On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space |
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Authors: | Daniel Leven Micha Sharir |
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Institution: | (1) School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel;(2) Courant Institute of Mathematical Sciences, New York University, 10012 New York, NY, USA |
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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(kn
s
(kn)) for somes 6, wherek is the number of boundary segments ofB,n is the number of wall segments, and
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 (k
2
n
2), showing that in the worst case our upper bound is almost optimal.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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Keywords: | |
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