Nonobtuse Triangulations of PSLGs

We show that any planar PSLG with n vertices has a conforming triangulation by O(n) nonobtuse triangles, answering the question of whether a polynomial bound exists. The triangles may be chosen to be all acute or all right. A nonobtuse triangulation is Delaunay, so this result improves a previous O(n) bound of Eldesbrunner and Tan for conforming Delaunay triangulations. In the special case that the PSLG is the triangulation of a simple polygon, we will show that only O(n) elements are needed, improving an O(n) bound of Bern and Eppstein. We also show that for any ǫ > 0, every PSLG has a conforming triangulation with O(n/ǫ) elements and with all angles bounded above by 90◦ + ǫ. This improves a result of S. Mitchell when ǫ = 3 8 π = 67.5◦ and Tan when ǫ = 7 30 π = 42◦. Date: March 1, 2011. 1991 Mathematics Subject Classification. Primary: 30C62 Secondary: