). @kwarg radius: Mean earth radius (C{meter}). @kwarg height: Optional height at the trilaterated point, overriding the IDW height (C{meter}, same units as B{C{radius}}). @kwarg useZ: Include Z component iff non-NaN, non-zero (C{bool}). @kwarg LatLon: Optional class to return the trilaterated @kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if B{C{LatLon=None}}. @return: Trilaterated point (B{C{LatLon}}). @raise TypeError: If B{C{point1}}, B{C{point2}} or B{C{point3}} is not L{LatLon}. @raise ValueError: Invalid B{C{distance1}}, B{C{distance2}}, B{C{distance3}} or B{C{radius}}, or some B{C{distances}} exceed trilateration or some B{C{points}} coincide. ''' def _nd2(p, d, r, i, *qs): # return Nvector and angular distance squared _Nvll.others(p, name='point' + i) for q in qs: if p.isequalTo(q, EPS): raise _ValueError(points=p, txt='coincident') return p.toNvector(), (Scalar(d, name='distance' + i) / r)**2 r = Radius_(radius) n1, d12 = _nd2(point1, distance1, r, '1') n2, d22 = _nd2(point2, distance2, r, '2', point1) n3, d32 = _nd2(point3, distance3, r, '3', point1, point2) # the following uses x,y coordinate system with origin at n1, x axis n1->n2 y = n3.minus(n1) x = n2.minus(n1) d = x.length # distance n1->n2 if d > EPS_2: # and y.length > EPS_2: X = x.unit() # unit vector in x direction n1->n2 i = X.dot(y) # signed magnitude of x component of n1->n3 Y = y.minus(X.times(i)).unit() # unit vector in y direction j = Y.dot(y) # signed magnitude of y component of n1->n3 if abs(j) > EPS_2: # courtesy Carlos Freitas <https://GitHub.com/mrJean1/PyGeodesy/issues/33> x = fsum_(d12, -d22, d**2) / (2 * d) # n1->intersection x- and ... y = fsum_(d12, -d32, i**2, j**2) / (2 * j) - (x * i / j) # ... y-component n = n1.plus(X.times(x)).plus(Y.times(y)) # .plus(Z.times(z)) if useZ: # include non-NaN, non-zero Z component z = fsum_(d12, -(x**2), -(y**2)) if z > EPS: Z = X.cross(Y) # unit vector perpendicular to plane n = n.plus(Z.times(sqrt(z))) if height is None: h = fidw((point1.height, point2.height, point3.height), map1(fabs, distance1, distance2, distance3)) else: h = Height(height) kwds = _xkwds(LatLon_kwds, height=h, LatLon=LatLon) return n.toLatLon(**kwds) # Nvector(n.x, n.y, n.z).toLatLon(...) # no intersection, d < EPS_2 or abs(j) < EPS_2 t = unstr(trilaterate.__name__, point1, distance1, point2, distance2, point3, distance3, useZ=useZ, d=d) raise _ValueError('no intersection', txt=t) # **) MIT License # # Copyright (C) 2016-2020 -- 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.