Smooth Morphing Delaunay Triangulation

A variety of applications nowadays deal with complex dynamical problems where the data sets and interactions change over time. One of the ways to effectively deal with such problems is to employ Delaunay triangulation (DT). The structure however is well known to undergo significant changes when vertices are inserted or removed. A DT with dynamical updates displays visualization artifacts with non-smooth motions when viewed over time. The topic of this paper is smooth morphing of a DT under such circumstances. We address the issue by a series of simple operations carried out over time. Specifically, when a vertex is inserted at certain point, we split an existing vertex into two and slide one of them to the point. When a vertex is removed, we slide it towards one of its neighbors, and then merge the two vertices. For both cases, the DT properties are restored by collecting and then flipping illegal edges. This is performed by sliding a temporary vertex in two phases along two diagonals of the quadrilateral incident to the edge. The proposed algorithm has applications in dynamical computer graphics where temporal continuity is important. We validate the proposed algorithm by experiments.