Fast Triangulation of Simple Polygons

We present a new algorithm for triangulating simple polygons that has four advantages over previous solutions [GJPT, Ch]. a) It is faster: Whilst previous solutions worked in time O(nlogn), the new algorithm only needs time O(n+rlogr) where r is the number of concave angles of the polygon. b) It works for a larger class of inputs: Whilst previous solutions worked for simple polygons, the new algorithm handles simple polygons with polygonal holes. c) It does more: Whilst previous solutions only triangulated the interior of a simple polygon, the new algorithm triangulates both the interior and the exterior region. d) It is simpler: The algorithm is based on the plane-sweep paradigm and is at least in its O(nlogn) version very simple. In addition to the new triangulation algorithm, we present two new applications of triangulation. a) We show that one can compute the intersection of a convex m-gon Q and a triangulated simple n-gon P in time O(n+m). This improves a result by Shamos [Sh] stating that the intersection of two convex polygons can be computed in time O(n). b) Given the triangulation of a simple n-gon P, we show how to compute in time O(n) a convex decomposition of P into at most 4.OPT pieces. Here OPT denotes the minimum number of pieces in any convex decomposition. The best factor known so far was 4.333 (Chazelle[Ch]).