Math @ Duke

Publications [#235432] of Pankaj K. Agarwal
Papers Published
 Agarwal, PK; Sharir, M, Pipes, Cigars, and Kreplach: The Union of Minkowski Sums in Three Dimensions,
Discrete and Computanional Geometry, vol. 24 no. 4
(2000),
pp. 645685
(last updated on 2017/12/13)
Abstract: Let Ω be a set of pairwisedisjoint polyhedral obstacles in ℝ3 with a total of n vertices, and let B be a ball in ℝ3. 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(n2+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(n2+ε), for any ε > 0.


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