Overlap of convex polytopes under rigid motion

We present an algorithm to compute a rigid motion that approximately maximizes the volume of the intersection of two convex polytopes P-1 and P-2 in R-3. For all epsilon is an element of (0, 1/2] and for all n >= 1/epsilon, our algorithm runs in O(epsilon(-3) n log(3.5) n) time with probability 1 - n(-O(1)). The volume of the intersection guaranteed by the output rigid motion is a (1 - epsilon)-approximation of the optimum, provided that the optimum is at least lambda . max{vertical bar P-1 vertical bar . vertical bar P-2 vertical bar} for some given constant lambda is an element of (0, 1]. (C) 2013 Elsevier B.V. All rights reserved.X1155Ysciescopu

Approximating the Maximum Overlap of Polygons under Translation

Let $P$ and $Q$ be two simple polygons in the plane of total complexity $n$, each of which can be decomposed into at most $k$ convex parts. We present an $(1-\varepsilon)$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O(c n)$ time, where $c$ is a constant that depends only on $k$ and $\varepsilon$. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

Overlap of Convex Polytopes under Rigid Motion ∗

We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R 3 under rigid motion. Given any ε ∈ (0, 1/2], our algorithm runs in O(ε −3 n log 3.5 n) time with probability 1 − n −O(1) and returns a (1 − ε)-approximate maximum overlap, provided that the maximum overlap is at least λ · max{|P1|, |P2|} for some given constant λ ∈ (0, 1]

Overlap of Convex Polytopes under Rigid Motion

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Overlap of convex polytopes under rigid motion

We present an algorithm to compute an approximate overlap of two convex polytopes P1 and P2 in R under rigid motion. Given any ε ∈ (0,1/2], our algorithm runs in O(ε-3 n log 3.5 n) time with probability 1 - n-O(1) and returns a (1 - ε) -approximate maximum overlap, provided that the maximum overlap is at least λ · max{\P1\, \P2\} for some given constant λ ∈ (0,1]. © H.-K. Ahn, S.-W. Cheng, H.J. Kwon, and J. Yon