# Coverage for pygeodesy/vector3d.py : 96%

## 241 statements 231 run 10 missing 36 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 | # -*- coding: utf-8 -*- u'''Extended 3-D vector class L{Vector3d} and functions. Function L{intersection3d3}, L{intersections2}, L{parse3d}, L{sumOf}, L{trilaterate2d2} and L{trilaterate3d2}. ''' from pygeodesy.basics import len2, map2 from pygeodesy.errors import IntersectionError, _TypeError, _ValueError, \ VectorError, _xError, _xkwds, _xkwds_popitem from pygeodesy.fmath import fdot, fsum, fsum1_ from pygeodesy.formy import _radical2 from pygeodesy.interns import EPS, EPS0, EPS1, EPS4, MISSING, NN, \ _COMMA_, _concentric_, _datum_, _h_, _height_, \ _intersection_, _name_, _near_, _negative_, \ _no_, _too_, _xyz_, _y_, _z_, _0_0, _1_0 from pygeodesy.iters import Fmt, PointsIter from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import _xnamed, _xotherError from pygeodesy.namedTuples import Intersection3Tuple, NearestOn2Tuple, \ NearestOn6Tuple, Vector3Tuple # Vector4Tuple # from pygeodesy.streprs import Fmt # from .iters from pygeodesy.units import _fi_j2, Radius, Radius_ from pygeodesy.vector3dBase import Vector3dBase from math import sqrt __all__ = _ALL_LAZY.vector3d __version__ = '21.10.05' class Vector3d(Vector3dBase): '''Extended 3-D vector. In a geodesy context, these may be used to represent: - n-vector representing a normal to a point on earth's surface - earth-centered, earth-fixed cartesian (ECEF) - great circle normal to vector - motion vector on earth's surface - etc. ''' def circin6(self, point2, point3, eps=EPS4): '''Return the radius and center of the I{inscribed} aka I{In- circle} of a (3-D) triangle formed by this and two other points. @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}. @return: L{Circin6Tuple}C{(radius, center, deltas, cA, cB, cC)}. The C{center} and contact points C{cA}, C{cB} and C{cC}, each an instance of this (sub-)class, are co-planar with this and the two given points. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-coincident or colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.circin6}, U{Incircle <https://MathWorld.Wolfram.com/Incircle.html>} and U{Contact Triangle<https://MathWorld.Wolfram.com/ContactTriangle.html>}. ''' from pygeodesy.vector2d import _circin6 try: return _circin6(self, point2, point3, eps=eps, useZ=True) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, point=self, point2=point2, point3=point3) def circum3(self, point2, point3, circum=True, eps=EPS4): '''Return the radius and center of the smallest circle I{through} or I{containing} this and two other (3-D) points. @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter}, always, ignoring the I{Meeus}' Type I case (C{bool}). @kwarg eps: Tolerance passed to function L{pygeodesy.trilaterate3d2}. @return: A L{Circum3Tuple}C{(radius, center, deltas)}. The C{center}, an instance of this (sub-)class, is co-planar with this and the two given points. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-concentric, coincident or colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.circum3} and methods L{circum4_} and L{meeus2}. ''' from pygeodesy.vector2d import _circum3 try: return _circum3(self, point2, point3, circum=circum, eps=eps, useZ=True, clas=self.classof) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, point=self, point2=point2, point3=point3, circum=circum) def circum4_(self, *points): '''Best-fit a sphere through this and two or more other (3-D) points. @arg points: Other points (each a C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @return: L{Circum4Tuple}C{(radius, center, rank, residuals)} with C{center} an instance if this (sub-)class. