Discrete & Computational Geometry

, Volume 56, Issue 1, pp 43–92 | Cite as

Nonobtuse Triangulations of PSLGs

Article

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 

Notes

Acknowledgments

Many thanks to Joe Mitchell and Estie Arkin for numerous conversations about computational geometry in general and the results of this paper in particular. Also thanks to two anonymous referees for many helpful comments and suggestions on two earlier versions of the paper; their efforts greatly improved the presentation in this version. The author was partially supported by NSF Grant DMS 13-05233.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematics DepartmentStony Brook UniversityStony BrookUSA

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