Capturing a convex object with three discs

Abstract

This paper addresses the problem of capturing an arbitrary convex object P in the plane with three congruent disc-shaped robots. Given two stationary robots in contact with P, we characterize the set of positions of a third robot that prevent P from escaping to infinity and show that the computation of this so-called capture region reduces to the resolution of a visibility problem. We present two algorithms for solving this problem and computing the capture region when P is a polygon and the robots are points (zero-radius discs). The first algorithm is exact and has polynomial-time complexity. The second one uses simple hidden-surface removal techniques from computer graphics to output an arbitrarily accurate approximation of the capture region; it has been implemented and examples are presented.