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise NumPyError: Some C{numpy} issue. @raise PointsError: Too few B{C{points}}. @raise TypeError: One of the B{C{points}} invalid. @see: Function L{pygeodesy.circum4_} and methods L{circum3} and L{meeus2}. ''' from pygeodesy.vector2d import circum4_ return circum4_(self, *points, Vector=self.classof) def iscolinearWith(self, point1, point2, eps=EPS): '''Check whether this and two other (3-D) points are colinear. @arg point1: One point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: An other point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. @return: C{True} if this point is colinear with B{C{point1}} and B{C{point2}}, C{False} otherwise. @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. @see: Method L{nearestOn}. ''' from pygeodesy.vector2d import _iscolinearWith v = self if self.name else _otherV3d(NN_OK=False, this=self) return _iscolinearWith(v, point1, point2, eps=eps) def meeus2(self, point2, point3, circum=False): '''Return the radius and I{Meeus}' Type of the smallest circle I{through} or I{containing} this and two other (3-D) points. @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg circum: If C{True} return the C{circumradius} and C{circumcenter} always, ignoring the I{Meeus}' Type I case (C{bool}). @return: L{Meeus2Tuple}C{(radius, Type)}. @raise IntersectionError: Coincident or colinear points, iff C{B{circum}=True}. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.meeus2} and methods L{circum3} and L{circum4_}. ''' from pygeodesy.vector2d import _meeus4, Meeus2Tuple try: r, t, _, _ = _meeus4(self, point2, point3, circum=circum, clas=self.classof) except (TypeError, ValueError) as x: raise _xError(x, point=self, point2=point2, point3=point3, circum=circum) return Meeus2Tuple(r, t) def nearestOn(self, point1, point2, within=True): '''Locate the point between two points closest to this point. @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg within: If C{True} return the closest point between the given points, otherwise the closest point on the extended line through both points (C{bool}). @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance of this (sub-)class. @raise TypeError: Invalid B{C{point1}} or B{C{point2}}. @see: Method L{sphericalTrigonometry.LatLon.nearestOn3} and U{3-D Point-Line Distance<https://MathWorld.Wolfram.com/ Point-LineDistance3-Dimensional.html>}. ''' return _nearestOn2(self, point1, point2, within=within).closest def nearestOn6(self, points, closed=False, useZ=True): # eps=EPS '''Locate the point on a path or polygon closest to this point. The closest point is either on and within the extent of a polygon edge or the nearest of that edge's end points. @arg points: The path or polygon points (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}[]). @kwarg closed: Optionally, close the path or polygon (C{bool}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (C{bool}). @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the C{closest}, the C{start} and the C{end} point each an instance of this point's (sub-)class. @raise PointsError: Insufficient number of B{C{points}} @raise TypeError: Non-cartesian B{C{points}}. @note: Distances measured with method L{Vector3d.equirectangular}. @see: Function L{nearestOn6}. ''' return nearestOn6(self, points, closed=closed, useZ=useZ) # Vector=self.classof def parse(self, str3d, sep=_COMMA_, name=NN): '''Parse an C{"x, y, z"} string to a L{Vector3d} instance. @arg str3d: X, y and z string (C{str}), see function L{parse3d}. @kwarg sep: Optional separator (C{str}). @kwarg name: Optional instance name (C{str}), overriding this name. @return: The instance (L{Vector3d}). @raise VectorError: Invalid B{C{str3d}}. ''' return parse3d(str3d, sep=sep, Vector=self.classof, name=name or self.name) def radii11(self, point2, point3): '''Return the radii of the C{Circum-}, C{In-}, I{Soddy} and C{Tangent} circles of a (3-D) triangle. @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @return: L{Radii11Tuple}C{(rA, rB, rC, cR, rIn, riS, roS, a, b, c, s)}. @raise IntersectionError: Near-coincident or colinear points. