Abstract: We present two new Delaunay refinement algorithms, the second an extension of the first. For a given input domain (a set of points in the plane or a planar straight line graph), and a threshold angle a, the Delaunay refinement algorithms compute triangulations that have all angles at least a. 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 a = 20.7 degrees. In practice, it generally works for a = 34 degrees and fails to terminate for larger constraint angles. The new Delaunay refinement algorithm generally terminates for constraint angles up to 42 degrees. 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.