On the Non-Termination of Ruppert's Algorithm

TL;DR

This paper presents specific examples of planar straight-line graphs that cause Ruppert's and Chew's algorithms to fail to terminate at certain angle thresholds, highlighting limitations unrelated to small input angles.

Contribution

The paper provides explicit counterexamples demonstrating non-termination of Ruppert's and Chew's algorithms at specific angle thresholds, challenging assumptions about their robustness.

Findings

Ruppert's algorithm fails to terminate for thresholds above 29.5 degrees with a 74.5-degree input angle.

Chew's second algorithm does not terminate for thresholds above 30.7 degrees with a non-acute input.

Failure is not caused by small input angles, but by specific graph configurations.

Abstract

A planar straight-line graph which causes the non-termination Ruppert's algorithm for a minimum angle threshold larger than about 29.5 degrees is given. The minimum input angle of this example is about 74.5 degrees 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 larger than about 30.7 degrees.