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.radii11}, U{Incircle <https://MathWorld.Wolfram.com/Incircle.html>}, U{Soddy Circles <https://MathWorld.Wolfram.com/SoddyCircles.html>} and U{Tangent Circles<https://MathWorld.Wolfram.com/TangentCircles.html>}. ''' from pygeodesy.vector2d import _radii11ABC try: return _radii11ABC(self, point2, point3, useZ=True)[0] except (TypeError, ValueError) as x: raise _xError(x, point=self, point2=point2, point3=point3) def soddy4(self, point2, point3, eps=EPS4): '''Return the radius and center of the C{inner} I{Soddy} circle of a (3-D) triangle. @arg point2: Second point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @arg point3: Third point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple}, C{Vector4Tuple} or C{Vector2Tuple} if C{B{useZ}=False}). @kwarg eps: Tolerance for function L{pygeodesy.trilaterate3d2} if C{B{useZ} is True} otherwise L{pygeodesy.trilaterate2d2}. @return: L{Soddy4Tuple}C{(radius, center, deltas, outer)}. The C{center}, an instance of B{C{point1}}'s (sub-)class, is co-planar with the three given points. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-coincident or colinear points or a trilateration or C{numpy} issue. @raise TypeError: Invalid B{C{point2}} or B{C{point3}}. @see: Function L{pygeodesy.soddy4}. ''' from pygeodesy.vector2d import soddy4 return soddy4(self, point2, point3, eps=eps, useZ=True) def trilaterate2d2(self, radius, center2, radius2, center3, radius3, eps=EPS, z=0): '''Trilaterate this and two other circles, each given as a (2-D) center and a radius. @arg radius: Radius of this circle (same C{units} as this C{x} and C{y}. @arg center2: Center of the 2nd circle (C{Cartesian}, L{Vector3d}, C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius2: Radius of this circle (same C{units} as this C{x} and C{y}. @arg center3: Center of the 3rd circle (C{Cartesian}, L{Vector3d}, C{Vector2Tuple}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius3: Radius of the 3rd circle (same C{units} as this C{x} and C{y}. @kwarg eps: Check the trilaterated point I{delta} on all 3 circles (C{scalar}) or C{None}. @kwarg z: Optional Z component of the trilaterated point (C{scalar}). @return: Trilaterated point, an instance of this (sub-)class with C{z=B{z}}. @raise IntersectionError: No intersection, colinear or near-concentric centers, trilateration failed some other way or the trilaterated point is off one circle by more than B{C{eps}}. @raise TypeError: Invalid B{C{center2}} or B{C{center3}}. @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. @see: Function L{pygeodesy.trilaterate2d2}. ''' from pygeodesy.vector2d import _trilaterate2d2 def _xyr3(r, **name_v): v = _otherV3d(useZ=False, **name_v) return v.x, v.y, r try: return _trilaterate2d2(*(_xyr3(radius, center=self) + _xyr3(radius2, center2=center2) + _xyr3(radius3, center3=center3)), eps=eps, Vector=self.classof, z=z) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, center=self, radius=radius, center2=center2, radius2=radius2, center3=center3, radius3=radius3) def trilaterate3d2(self, radius, center2, radius2, center3, radius3, eps=EPS): '''Trilaterate this and two other spheres, each given as a (3-D) center and a radius. @arg radius: Radius of this sphere (same C{units} as this C{x}, C{y} and C{z}). @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius2: Radius of this sphere (same C{units} as this C{x}, C{y} and C{z}). @arg center3: Center of the 3rd sphere (C{Cartesian}, , L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius3: Radius of the 3rd sphere (same C{units} as this C{x}, C{y} and C{z}). @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. @return: 2-Tuple with two trilaterated points, each an instance of this (sub-)class. Both points are the same instance if all three spheres intersect or abut in a single point. