Triangulations with locally optimal Steiner points

We present two new Delaunay refinement algorithms, second an extension of the first. For a given input domain (a set of points or a planar straight line graph), and a threshold angle α, the Delaunay refinement algorithms compute triangulations that have all angles at least α. Our algorithms have the same theoretical guarantees as the previous Delaunay refinement algorithms. The original Delaunay refinement algorithm of Ruppert is proven to terminate with size-optimal quality triangulations for α ≤ 20.7. In practice, it generally works for α ≤ 34 and fails to terminate for larger constraint angles. The new variant of the Delaunay refinement algorithm generally terminates for constraint angles up to 42. Experiments also indicate that our algorithm computes significantly (almost by a factor of two) smaller triangulations than the output of the previous Delaunay refinement algorithms.