Abstract
We show that the number of critical positions of a convex polygonal objectB moving amidst polygonal barriers in two-dimensional space, at which it makes three simultaneous contacts with the obstacles but does not penetrate into any obstacle isO(knλ s (kn)) for somes≤6, wherek is the number of boundary segments ofB,n is the number of wall segments, andλ s (q) is an almost linear function ofq yielding the maximal number of “breakpoints” along the lower envelope (i.e., pointwise minimum) of a set ofq continuous functions each pair of which intersect in at mosts points (here a breakpoint is a point at which two of the functions simultaneously attain the minimum). We also present an example where the number of such critical contacts is Ω(k 2 n 2), showing that in the worst case our upper bound is almost optimal.
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M. Atallah, Dynamic computational geometry,Proceedings of the 24th Symposium on Foundations of Computer Science, 92–99, 1983.
S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes,Combinatorica 6 (1986), 175–201.
K. Kedem and M. Sharir, An Efficient Motion-Planning Algorithm for a Convex Polygonal Object in Two-Dimensional Polygonal Space, Technical Report 253, Computer Science Department, Courant Institute, 1986.
K. Kedem, R. Livne, J. Pach, and M. Sharir, On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles,Discrete Comput. Geom. 1 (1986), 59–71.
D. Leven and M. Sharir, An efficient and simple motion-planning algorithm for a ladder moving in two-dimensional space amidst polygonal barriers,Proceedings of the ACM Symposium on Computational Geometry, 221–227, 1985 (also to appear inJ. Algorithms).
D. Leven and M. Sharir, Planning a Purely Translational Motion for a Convex Object in Two-Dimensional Space Using Generalized Voronoi Diagrams, Technical Report 34/85, The Eskenasy Institute of Computer Science, Tel Aviv University, 1985 (also to appear inDiscrete Comput. Geom.).
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 Appl. Math. 36 (1983), 345–398.
M. Sharir, Almost Linear Upper Bounds on the Length of Generalized Davenport-Schinzel Sequences, Technical Report 29/85, The Eskenasy Institure of Computer Science, Tel-Aviv University, 1985 (also to appear inCombinatorica 7 (1987).)
M. Sharir, Improved Lower Bounds on the Length of Davenport-Schinzel Sequences, Technical Report 204, Computer Science Department, Courant Institute, 1986.
E. Szemeredi, On a problem by Davenport and Schinzel,Acta Arith. 25 (1974), 213–224.
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Work on this paper by the second author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation.
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Leven, D., Sharir, M. On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space. Discrete Comput Geom 2, 255–270 (1987). https://doi.org/10.1007/BF02187883
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DOI: https://doi.org/10.1007/BF02187883