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-concentric, colinear, too distant or non-intersecting spheres or C{numpy} issue. @raise TypeError: Invalid B{C{center2}} or B{C{center3}}. @raise UnitError: Invalid B{C{radius}}, B{C{radius2}} or B{C{radius3}}. @note: Package U{numpy<https://pypi.org/project/numpy>} is required, version 1.10 or later. @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration Problem}<https://www.ResearchGate.net/publication/ 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>} and U{I{implementation}<https://www.ResearchGate.net/publication/ 288825016_Trilateration_Matlab_Code>}. ''' from pygeodesy.vector2d import _trilaterate3d2 try: return _trilaterate3d2(_otherV3d(center=self, NN_OK=False), Radius_(radius, low=eps), center2, radius2, center3, radius3, eps=eps, clas=self.classof) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, center=self, radius=radius, center2=center2, radius2=radius2, center3=center3, radius3=radius3) def _intersect3d3(start1, end1, start2, end2, eps=EPS, useZ=False): # (INTERNAL) Intersect two lines, see L{intersection} above, # separated to allow callers to embellish any exceptions def _outside(t, d2): # -1 before start#, +1 after end# return -1 if t < 0 else (+1 if t > d2 else 0) # XXX d2 + eps? s1 = _otherV3d(useZ=useZ, start1=start1) e1 = _otherV3d(useZ=useZ, end1=end1) s2 = _otherV3d(useZ=useZ, start2=start2) e2 = _otherV3d(useZ=useZ, end2=end2) a = e1.minus(s1) b = e2.minus(s2) c = s2.minus(s1) ab = a.cross(b) d = abs(c.dot(ab)) e = max(EPS0, eps or _0_0) if d > EPS0 and ab.length > e: d = d / ab.length if d > e: # argonic, skew lines distance raise IntersectionError(skew_d=d, txt=_no_(_intersection_)) # co-planar, non-skew lines ab2 = ab.length2 if ab2 < e: # colinear, parallel or null line(s) x = a.length2 > b.length2 if x: # make a shortest a, b = b, a s1, s2 = s2, s1 e1, e2 = e2, e1 if b.length2 < e: # both null if c.length < e: return s1, 0, 0 elif e2.minus(e1).length < e: return e1, 0, 0 elif a.length2 < e: # null (s1, e1), non-null (s2, e2) # like _nearestOn2(s1, s2, e2, within=False, eps=e) t = s1.minus(s2).dot(b) v = s2.plus(b.times(t / b.length2)) if s1.minus(v).length < e: o = _outside(t, b.length2) return (v, o, 0) if x else (v, 0, (o * 2)) raise IntersectionError(length2=ab2, txt=_no_(_intersection_)) cb = c.cross(b) t = cb.dot(ab) o1 = _outside(t, ab2) v = s1.plus(a.times(t / ab2)) o2 = _outside(v.minus(s2).dot(b), b.length2) * 2 return v, o1, o2 def intersection3d3(start1, end1, start2, end2, eps=EPS, useZ=True, **Vector_and_kwds): '''Compute the intersection point of two lines, each defined by or through a start and end point (3-D). @arg start1: Start point of the first line (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg end1: End point of the first line (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg start2: Start point of the second line (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg end2: End point of the second line (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg eps: Tolerance for skew line distance and length (C{EPS}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (C{bool}). @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the intersection points and optional, additional B{C{Vector}} keyword arguments, otherwise B{C{start1}}'s (sub-)class. @return: An L{Intersection3Tuple}C{(point, outside1, outside2)} with C{point} an instance of B{C{Vector}} or B{C{start1}}'s (sub-)class. @raise IntersectionError: Invalid, skew, non-co-planar or otherwise non-intersecting lines. @see: U{Line-line intersection<https://MathWorld.Wolfram.com/Line-LineIntersection.html>} and U{line-line distance<https://MathWorld.Wolfram.com/Line-LineDistance.html>}, U{skew lines<https://MathWorld.Wolfram.com/SkewLines.html>} and U{point-line distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}. ''' try: v, o1, o2 = _intersect3d3(start1, end1, start2, end2, eps=eps, useZ=useZ) except (TypeError, ValueError) as x: raise _xError(x, start1=start1, end1=end1, start2=start2, end2=end2) v = _nVc(v, **_xkwds(Vector_and_kwds, clas=start1.classof, name=intersection3d3.