Maximizing the overlap of two planar convex sets under rigid motions

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 that are given in an array or balanced tree, the total running time is O((1/ε)logn+(1/ε2)log(1/ε)) for rigid motions and O((1/ε)logn+(1/ε)log(1/ε)) for translations.