On the number of critical free contacts of a convex polygonal object moving in two-dimensional polygonal space

Abstract

We show that the number of critical positions of a convex polygonal object B moving amidst polygonal barriers in two-dimensional space, at which it makes three simultaneous contacts with the obstacles but does not penetrate into any obstacle is O(knλs(kn)) for some s≤6, where k is the number of boundary segments of B, n is the number of wall segments, and λs(q) is an almost linear function of q yielding the maximal number of "breakpoints" along the lower envelope (i.e., pointwise minimum) of a set of q continuous functions each pair of which intersect in at most s points (here a breakpoint is a point at which two of the functions simultaneously attain the minimum). We also present an example where the number of such critical contacts is Ω(k2n2), showing that in the worst case our upper bound is almost optimal. © 1987 Springer-Verlag New York Inc.