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Discrete & Computational Geometry

, Volume 1, Issue 1, pp 59–71 | Cite as

On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles

  • Klara Kedem
  • Ron Livne
  • János Pach
  • Micha Sharir
Article

Abstract

Let γ1,..., γ m bem simple Jordan curves in the plane, and letK1,...,K m be their respective interior regions. It is shown that if each pair of curves γ i , γ j ,ij, intersect one another in at most two points, then the boundary ofK=∩ i =1m K i contains at most max(2,6m − 12) intersection points of the curvesγ1, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygonB amidst several (convex) polygonal obstaclesA1,...,A m . Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log2n), wheren is the total number of corners of theA i 's.

Keywords

Voronoi Diagram Jordan Curve Simple Polygon Convex Corner Motion Planning Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    T. Asano, T. Asano, L. Guibas, J. Hershberger, and H. Imai, Visibility polygon search and Euclidean shortest paths, Proc. 26th IEEE Symp. on Foundations of Computer Science, 1985, pp. 155–164.Google Scholar
  2. 2.
    R. V. Benson, Euclidean Geometry and Convexity, McGraw-Hill, 1966, pp. 97–113.Google Scholar
  3. 3.
    J. L. Bentley and A. Ottmann, Algorithms for reporting and counting geometric intersections, IEEE Trans. on Computers, Vol. C-28 (1979), pp. 643–647.CrossRefGoogle Scholar
  4. 4.
    P. Erdős and B. Grünbaum, Osculation vertices in arrangements of curves, Geometriae Dedicata 1 (1973) 322–333.MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Grünbaum, Arrangements and Spreads, Regional Conference Series in Mathematics, Vol. 10, Conference Board of the Mathematical Sciences, American Mathematical Society, Providence R.I. 1972.zbMATHGoogle Scholar
  6. 6.
    M. D. Guay and D. C. Kay, On sets having finitely many points of local nonconvexity, Israel J. Math. 8 (1970), 39–52.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    L. Guibas, L. Ramshaw, and J. Stolfi, A kinetic approach to computational geometry, Proc. 24th IEEE Symp. on Foundations of Computer Science, 1983, 100–111.Google Scholar
  8. 8.
    H. Imai, M. Iri, and K. Murota, Voronoi diagram in the Laguerre geometry and its applications, Tech. Rept. RMI 83-02, Dept. of Mathematical Engineering and Instrumentation Physics, University of Tokyo.Google Scholar
  9. 9.
    D. G. Kirkpatrick, Optimal search in planar subdivisions, SIAM J. Computing 12 (1983), 28–35.MathSciNetCrossRefGoogle Scholar
  10. 10.
    D. Leven and M. Sharir, An efficient and simple motion planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers, Tech. Rept. 84-014, The Eskenasy Institute of Computer Science, Tel Aviv University, November 1984.Google Scholar
  11. 11.
    T. Lozano-Perez and M. Wesley, An algorithm for planning collision-free paths among polyhedral obstacles, Comm. ACM 22 (1979), 560–570.CrossRefGoogle Scholar
  12. 12.
    E. E. Moise, Geometric Topology in Dimension 2 and 3, Springer-Verlag, New York, 1977.CrossRefGoogle Scholar
  13. 13.
    J. Nievergelt and F. P. Preparata, Plane sweeping algorithms for intersecting geometric figures, Comm. ACM 25 (1982), 739–747.CrossRefzbMATHGoogle Scholar
  14. 14.
    C. Ó'Dúnlaing, M. Sharir, and C. Yap, Generalized Voronoi diagrams for a ladder: I. Topological analysis, Tech. Rept. 139, Computer Science Dept., Courant Institute, November 1984.Google Scholar
  15. 15.
    C. Ó'Dúnlaing, M. Sharir, and C. Yap, Generalized Voronoi diagrams for a ladder: II. Efficient construction of the diagram, Tech. Rept. 140, Computer Science Dept., Courant Institute, November 1984.Google Scholar
  16. 16.
    C. Ó'Dúnlaing and C. K. Yap, A ‘retraction’ method for planning the motion of a disc, J. of Algorithms 6 (1985) 104–111.CrossRefGoogle Scholar
  17. 17.
    T. Ottmann, P. Widmeyer, and D. Wood, A fast algorithm for boolean mask operations, Inst. f. Angewandte Mathematik und Formale Beschreibungsverfahren, D-7500 Karlsruhe, Rept. No. 112, 1982.zbMATHGoogle Scholar
  18. 18.
    J. T. Schwartz and M. Sharir, On the “Piano Movers” problem: I. The case of a two dimensional rigid polygonal body moving amidst polygonal barriers, Comm. Pure and Appl. Math. 36 (1983), 345–398.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    M. Sharir, Intersection and closest-pair problems for a set of planar discs, SIAM J. Computing 14 (1985), 448–468.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    S. Sifrony and M. Sharir, A new efficient motion planning algorithm for a rod in two-dimensional polygonal space, Tech. Rept. 40/85, The Eskenasy Institute of Computer Sciences, Tel Aviv University, August 1985.Google Scholar
  21. 21.
    G. T. Toussaint, On translating a set of spheres, Tech. Rept. SOCS-84.4, School of Computer Science, McGill University, March 1984.Google Scholar
  22. 22.
    E. Welzl, Constructing the visibility graph forn line segments inO(n 2) time, Inf. Proc. Letters 20 (1985), 167–172.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    F. Aurenhammer, Power diagrams: properties, algorithms, and applications, Tech. Rept. F120, IIG, Tech. Univ. of Graz, Austria, 1983.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Klara Kedem
    • 1
  • Ron Livne
    • 1
  • János Pach
    • 3
  • Micha Sharir
    • 1
    • 2
  1. 1.School of Mathematical SciencesTel Aviv UniversityIsrael
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Mathematical Institute of the Hungarian Academy of SciencesHungary

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