Pipes, Cigars, and Kreplach: the Union of Minkowski Sums in Three Dimensions |
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Authors: | PK Agarwal M Sharir |
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Institution: | (1) Center for Geometric Computing, Department of Computer Science, Box 90129, Duke University, Durham, NC 27708-0129, USA pankaj@cs.duke.edu , US;(2) School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel sharir@math.tau.ac.il , IL;(3) Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA , US |
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Abstract: | Let Ω be a set of pairwise-disjoint polyhedral obstacles in R
3 with a total of n vertices, and let B be a ball in R
3. We show that the combinatorial complexity of the free configuration space F of B amid Ω:, i.e., (the closure of) the set of all placements of B at which B does not intersect any obstacle, is O(n
2+ε
), for any ε >0; the constant of proportionality depends on ε. This upper bound almost matches the known quadratic lower bound
on the maximum possible complexity of F . The special case in which Ω is a set of lines is studied separately. We also present a few extensions of this result, including
a randomized algorithm for computing the boundary of F whose expected running time is O(n
2+ε
).
Received July 6, 1999, and in revised form April 25, 2000. Online publication August 18, 2000. |
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Keywords: | |
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