On Topological Changes in the Delaunay Triangulation of Moving Points

Let P be a collection of n points moving along pseudo-algebraic trajectories in the plane. (So, in particular, there are constants s,c>0 such that any four points are co-circular at most s times, and any three points are collinear at most c times.) One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a sub-cubic bound, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during the motion of the points of P. In this paper, we obtain an upper bound of O(n^2+ε), for any ε>0, under the assumptions that (i) any four points can be co-circular at most twice and (ii) either no triple of points can be collinear more than twice or no ordered triple of points can be collinear more than once.