On Kinetic Delaunay Triangulations: A Near-Quadratic Bound for Unit Speed Motions

Let P be a collection of n points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of O ( n 2+ϵ ), for any ϵ > 0, on the maximum number of discrete changes that the Delaunay triangulation DT( P ) of P experiences during this motion. Our analysis is cast in a purely topological setting, where we only assume that (i) any four points can be co-circular at most three times, and (ii) no triple of points can be collinear more than twice; these assumptions hold for unit speed motions.