On the Non-Termination of Ruppert's Algorithm

A planar straight-line graph which causes the non-termination Ruppert’s algorithm for a minimum angle threshold α ' 29.5 is given. The minimum input angle of this example is about 74.5 meaning that failure is not due to small input angles. Additionally, a similar non-acute input is given for which Chew’s second algorithm does not terminate for a minimum angle threshold α ' 30.7. For a non-acute planar straight-line graph, Ruppert’s algorithm produces a conforming Delaunay triangulation composed of triangles containing no angles less than α. Ruppert proved the algorithm terminates for all α / 20.7◦ [3] and a minor addition to the analysis extends the results to input with all angles larger than 60◦. In practice, the constraint α / 20.7◦ has been seen to be overly conservative. Ruppert observed that the minimum angle reaches 30◦ during typical runs of the algorithm. Further experimentation by Shewchuk [4] suggested that even higher values are admissible: “In practice, the algorithm generally halts with an angle constraint of 33.8◦, but often fails [at] 33.9◦.” In this note, we demonstrate an input (the upcoming Example 2) for which Ruppert’s algorithm does not terminate for some minimum angle parameter α less than 30◦. We begin by revisiting the best known example which causes non-termination for any α > 30◦. Pav Example Steven Pav gave an example demonstrating that Ruppert’s algorithm can fail to terminate for any α > 30◦ [2]. This example, depicted in Figure 1, involves two adjacent segments with lengths 1 and √ 2 such that they form a triangle with a 30◦ angle. Pav observed that the circumcenter of this triangle lies on the boundary of the diametral ball of the longer segment. This causes the longer segment to split, This note is an extension of “On the Termination of Ruppert’s Algorithm” which appeared in the Research Notes of the 19th International Roundtable, 2010. While generated after the submission deadline, these results were presented at the conference with the original research note. University of Texas-Austin, [email protected]