__name__)) return Intersection3Tuple(v, o1, o2) def intersections2(center1, radius1, center2, radius2, sphere=True, **Vector_and_kwds): '''Compute the intersection of two spheres or circles, each defined by a (3-D) center point and a radius. @arg center1: Center of the first sphere or circle (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius1: Radius of the first sphere or circle (same units as the B{C{center1}} coordinates). @arg center2: Center of the second sphere or circle (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius2: Radius of the second sphere or circle (same units as the B{C{center1}} and B{C{center2}} coordinates). @kwarg sphere: If C{True} compute the center and radius of the intersection of two spheres. If C{False}, ignore the C{z}-component and compute the intersection of two circles (C{bool}). @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the intersection points and optional, additional B{C{Vector}} keyword arguments, otherwise B{C{center1}}'s (sub-)class. @return: If B{C{sphere}} is C{True}, a 2-tuple of the C{center} and C{radius} of the intersection of the I{spheres}. The C{radius} is C{0.0} for abutting spheres (and the C{center} is aka I{radical center}). If B{C{sphere}} is C{False}, a 2-tuple with the two intersection points of the I{circles}. For abutting circles, both points are the same instance, aka I{radical center}. @raise IntersectionError: Concentric, invalid or non-intersecting spheres or circles. @raise TypeError: Invalid B{C{center1}} or B{C{center2}}. @raise UnitError: Invalid B{C{radius1}} or B{C{radius2}}. @see: U{Sphere-Sphere<https://MathWorld.Wolfram.com/Sphere-SphereIntersection.html>} and U{Circle-Circle<https://MathWorld.Wolfram.com/Circle-CircleIntersection.html>} Intersection. ''' try: return _intersects2(center1, Radius_(radius1=radius1), center2, Radius_(radius2=radius2), sphere=sphere, clas=center1.classof, **Vector_and_kwds) except (TypeError, ValueError) as x: raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2) def _intersects2(center1, r1, center2, r2, sphere=True, too_d=None, # in CartesianEllipsoidalBase.intersections2, **clas_Vector_and_kwds): # .ellipsoidalBaseDI._intersections2 # (INTERNAL) Intersect two spheres or circles, see L{intersections2} # above, separated to allow callers to embellish any exceptions def _nV3(x, y, z): v = Vector3d(x, y, z) n = intersections2.__name__ return _nVc(v, **_xkwds(clas_Vector_and_kwds, name=n)) def _xV3(c1, u, x, y): xy1 = x, y, _1_0 # transform to original space return _nV3(fdot(xy1, u.x, -u.y, c1.x), fdot(xy1, u.y, u.x, c1.y), _0_0) c1 = _otherV3d(useZ=sphere, center1=center1) c2 = _otherV3d(useZ=sphere, center2=center2) if r1 < r2: # r1, r2 == R, r c1, c2 = c2, c1 r1, r2 = r2, r1 m = c2.minus(c1) d = m.length if d < max(r2 - r1, EPS): raise IntersectionError(_near_(_concentric_)) o = fsum1_(-d, r1, r2) # overlap == -(d - (r1 + r2)) # compute intersections with c1 at (0, 0) and c2 at (d, 0), like # <https://MathWorld.Wolfram.com/Circle-CircleIntersection.html> if o > EPS: # overlapping, r1, r2 == R, r x = _radical2(d, r1, r2).xline y = _1_0 - (x / r1)**2 if y > EPS: y = r1 * sqrt(y) # y == a / 2 elif y < 0: raise IntersectionError(_negative_) else: # abutting y = _0_0 elif o < 0: t = d if too_d is None else too_d raise IntersectionError(_too_(Fmt.distant(t))) else: # abutting x, y = r1, _0_0 u = m.unit() if sphere: # sphere center and radius c = c1 if x < EPS else ( c2 if x > EPS1 else c1.plus(u.times(x))) t = _nV3(c.x, c.y, c.z), Radius(y) elif y > 0: # intersecting circles t = _xV3(c1, u, x, y), _xV3(c1, u, x, -y) else: # abutting circles t = _xV3(c1, u, x, 0) t = t, t return t def iscolinearWith(point, point1, point2, eps=EPS, useZ=True): '''Check whether a point is colinear with two other (2- or 3-D) points. @arg point: The point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point1: First point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: