Refining a Triangulation of a Planar Straight-Line Graph to Eliminate Large Angles

We show that any PSLG with v vertices can be tri angulated with no angle larger than by adding O v log v Steiner points in O v log v time We rst triangulate the PSLG with an arbitrary con strained triangulation and then re ne that triangula tion by adding additional vertices and edges We fol low a lazy strategy of starting from an obtuse angle and exploring the triangulation in search of a sequence of Steiner points that will satisfy a local angle condi tion Explorations may either terminate successfully for example at a triangle vertex or merge Some PSLGs require v Steiner points in any triangulation achieving any largest angle bound less than Hence the number of Steiner points added by our algorithm is within a logv factor of worst case optimal For most inputs the number of Steiner points and running time would be considerably smaller than in the worst case