Quality Triangulations with Locally Optimal Steiner Points

We propose two novel ideas to improve the performance of Delaunay refinement algorithms which are used for computing quality triangulations. The first idea is an effective use of the Voronoi diagram and unifies previously suggested Steiner point insertion schemes (circumcenter, sink, off-center) together with a new strategy. The second idea is the integration of a new local smoothing strategy into the refinement process. These lead to two new versions of Delaunay refinement, where the second is simply an extension of the first. For a given input domain and a constraint angle $\alpha$, Delaunay refinement algorithms aim to compute triangulations that have all angles at least $\alpha$. The original Delaunay refinement algorithm of Ruppert is proven to terminate with size-optimal quality triangulations for $\alpha\le20.7^\circ$. In practice, the original and the consequent Delaunay refinement algorithms generally work for $\alpha\le34^\circ$ and fail to terminate for larger constraint angles. Our algorithms provide the same theoretical guarantees as the previous Delaunay refinement algorithms. The second of the proposed algorithms generally terminates for constraint angles up to $42^\circ$. Experiments also indicate that our algorithm computes significantly (about by a factor of two) smaller triangulations than the output of the previous Delaunay refinement algorithms. Moreover, the new algorithms are experimentally shown to outperform the previous algorithms even in the existence of additional constraints, such as the maximum area triangle constraint which is commonly used for computing uniform triangulations.