Quasi-Delaunay Triangulations Using GPU-Based Edge-Flips

Abstract

The edge-flip technique has been widely used for transforming any existing triangular mesh into a Delaunay mesh. Although several tools for generating Delaunay triangulations are known, there is no one that offers a realtime solution capable of maintaining the Delaunay condition on dynamically changing triangulations and, in particular, one integrable with the OpenGL rendering pipeline. In this paper we present an iterative GPGPU-based method capable of improving triangulations under the Delaunay criterion. Since the algorithm uses an $\epsilon$ value to handle co-circular or close to co-circular point configurations, a low percentage of triangles do not fulfill the Delaunay condition. We have compared the triangulations generated by our method with the ones generated by the Triangle software and by the CGAL library and we obtained less than 0.05 % different triangles for full random meshes and less than 1 % for noise based ones. Based on our experimental results, we report speedups from 14$\times$ to 50$\times$ against Lawson’s sequential algorithm and of approximately 3$\times$ against the CGAL’s and Triangle’s constructive algorithms when processing full random triangulations. In our noise based tests we report up to 36$\times$ and 27$\times$ of speedup against CGAL and Triangle, respectively.