Second point (L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y} and C{z}. @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (C{bool}). @return: C{True} if B{C{point}} is colinear B{C{point1}} and B{C{point2}}, C{False} otherwise. @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}. @see: Function L{nearestOn}. ''' p = _otherV3d(useZ=useZ, point=point) from pygeodesy.vector2d import _iscolinearWith return _iscolinearWith(p, point1, point2, eps=eps, useZ=useZ) def nearestOn(point, point1, point2, within=True, useZ=True, Vector=None, **Vector_kwds): '''Locate the point between two points closest to a reference (2- or 3-D). @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point1: Start point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg point2: End point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @kwarg within: If C{True} return the closest point between both given points, otherwise the closest point on the extended line through both points (C{bool}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (C{bool}). @kwarg Vector: Class to return closest point (C{Cartesian}, L{Vector3d} or C{Vector3Tuple}) or C{None}. @kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword arguments, ignored if C{B{Vector} is None}. @return: Closest point, either B{C{point1}} or B{C{point2}} or an instance of the B{C{point}}'s (sub-)class or B{C{Vector}} if not C{None}. @raise TypeError: Invalid B{C{point}}, B{C{point1}} or B{C{point2}}. @see: U{3-D Point-Line Distance<https://MathWorld.Wolfram.com/Point-LineDistance3-Dimensional.html>}, C{Cartesian} and C{LatLon} methods C{nearestOn}, method L{sphericalTrigonometry.LatLon.nearestOn3} and function L{sphericalTrigonometry.nearestOn3}. ''' p0 = _otherV3d(useZ=useZ, point =point) p1 = _otherV3d(useZ=useZ, point1=point1) p2 = _otherV3d(useZ=useZ, point2=point2) p, _ = _nearestOn2(p0, p1, p2, within=within) if Vector is not None: p = Vector(p.x, p.y, **_xkwds(Vector_kwds, z=p.z, name=nearestOn.__name__)) elif p is p1: p = point1 elif p is p2: p = point2 else: # ignore Vector_kwds p = point.classof(p.x, p.y, Vector_kwds.get(_z_, p.z), name=nearestOn.__name__) return p def _nearestOn2(p0, p1, p2, within=True, eps=EPS): # (INTERNAL) Closest point and fraction, see L{nearestOn} above, # separated to allow callers to embellish any exceptions p21 = p2.minus(p1) d2 = p21.length2 if d2 < eps: # coincident p = p1 # ~= p2 t = 0 else: # see comments in .points.nearestOn5 t = p0.minus(p1).dot(p21) / d2 if within and t < eps: p = p1 t = 0 elif within and t > (_1_0 - eps): p = p2 t = 1 else: p = p1.plus(p21.times(t)) return NearestOn2Tuple(p, t) def nearestOn6(point, points, closed=False, useZ=True, **Vector_and_kwds): # eps=EPS '''Locate the point on a path or polygon closest to a reference point. The closest point is either on and within the extent of a polygon edge or the nearest of that edge's end points. @arg point: Reference point (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg points: The path or polygon points (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}[]). @kwarg closed: Optionally, close the path or polygon (C{bool}). @kwarg useZ: If C{True}, use the Z components, otherwise force C{z=0} (C{bool}). @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the closest point and optional, additional B{C{Vector}} keyword arguments, otherwise B{C{point}}'s (sub-)class. @return: A L{NearestOn6Tuple}C{(closest, distance, fi, j, start, end)} with the C{closest}, the C{start} and the C{end} point each an instance of the B{C{Vector}} keyword argument of if {B{Vector}=None} or not specified, an instance of the B{C{point}}'s (sub-)class. @raise PointsError: Insufficient number of B{C{points}} @raise TypeError: Non-cartesian B{C{point}} and B{C{points}}. @note: Distances measured with method L{Vector3d.equirectangular}. @see: Method C{LatLon.nearestOn6} or function L{nearestOn5} for geodetic points. ''' r = _otherV3d(useZ=useZ, point=point) D2 = r.equirectangular # distance squared Ps = PointsIter(points, loop=1, name=nearestOn6.