Maximizing the overlap of two planar convex sets under rigid motions

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 / ε 2 ) 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 ) log n + ( 1 / ε 2 ) log ( 1 / ε ) ) for rigid motions and O ( ( 1 / ε ) log n + ( 1 / ε ) log ( 1 / ε ) ) for translations.