Simultaneous Containment of Several Polygons: Analysis of the Contact Configurations

Abstract : The main concern of this paper is the detection of double contact configurations for some polygons moving in translation in a polygonal environment. We first establish some general properties about such configurations and give conditions of existence of double contacts for two or three objects. For three convex polygons moving in a polygonal environment or three simple polygons moving in a rectangle there always exists a double contact. Two examples without possibility of double contacts are given, one with three polygons (not convex) moving in a polygonal environment, and one with four convex polygons moving in a rectangle. We deduce an algorithm detecting a double contact position in time O(n2) (resp. O(n3)) for two (resp three) convex polygons of constant sizes moving in a non-convex polygon of size n.