# Coverage for pygeodesy/sphericalTrigonometry.py : 91%

## 275 statements 249 run 26 missing 10 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 | # -*- coding: utf-8 -*- u'''Trigonometric classes geodetic (lat-/longitude) L{LatLon} and geocentric (ECEF) L{Cartesian} and functionsL{areaOf}, L{intersection}, L{isPoleEnclosedBy}, L{meanOf}, L{nearestOn2} and L{perimeterOf}, I{all spherical}. Pure Python implementation of geodetic (lat-/longitude) methods using spherical trigonometry, transcribed 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>}. @newfield example: Example, Examples ''' from pygeodesy.datum import R_M from pygeodesy.fmath import EPS, acos1, favg, fdot, fmean, fsum, \ isscalar, map1 from pygeodesy.formy import antipode, bearing_, haversine_ from pygeodesy.lazily import _ALL_LAZY, _2kwds from pygeodesy.named import LatLon3Tuple, NearestOn3Tuple, _xnamed from pygeodesy.points import _imdex2, ispolar, nearestOn5 as _nearestOn5 from pygeodesy.sphericalBase import CartesianSphericalBase, LatLonSphericalBase from pygeodesy.utily import PI2, PI_2, PI_4, degrees90, degrees180, \ degrees2m, iterNumpy2, radiansPI2, \ sincos2, tan_2, unrollPI, wrapPI from pygeodesy.vector3d import CrossError, crosserrors, Vector3d, sumOf from math import asin, atan2, copysign, cos, degrees, hypot, \ radians, sin # all public contants, classes and functions __all__ = _ALL_LAZY.sphericalTrigonometry + ( 'Cartesian', 'LatLon', # classes 'areaOf', # functions 'intersection', 'ispolar', 'isPoleEnclosedBy', # DEPRECATED, use ispolar 'meanOf', 'nearestOn2', 'nearestOn3', 'perimeterOf', 'sumOf') # == vector3d.sumOf __version__ = '19.10.21' class Cartesian(CartesianSphericalBase): '''Extended to convert geocentric, L{Cartesian} points to spherical, geodetic L{LatLon}. ''' def toLatLon(self, **kwds): # PYCHOK LatLon=LatLon '''Convert this cartesian point to an C{Nvector}-based geodetic point. @keyword kwds: Optional, additional B{C{LatLon}} keyword arguments, ignored if C{B{LatLon}=None}. For example, use C{LatLon=...} to override the L{LatLon} (sub-)class or specify @return: The B{C{LatLon}} point (L{LatLon}) or if C{B{LatLon}=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}}, B{C{datum}} or B{C{kwds}}. ''' kwds = _2kwds(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 _trackDistanceTo3(self, start, end, radius, wrap): '''(INTERNAL) Helper for along-/crossTrackDistanceTo. ''' self.others(start, name='start') self.others(end, name='end') r = float(radius) if r < EPS: raise ValueError('%s invalid: %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 alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False): '''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. @param start: Start point of great circle path (L{LatLon}). @param end: End point of great circle path (L{LatLon}). @keyword radius: Optional, mean earth radius (C{meter}). @keyword wrap: Wrap and unroll longitudes (C{bool}). @return: Distance along the great circle path (C{meter}, same units as B{C{radius}}), 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: The B{C{start}} or B{C{end}} point is not L{LatLon}. @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) if abs(cx) > EPS: return copysign(acos1(cos(r) / cx), cos(b)) * radius else: return 0.0 def bearingTo(self, other, wrap=False, raiser=False): '''DEPRECATED, use method C{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. @param other: The other point defining great circle (L{LatLon}). @param lat: Latitude at the crossing (C{degrees}). @keyword 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.to2ab() a2, b2 = other.to2ab() 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 (signed) distance from this point to the great circle defined by a start and an end point. @param start: Start point of great circle path (L{LatLon}). @param end: End point of great circle path (L{LatLon}). @keyword radius: Optional, mean earth radius (C{meter}). @keyword 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). @raise TypeError: The B{C{start}} or B{C{end}} point is not L{LatLon}. @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 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. @param distance: Distance travelled (C{meter}, same units as B{C{radius}}). @param bearing: Bearing from this point (compass C{degrees360}). @keyword radius: Optional, mean