Motion planning in the CL-environment

Extended abstract
  • Chee-Keng Yap
  • Helmut Alt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 382)


Most motion planning work assumes the piecewise-linear or PL-environment. Here we consider the piecewise curvi-linear or CL-environment. We describe extensions to current techniques suitable for the CL-environment. In particular, we show that various techniques for computing Voronoi diagrams and for motion planning generalize in a satisfactory way: there is no asymptotic increase in complexity when the algebraic complexity is kept constant. An underlying premise of our approach is that in the CL-environment, convex chains play the role of convex polygons in PL-environments.


Motion Planning Voronoi Diagram Convex Polygon Algebraic Curve Medial Axis 
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  1. [1]
    A. Aggarwal, L. J. Guibas, J. Saxe, and P. W. Shor. A linear time algorithm for computing the Voronoi diagram of a convex polygon. In ACM Symposium on Theory of Computing, pages 39–45, 1987.Google Scholar
  2. [2]
    J. H. Davenport, Y. Siret, and E. Tournier. Computer Algebra: systems and algorithms for algebraic computation. Academic Press, 1988.Google Scholar
  3. [3]
    John Johnstone and Chanderjit Bajaj. On the sorting of points along an algebraic curve. 1988. Submitted, SIAM J. Computing.Google Scholar
  4. [4]
    John K. Johnstone. The Sorting of Points along an Algebraic Curve. Technical Report 87-841, Department of Computer Science, Cornell University, 1987. Ph.D. thesis.Google Scholar
  5. [5]
    Klara Kedem and Micha Sharir. An automatic motion planning system for a convex polygonal mobile robot in 2-dimensional polygonal space. In ACM Symp. on Comp. Geo., pages 329–340, 1988.Google Scholar
  6. [6]
    Daniel Leven and Micha Sharir. Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams. Discrete and Comput. Geo., 2:9–31, 1987.Google Scholar
  7. [7]
    Jacob T. Schwartz and Micha Sharir. On the piano movers' problem: II. General techniques for computing topological properties of real algebraic manifolds. Advances in Appl. Math., 4:298–351, 1983.CrossRefGoogle Scholar
  8. [8]
    Chee-Keng Yap. Algorithmic Motion Planning, chapter 3. Lawrence Erlbaum Associates, 1987.Google Scholar
  9. [9]
    Chee-Keng Yap. An O(n log n) algorithm for the Voronoi diagram of a set of simple curve segments. Discrete and Comput. Geo., 2:365–394, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Chee-Keng Yap
    • 1
  • Helmut Alt
    • 2
  1. 1.Courant InstituteNew York UniversityNew YorkUSA
  2. 2.Fachbereich MathematikFreie Universität BerlinBerlin 33FRG

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