# Computing the Maximum Overlap of Two Convex Polygons Under Translations

1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
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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Type de document :
Rapport
RR-2832, INRIA. 1996
Domaine :
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https://hal.inria.fr/inria-00073859
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 13:55:42
Dernière modification le : jeudi 11 janvier 2018 - 16:43:47
Document(s) archivé(s) le : dimanche 4 avril 2010 - 22:08:29

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• HAL Id : inria-00073859, version 1

### Citation

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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