Computing the Minimum Directional Distance between Two Convex Polyhedra

Given two convex polyhedra P and Q and a direction s, the minimum directional distance (MDD) is defined to be the shortest translational distance along the direction s that is required to make P and Q just in contact. In this paper we propose a novel method, called MDD-DUAL , for computing the MDD between two convex polyhedra. The MDD is equivalent to the shortest distance between the origin and the Minkowski difference M of the polyhedra in the direction s. Our idea is to reduce the MDD problem to seeking a vertex on the dual polyhedron of M with the maximum signed distance from a special plane by means of a duality transformation. We show that this is equivalent to locating a face on M with which a ray shooting from the origin in the direction s first intersects. The MDD can then easily be derived from the signed distance. Our algorithm constructs only a subset of the faces on M along the search path. By further breaking down the search into three phases, each on a different type of faces on M , MDDDUAL reports the MDD between two convex polyhedra efficiently.