__name__) p1 = c = s = e = _otherV3d(useZ=useZ, i=0, points=Ps[0]) c2 = D2(c) # == r.minus(c).length2 f = i = 0 # p1..p2 == points[i]..[j] for j, p2 in Ps.enumerate(closed=closed): p2 = _otherV3d(useZ=useZ, i=j, points=p2) p, t = _nearestOn2(r, p1, p2) # within=True, eps=EPS d2 = D2(p) # == r.minus(p).length2 if d2 < c2: c2, c, s, e, f = d2, p, p1, p2, (i + t) p1, i = p2, j f, j = _fi_j2(f, len(Ps)) # like .ellipsoidalBaseDI._nearestOn2_ kwds = _xkwds(Vector_and_kwds, clas=point.classof, name=Ps.name) v = _nVc(c, **kwds) s = _nVc(s, **kwds) if s is not c else v e = _nVc(e, **kwds) if e is not c else v return NearestOn6Tuple(v, sqrt(c2), f, j, s, e) def _nVc(v, clas=None, name=NN, Vector=None, **Vector_kwds): # in .vector2d # return a named C{Vector} or C{clas} instance if Vector is not None: v = Vector(v.x, v.y, v.z, **Vector_kwds) elif clas is not None: v = clas(v.x, v.y, v.z) # ignore Vector_kwds return _xnamed(v, name) if name else v def _otherV3d(useZ=True, NN_OK=True, i=None, **name_v): # in .CartesianEllipsoidalBase.intersections2, # check named vector instance, return Vector3d .Ellipsoid.height4, .formy.hartzell, .vector2d def _name_i(name, i): return name if i is None else Fmt.SQUARE(name, i) name, v = _xkwds_popitem(name_v) if useZ and isinstance(v, Vector3dBase): return v if NN_OK or v.name else v.copy(name=_name_i(name, i)) try: return Vector3d(v.x, v.y, (v.z if useZ else _0_0), name=_name_i(name, i)) except AttributeError: # no _x_ or _y_ attr pass raise _xotherError(Vector3d(0, 0, 0), v, name=_name_i(name, i), up=2) def parse3d(str3d, sep=_COMMA_, Vector=Vector3d, **Vector_kwds): '''Parse an C{"x, y, z"} string. @arg str3d: X, y and z values (C{str}). @kwarg sep: Optional separator (C{str}). @kwarg Vector: Optional class (L{Vector3d}). @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, ignored if C{B{Vector} is None}. @return: A B{C{Vector}} instance or if B{C{Vector}} is C{None}, a named L{Vector3Tuple}C{(x, y, z)}. @raise VectorError: Invalid B{C{str3d}}. ''' try: v = [float(v.strip()) for v in str3d.split(sep)] n = len(v) if n != 3: raise _ValueError(len=n) except (TypeError, ValueError) as x: raise VectorError(str3d=str3d, txt=str(x)) return _xnamed((Vector3Tuple(*v) if Vector is None else Vector(*v, **Vector_kwds)), parse3d.__name__) def sumOf(vectors, Vector=Vector3d, **Vector_kwds): '''Compute the vectorial sum of several vectors. @arg vectors: Vectors to be added (L{Vector3d}[]). @kwarg Vector: Optional class for the vectorial sum (L{Vector3d}). @kwarg Vector_kwds: Optional B{C{Vector}} keyword arguments, ignored if C{B{Vector} is None}. @return: Vectorial sum as B{C{Vector}} or if B{C{Vector}} is C{None}, a named L{Vector3Tuple}C{(x, y, z)}. @raise VectorError: No B{C{vectors}}. ''' n, vectors = len2(vectors) if n < 1: raise VectorError(vectors=n, txt=MISSING) v = Vector3Tuple(fsum(v.x for v in vectors), fsum(v.y for v in vectors), fsum(v.z for v in vectors)) return _xnamed((v if Vector is None else Vector(*v, **Vector_kwds)), sumOf.__name__) def trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, eps=None, **Vector_and_kwds): '''Trilaterate three circles, each given as a (2-D) center and a radius. @arg x1: Center C{x} coordinate of the 1st circle (C{scalar}). @arg y1: Center C{y} coordinate of the 1st circle (C{scalar}). @arg radius1: Radius of the 1st circle (C{scalar}). @arg x2: Center C{x} coordinate of the 2nd circle (C{scalar}). @arg y2: Center C{y} coordinate of the 2nd circle (C{scalar}). @arg radius2: Radius of the 2nd circle (C{scalar}). @arg x3: Center C{x} coordinate of the 3rd circle (C{scalar}). @arg y3: Center C{y} coordinate of the 3rd circle (C{scalar}). @arg radius3: Radius of the 3rd circle (C{scalar}). @kwarg eps: Check the trilaterated point I{delta} on all 3 circles (C{scalar}) or C{None}. @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the