Discrete & Computational Geometry

, Volume 56, Issue 1, pp 43–92

Nonobtuse Triangulations of PSLGs

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Abstract

We show that any planar straight line graph with n vertices has a conforming triangulation by \(O(n^{2.5})\) nonobtuse triangles (all angles \(\le 90^\circ \)), 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  Edelsbrunner 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 \(\varepsilon >0\), every PSLG has a conforming triangulation with \(O(n^2 /\varepsilon ^2)\) elements and with all angles bounded above by \(90^\circ + \varepsilon \). This improves a result of S. Mitchell when \(\varepsilon = \frac{3}{8} \pi =67.5^\circ \) and Tan when \(\varepsilon = \frac{7}{30} \pi = 42^\circ \).

Keywords

nonobtuse triangulations Delaunay triangulations Gabriel condition Thick/Thin decomposition 

Mathematics Subject Classification

Primary 68U05 Secondary 52B55 68Q25 

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematics DepartmentStony Brook UniversityStony BrookUSA

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