earth radius (C{meter}). @keyword height: Optional height at destination (C{meter}, same units a B{C{radius}}). @return: Destination point (L{LatLon}). @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.to2ab() r = float(distance) / float(radius) # angular distance in radians t = radians(bearing) a, b = _destination2(a, b, r, t) h = self.height if height is None else height return self.classof(a, b, height=h) def distanceTo(self, other, radius=R_M, wrap=False): '''Compute the distance from this to an other point. @param other: The other point (L{LatLon}). @keyword radius: Optional, mean earth radius (C{meter}). @keyword 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}}). @raise TypeError: The B{C{other}} point is not L{LatLon}. @example: >>> p1 = LatLon(52.205, 0.119) >>> p2 = LatLon(48.857, 2.351); >>> d = p1.distanceTo(p2) # 404300 ''' self.others(other) a1, b1 = self.to2ab() a2, b2 = other.to2ab() db, b2 = unrollPI(b1, b2, wrap=wrap) r = haversine_(a2, a1, db) return r * float(radius) 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. @param bearing: Bearing from this point (compass C{degrees360}). @return: Vector representing great circle (L{Vector3d}). @example: >>> p = LatLon(53.3206, -1.7297) >>> g = p.greatCircle(96.0) >>> g.toStr() # (-0.794, 0.129, 0.594) ''' a, b = self.to2ab() t = radians(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. @param other: The other point (spherical L{LatLon}). @keyword wrap: Wrap and unroll longitudes (C{bool}). @keyword raiser: Optionally, raise L{CrossError} (C{bool}), use B{C{raiser}}=C{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 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.to2ab() a2, b2 = other.to2ab() # XXX behavior like sphericalNvector.LatLon.initialBearingTo if raiser and crosserrors() and max(abs(a2 - a1), abs(b2 - b1)) < EPS: raise CrossError('%s %s: %r' % ('coincident', 'points', other)) 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. @param other: The other point (L{LatLon}). @param fraction: Fraction between both points (float, 0.0 = this point, 1.0 = the other point). @keyword height: Optional height, overriding the fractional height (C{meter}). @keyword wrap: Wrap and unroll longitudes (C{bool}). @return: Intermediate point (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) >>> p = p1.intermediateTo(p2, 0.25) # 51.3721°N, 000.7073°E @JSname: I{intermediatePointTo}. ''' self.others(other) a1, b1 = self.to2ab() a2, b2 = other.to2ab() db, b2 = unrollPI(b1, b2, wrap=wrap) r = haversine_(a2, a1, db) sr = sin(r) if abs(sr) > EPS: sa1, ca1, sa2, ca2, \ sb1, cb1, sb2, cb2 = sincos2(a1, a2, b1, b2) A = sin((1 - fraction) * r) / sr B = sin( fraction * 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=fraction) b = favg(b1, b2, f=fraction) if height is None: h = self._havg(other, f=fraction) else: h = height return self.classof(degrees90(a), degrees180(b), height=h) def intersection(self, end1, start2, end2, height=None, wrap=False): '''Locate the intersection point of two paths both defined by two points or a start point and bearing from North. @param end1: End point of the first path (L{LatLon}) or the initial bearing at this point (compass C{degrees360}). @param start2: Start point of the second path (L{LatLon}). @param end2: End point of the second path (L{LatLon}) or the initial bearing at the second point (compass C{degrees360}). @keyword height: Optional height for intersection point, overriding the mean height (C{meter}). @keyword 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 TypeError: The B{C{start2}}, B{C{end1}} or B{C{end2}} 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) >>> s = LatLon(49.0034, 2.5735) >>> i = p.intersection(108.547, s, 32.435) # '50.9078°N, 004.5084°E' ''' return intersection(self, end1, start2, end2, height=height, wrap=wrap, LatLon=self.classof) def isenclosedBy(self, points): '''Check whether a (convex) polygon encloses this point. @param points: The polygon points (L{LatLon}[]). @return: C{True} if the polygon encloses this point, C{False} otherwise. @raise ValueError: Insufficient number of B{C{points}} or non-convex polygon. @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 ''' n, points = self.points2(points, closed=True) n0 = self.toVector3d() if iterNumpy2(points): v1 = points[n-1].toVector3d() v2 = points[n-2].toVector3d() gc1 = v2.cross(v1) t0 = gc1.angleTo(n0) > PI_2 for i in range(n): v2 = points[i].toVector3d() gc = v1.cross(v2) v1 = v2 ti = gc.angleTo(n0) > PI_2 if ti != t0: return False # outside if gc1.angleTo(gc, vSign=n0) < 0: raise ValueError('non-convex: %r...' % (points[:2],)) gc1 = gc else: # get great-circle vector for each edge gc, v1 = [], points[n-1].toVector3d() for i in range(n): v2 = points[i].toVector3d() gc.append(v1.cross(v2)) v1 = v2 # check whether this point on same side of all # polygon edges (to the left or right depending # on anti-/clockwise polygon direction) t0 = gc[0].angleTo(n0) > PI_2 # True if on the right for i in range(1, n): ti = gc[i].angleTo(n0) > PI_2 if ti != t0: # different sides of edge i return False # outside # check for convex polygon (otherwise # the test above is not reliable) gc1 = gc[n-1] for gc2 in gc: # angle between gc vectors, signed by direction of n0 if gc1.angleTo(gc2, vSign=n0) < 0: raise ValueError('non-convex: %r...' % (points[:2],)) gc1 = gc2 return True # inside def isEnclosedBy(self, points): '''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. @param other: The other point (L{LatLon}). @keyword height: Optional height for midpoint, overriding the mean height (C{meter}). @keyword wrap: Wrap and unroll longitudes (C{bool}). @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) # see <https://MathForum.org/library/drmath/view/51822.html> a1, b1 = self.to2ab() a2, b2 = other.to2ab() 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 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 and this point. Distances are approximated by function L{equirectangular_}, subject to the supplied B{C{options}}. @param point1: Start point (L{LatLon}). @param point2: End point (L{LatLon}). @keyword radius: Optional, mean earth radius (C{meter}). @keyword options: Optional keyword arguments for function L{equirectangular_}. @return: Closest point on the arc (L{LatLon}). @raise LimitError: Lat- and/or longitudinal delta exceeds B{C{limit}}, see function L{equirectangular_}. @raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}. @see: Functions L{equirectangular_} and L{nearestOn5} and method L{sphericalTrigonometry.LatLon.nearestOn3}. ''' return self.nearestOn3([point1, point2], closed=False, radius=radius, **options)[0] def nearestOn2(self, points, closed=False, radius=R_M, **options): '''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 ... ''' return self.nearestOn3(points, closed=closed, radius=radius, **options)[:2] 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}}. @param points: The polygon points (L{LatLon}[]). @keyword closed: Optionally, close the polygon (C{bool}). @keyword radius: Optional, mean earth radius (C{meter}). @keyword options: Optional keyword arguments for function L{equirectangular_}. @return: A L{NearestOn3Tuple}C{(closest, distance, angle)} where C{distance} is the L{equirectangular_} 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 TypeError: Some B{C{points}} are not C{LatLon}. @raise ValueError: Insufficient number of B{C{points}}. @see: Functions L{compassAngle}, L{equirectangular_} and L{nearestOn5}. ''' a, b, d, c, h = _nearestOn5(self, points, closed=closed, **options) return NearestOn3Tuple(self.classof(a, b, height=h), degrees2m(d, radius=radius), c) def toCartesian(self, **kwds): # PYCHOK Cartesian=Cartesian '''Convert this point to C{Karney}-based cartesian (ECEF) coordinates. @keyword kwds: Optional, additional B{C{Cartesian}} keyword arguments, ignored if C{B{Cartesian}=None}. For example, use C{Cartesian=...} to override the L{Cartesian} (sub-)class or specify @return: The B{C{Cartesian}} point (L{Cartesian}) or if C{B{Cartesian}=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{datum}} or B{C{kwds}}. ''' kwds = _2kwds(kwds, Cartesian=Cartesian, datum=self.datum) return LatLonSphericalBase.toCartesian(self, **kwds) _Trll = LatLon(0, 0) #: (INTERNAL) Reference instance (L{LatLon}). def _destination2_(a, b, r, t): '''(INTERNAL) Computes destination lat- and longitude. @param a: Latitude (C{radians}). @param b: Longitude (C{radians}). @param r: Angular distance (C{radians}). @param t: Bearing (compass C{radians}). @return: 2-Tuple (lat, lon) of (radians, radians). ''' # see <https://www.EdWilliams.org/avform.htm#LL> sa, ca, sr, cr, st, ct = sincos2(a, r, t) a = asin(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 _destination2(a, b, r, t): '''(INTERNAL) Computes destination lat- and longitude. @param a: Latitude (C{radians}). @param b: Longitude (C{radians}). @param r: Angular distance (C{radians}). @param t: Bearing (compass C{radians}). @return: 2-Tuple (lat, lon) of (C{degrees90}, C{degrees180}). ''' a, b = _destination2_(a, b, r, t) return degrees90(a), degrees180(b) def areaOf(points, radius=R_M, wrap=True): '''Calculate the area of a (spherical) polygon (with great circle arcs joining the points). @param points: The polygon points (L{LatLon}[]). @keyword radius: Optional, mean earth radius (C{meter}). @keyword wrap: Wrap and unroll longitudes (C{bool}). @return: Polygon area (C{meter}, same units as B{C{radius}}, squared). @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Insufficient number of B{C{points}}. @note: The area is based on 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} and L{ellipsoidalKarney.areaOf}. @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 ''' n, points = _Trll.points2(points, closed=True) # Area method due to Karney: for each edge of the polygon, # # tan(Δλ/2) · (tan(φ1/2) + tan(φ2/2)) # tan(E/2) = ------------------------------------ # 1 + tan(φ1/2) · tan(φ2/2) # # where E is the spherical excess of the trapezium obtained by # extending the edge to the equator-circle vector for each edge def _exs(n, points): # iterate over spherical edge excess a1, b1 = points[n-1].to2ab() ta1 = tan_2(a1) for i in range(n): a2, b2 = points[i].to2ab() db, b2 = unrollPI(b1, b2, wrap=wrap) ta2, tdb = map1(tan_2, a2, db) yield atan2(tdb * (ta1 + ta2), 1 + ta1 * ta2) ta1, b1 = ta2, b2 s = fsum(_exs(n, points)) * 2 if isPoleEnclosedBy(points): s = abs(s) - PI2 return abs(s * radius**2) def _x3d2(start, end, wrap, n, hs): # see <https://www.EdWilliams.org/intersect.htm> (5) ff a1, b1 = start.to2ab() if isscalar(end): # bearing, make a point a2, b2 = _destination2_(a1, b1, PI_4, radians(end)) else: # must be a point _Trll.others(end, name='end' + n) hs.append(end.height) a2, b2 = end.to2ab() db, b2 = unrollPI(b1, b2, wrap=wrap) if max(abs(db), abs(a2 - a1)) < EPS: raise ValueError('intersection %s%s null: %r' % ('path', n, (start, end))) # 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 = Vector3d(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 _xb(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, radians(a), radians(b), wrap=wrap) return PI_2 - abs(wrapPI(r - radians(end))) def _xdot(d, a1, b1, a, b, wrap): # compute dot product d . (-b + b1, a - a1) db, _ = unrollPI(b1, radians(b), wrap=wrap) return fdot(d, db, radians(a) - a1) def intersection(start1, end1, start2, end2, height=None, wrap=False, LatLon=LatLon): '''Compute the intersection point of two paths both defined by two points or a start point and bearing from North. @param start1: Start point of the first path (L{LatLon}). @param end1: End point ofthe first path (L{LatLon}) or the initial bearing at the first start point (compass C{degrees360}). @param start2: Start point of the second path (L{LatLon}). @param end2: End point of the second path (L{LatLon}) or the initial bearing at the second start point (compass C{degrees360}). @keyword height: Optional height for the intersection point, overriding the mean height (C{meter}). @keyword wrap: Wrap and unroll longitudes (C{bool}). @keyword LatLon: Optional (sub-)class to return the intersection point (L{LatLon}) or C{None}. @return: The intersection point (B{C{LatLon}}) or a L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} is C{None}. An alternate intersection point might be the L{antipode} to the returned result. @raise TypeError: A B{C{start}} or B{C{end}} point 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) >>> s = LatLon(49.0034, 2.5735) >>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E' ''' _Trll.others(start1, name='start1') _Trll.others(start2, name='start2') hs = [start1.height, start2. height] a1, b1 = start1.to2ab() a2, b2 = start2.to2ab() db, b2 = unrollPI(b1, b2, wrap=wrap) r12 = haversine_(a2, a1, db) if abs(r12) < EPS: # [nearly] coincident points a, b = map1(degrees, 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 abs(x1) < EPS or abs(x2) < EPS: raise ValueError('intersection %s: %r vs %r' % ('parallel', (start1, end1), (start2, end2))) # 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 