trilateration and optional, additional B{C{Vector}} keyword arguments, otherwise (L{Vector3d}). @return: Trilaterated point as C{B{Vector}(x, y, **B{Vector_kwds})} or L{Vector2Tuple}C{(x, y)} if C{B{Vector} is None}.. @raise IntersectionError: No intersection, colinear or near-concentric centers, trilateration failed some other way or the trilaterated point is off one circle by more than B{C{eps}}. @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. @see: U{Issue #49<https://GitHub.com/mrJean1/PyGeodesy/issues/49>}, U{Find X location using 3 known (X,Y) location using trilateration <https://math.StackExchange.com/questions/884807>} and function L{pygeodesy.trilaterate3d2}. ''' from pygeodesy.vector2d import _trilaterate2d2 return _trilaterate2d2(x1, y1, radius1, x2, y2, radius2, x3, y3, radius3, eps=eps, **Vector_and_kwds) def trilaterate3d2(center1, radius1, center2, radius2, center3, radius3, eps=EPS, **Vector_and_kwds): '''Trilaterate three spheres, each given as a (3-D) center and a radius. @arg center1: Center of the 1st sphere (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius1: Radius of the 1st sphere (same C{units} as C{x}, C{y} and C{z}). @arg center2: Center of the 2nd sphere (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius2: Radius of this sphere (same C{units} as C{x}, C{y} and C{z}). @arg center3: Center of the 3rd sphere (C{Cartesian}, L{Vector3d}, C{Vector3Tuple} or C{Vector4Tuple}). @arg radius3: Radius of the 3rd sphere (same C{units} as C{x}, C{y} and C{z}). @kwarg eps: Tolerance (C{scalar}), same units as C{x}, C{y}, and C{z}. @kwarg Vector_and_kwds: Optional class C{B{Vector}=None} to return the trilateration and optional, additional B{C{Vector}} keyword arguments, otherwise B{C{center1}}'s (sub-)class. @return: 2-Tuple with two trilaterated points, each a B{C{Vector}} instance. Both points are the same instance if all three spheres abut/intersect in a single point. @raise ImportError: Package C{numpy} not found, not installed or older than version 1.10. @raise IntersectionError: Near-concentric, colinear, too distant or non-intersecting spheres. @raise NumPyError: Some C{numpy} issue. @raise TypeError: Invalid B{C{center1}}, B{C{center2}} or B{C{center3}}. @raise UnitError: Invalid B{C{radius1}}, B{C{radius2}} or B{C{radius3}}. @see: Norrdine, A. U{I{An Algebraic Solution to the Multilateration Problem}<https://www.ResearchGate.net/publication/ 275027725_An_Algebraic_Solution_to_the_Multilateration_Problem>}, the U{I{implementation}<https://www.ResearchGate.net/publication/ 288825016_Trilateration_Matlab_Code>} and function L{pygeodesy.trilaterate2d2}. ''' from pygeodesy.vector2d import _trilaterate3d2 try: return _trilaterate3d2(_otherV3d(center1=center1, NN_OK=False), Radius_(radius1=radius1, low=eps), center2, radius2, center3, radius3, eps=eps, clas=center1.classof, **Vector_and_kwds) except (AssertionError, TypeError, ValueError) as x: raise _xError(x, center1=center1, radius1=radius1, center2=center2, radius2=radius2, center3=center3, radius3=radius3) def _xyzn4(xyz, y, z, Error=_TypeError): # see: .ecef '''(INTERNAL) Get an C{(x, y, z, name)} 4-tuple. ''' try: t = xyz.x, xyz.y, xyz.z n = getattr(xyz, _name_, NN) except AttributeError: t = xyz, y, z n = NN try: x, y, z = map2(float, t) except (TypeError, ValueError) as x: d = dict(zip((_xyz_, _y_, _z_), t)) raise Error(txt=str(x), **d) return x, y, z, n def _xyzhdn6(xyz, y, z, height, datum, ll, Error=_TypeError): # by .cartesianBase, .nvectorBase '''(INTERNAL) Get an C{(x, y, z, h, d, name)} 6-tuple. ''' x, y, z, n = _xyzn4(xyz, y, z, Error=Error) h = height or getattr(xyz, _height_, None) \ or getattr(xyz, _h_, None) \ or getattr(ll, _height_, None) d = datum or getattr(xyz, _datum_, None) \ or getattr(ll, _datum_, None) return x, y, z, h, d, n __all__ += _ALL_DOCS(intersections2, sumOf, Vector3dBase) # **) 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.