Non-mutual captures in cyclic pursuit

• Authors: T. J. Richardson
• Venue: Annals of Mathematics and Artificial Intelligence
• Year: 2001

In cyclic pursuit in bugs chase each other in cyclic order, each moving at unit speed. Mathematical problems and puzzles of pursuit, and cyclic pursuit in particular, have attracted interest for many years. In 1971 Klamkin and Newman [17] showed that if in=3 and the initial positions of the bugs are not collinear, then all three bugs capture their prey simultaneously, i.e., no bug captures its prey prior to the moment when the pursuit collapses to a single point. They asked whether the result generalizes to more bugs. Behroozi and Gagnon [4] showed that it does generalize to in=4 if the bugs' initial positions form a convex polygon. In this paper we resolve the general question in ik dimensions: It is possible for bugs to capture their prey without all bugs simultaneously doing so even for non-collinear initial positions. The set of initial conditions which give rise to non-mutual captures is, however, a sub-manifold in the manifold of all possible initial conditions. Hence, if the initial positions are picked randomly according to a smooth probability distribution, then the probability that a non-mutual capture will occur is zero.