ValueError('intersection %s: %r vs %r' % ('infinite', (start1, end1), (start2, end2))) sx3 = sx1 * sx2 # if sx3 < 0: # raise ValueError('intersection %s: %r vs %r' % ('ambiguous', # (start1, end1), (start2, end2))) 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 _xb(a1, b1, end1, a, b, wrap) < 0 or \ _xb(a2, b2, end2, a, b, wrap) < 0: a, b = antipode(a, b) else: # end point(s) or bearing(s) x1, d1 = _x3d2(start1, end1, wrap, '1', hs) x2, d2 = _x3d2(start2, end2, wrap, '2', hs) x = x1.cross(x2) if x.length < EPS: # [nearly] colinear or parallel paths raise ValueError('intersection %s: %r vs %r' % ('colinear', (start1, end1), (start2, end2))) a, b = x.to2ll() # choose intersection similar to sphericalNvector d1 = _xdot(d1, a1, b1, a, b, wrap) d2 = _xdot(d2, a2, b2, a, b, wrap) if (d1 < 0 and d2 > 0) or (d1 > 0 and d2 < 0): a, b = antipode(a, b) h = fmean(hs) if height is None else height r = LatLon3Tuple(a, b, h) if LatLon is None else \ LatLon(a, b, height=h) return _xnamed(r, intersection.__name__) def isPoleEnclosedBy(points, wrap=False): '''DEPRECATED, use function L{ispolar}. ''' return ispolar(points, wrap=wrap) def meanOf(points, height=None, LatLon=LatLon): '''Compute the geographic mean of several points. @param points: Points to be averaged (L{LatLon}[]). @keyword height: Optional height at mean point, overriding the mean height (C{meter}). @keyword LatLon: Optional (sub-)class to return the mean point (L{LatLon}) or C{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}}. ''' # geographic mean n, points = _Trll.points2(points, closed=False) m = sumOf(points[i].toVector3d() for i in range(n)) a, b = m.to2ll() if height is None: h = fmean(points[i].height for i in range(n)) else: h = height r = LatLon3Tuple(a, b, h) if LatLon is None else \ LatLon(a, b, height=h) return _xnamed(r, meanOf.__name__) def nearestOn2(point, points, closed=False, radius=R_M, LatLon=LatLon, **options): '''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}. @return: ... C{closest} as B{C{LatLon}} or a 2-tuple C{(lat, lon)} without the height if B{C{LatLon}} is C{None} ... ''' a, b, d, _, h = _nearestOn5(point, points, closed=closed, **options) ll = (a, b) if LatLon is None else LatLon(a, b, height=h) return ll, degrees2m(d, radius=radius) 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}}. @param point: The other, reference point (L{LatLon}). @param points: The polygon points (L{LatLon}[]). @keyword closed: Optionally, close the polygon (C{bool}). @keyword radius: Optional, mean earth radius (C{meter}). @keyword LatLon: Optional (sub-)class to return the closest point (L{LatLon}) or C{None}. @keyword options: Optional keyword arguments for function L{equirectangular_}. @return: A L{NearestOn3Tuple}C{(closest, distance, angle)}. The C{distance} is the L{equirectangular_} distance between the C{closest} and reference B{C{point}} in C{meter}, same units as B{C{radius}}. The C{angle} from the reference 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 TypeError: Some I{points} are not C{LatLon}. @raise ValueError: Insufficient number of B{C{points}}. @see: Functions L{equirectangular_} and L{nearestOn5}. ''' a, b, d, c, h = _nearestOn5(point, points, closed=closed, LatLon=None, **options) r = LatLon3Tuple(a, b, h) if LatLon is None else \ LatLon(a, b, 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). @param points: The polygon points (L{LatLon}[]). @keyword closed: Optionally, close the polygon (C{bool}). @keyword radius: Optional, mean earth radius (C{meter}). @keyword wrap: Wrap and unroll longitudes (C{bool}). @return: Polygon perimeter (C{meter}, same units as B{C{radius}}). @raise TypeError: Some B{C{points}} are not L{LatLon}. @raise ValueError: Insufficient number of B{C{points}}. @note: This perimeter is based on the L{haversine} formula. @see: L{pygeodesy.perimeterOf}, L{sphericalNvector.perimeterOf} and L{ellipsoidalKarney.perimeterOf}. ''' n, points = _Trll.points2(points, closed=closed) def _rads(n, points, closed): # angular edge lengths in radians i, m = _imdex2(closed, n) a1, b1 = points[i].to2ab() for i in range(m, n): a2, b2 = points[i].to2ab() db, b2 = unrollPI(b1, b2, wrap=wrap) yield haversine_(a2, a1, db) a1, b1 = a2, b2 r = fsum(_rads(n, points, closed)) return r * float(radius) # **) 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.