Nonobtuse triangulations of PSLGs

TL;DR

This paper proves that any planar straight line graph can be triangulated with a polynomial number of nonobtuse triangles, improving previous bounds and providing angle-bounded triangulations with fewer elements.

Contribution

It establishes polynomial bounds for nonobtuse triangulations of PSLGs and improves existing bounds for special cases and angle-bounded triangulations.

Findings

Any PSLG has a conforming triangulation with O(n^{2.5}) nonobtuse triangles.

For simple polygon triangulations, only O(n^2) triangles are needed.

Existence of angle-bounded triangulations with O(n^2/ε^2) elements for any ε > 0.

Abstract

We show that any planar straight line graph (PSLG) with $n$ vertices has a conforming triangulation by $O(n^{2.5})$ nonobtuse triangles (all angles $\leq 90^\circ$), answering the question of whether any polynomial bound exists. A nonobtuse triangulation is Delaunay, so this result also improves a previous $O(n^3)$ bound of Eldesbrunner and Tan for conforming Delaunay triangulations of PSLGs. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only $O(n^2)$ triangles are needed, improving an $O(n^4)$ bound of Bern and Eppstein. We also show that for any $\epsilon >0$, every PSLG has a conforming triangulation with $O(n^2/\epsilon^2)$ elements and with all angles bounded above by $90^\circ + \epsilon$. This improves a result of S. Mitchell when $\epsilon = 3 \pi /8 = 67.5^\circ $ and Tan when $\epsilon = 7\pi/30 =42^\circ$.