Maximizing the overlap of two planar convex sets under rigid motions

Authors:
Hee-Kap Ahn;Otfried Cheong;Chong-Dae Park;Chan-Su Shin;Antoine Vigneron
Affiliations:
Korea Advanced Institute of Science and Technology, Daejeon, Korea;Korea Advanced Institute of Science and Technology, Daejeon, Korea;Korea Advanced Institute of Science and Technology, Daejeon, Korea;Hankuk University of Foreign Studies, Yongin, Korea;National University of Singapore, Singapore
Venue:
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Year:
2005

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Abstract

Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any ε 0, we compute a rigid motion such that the area of overlap is at least 1 - ε times the maximum possible overlap. Our algorithm uses O(1/ε) extreme point and line intersection queries on P and Q, plus O((1/ε2) log(1/ε)) running time. If only translations are allowed, the extra running time reduces to O((1/ε) log(1/ε)). If P and Q are convex polygons with n vertices in total, the total running time is O((1/ε) log n + (1/ε2) log(1/ε)) for rigid motions and O((1/ε) log n + (1/ε) log(1/ε)) for translations.