Computing the Maximum Overlap of Two Convex Polygons Under Translations

Abstract : Let $P$ be a convex polygon in the plane with $n$ vertices and let $Q$ be a convex polygon with $m$ vertices. We prove that the maximum number of combinatorially distinct placements of $Q$ with respect to $P$ under translations is $O(n^2+m^2+\min(nm^2+n^2m))$, and we give an example showing that this bound is tight in the worst case. Second, we present an $O((n+m)\log(n+m))$ algorithm for determining a translation of $Q$ that maximizes the area of overlap of $P$ and $Q$. We also prove that the position which translates the centroid of $Q$ on the centroid of $P$ always realizes an overlap of 9/25 of the maximum overlap and that this overlap may be as small as 4/9 of the maximum.
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https://hal.inria.fr/inria-00073859
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Submitted on : Wednesday, May 24, 2006 - 1:55:42 PM
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Mark De Berg, Olivier Devillers, Marc Van Kreveld, Otfried Schwarzkopf, Monique Teillaud. Computing the Maximum Overlap of Two Convex Polygons Under Translations. RR-2832, INRIA. 1996. ⟨inria-00073859⟩

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