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. ¹ One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a subcubic 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 ordered triple of points can be collinear more than once, or no triple of points can be collinear more than twice.