The Complexity of a Single Face of a Minkowski Sum

This note considers the complexity of a free region in the configuration space of a polygonal robot translating amidst polygonal obstacles in the plane. Specifically, given polygonal sets P and Q with k and n vertices, respectively (k ! n), the number of edges and vertices bounding a single face of the complement of the Minkowski sum P \Phi Q is \Theta(nkff(k)) in the worst case. The lower bound comes from a construction based on lower envelopes of line segments; the upper bound comes from a combinatorial bound on Davenport-Schinzel sequences that satisfy two alternation conditions. 1 Introduction and Background Let A and B be two sets in IR 2 . The Minkowski sum (or vector sum) of A and B, denoted A \Phi B, is the set fa + b j a 2 A; b 2 Bg. The Minkowski sum is a useful concept in robot motion planning and related areas [2, 11, 12, 13]. For example, consider an obstacle A and a robot B that moves by translation. We can choose a reference point r rigidly attached to B and suppo..