Nonobtuse triangulations of PSLGs

07/20/2020

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by   Christopher J. Bishop, et al.

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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 ≤ 90^∘), 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 ϵ >0, every PSLG has a conforming triangulation with O(n^2/ϵ^2) elements and with all angles bounded above by 90^∘ + ϵ. This improves a result of S. Mitchell when ϵ = 3 π /8 = 67.5^∘ and Tan when ϵ = 7π/30 =42^∘.