On topological changes in the Delaunay triangulation of moving points

TL;DR

This paper establishes a near-quadratic upper bound on the number of topological changes in the Delaunay triangulation of moving points in the plane, advancing understanding of geometric motion complexity.

Contribution

It provides a new upper bound of O(n^{2+ε}) on Delaunay triangulation changes under specific collinearity and co-circularity constraints.

Findings

Upper bound of O(n^{2+ε}) for Delaunay triangulation changes

Conditions on collinearity and co-circularity are crucial

Progress towards nearly quadratic bounds in geometric motion

Abstract

Let $P$ be a collection of $n$ points moving along pseudo-algebraic trajectories in the plane. 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 $O(n^{2+\eps})$, for any $\eps>0$, 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.