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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 @e0, we compute a rigid motion such that the area of overlap is at least 1-@e times the maximum possible overlap. Our algorithm uses O(1/@e) extreme point and line intersection queries on P and Q, plus O((1/@e^2)log(1/@e)) running time. If only translations are allowed, the extra running time reduces to O((1/@e)log(1/@e)). 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/@e)logn+(1/@e^2)log(1/@e)) for rigid motions and O((1/@e)logn+(1/@e)log(1/@e)) for translations.