Fast triangulation of the plane with respect to simple polygons

Let P₁,…, P_k be pairwise non-intersecting simple polygons with a total of n vertices and s start vertices. A start vertex, in general, is a vertex both of which neighbors have larger x coordinate. We present an algorithm for triangulating P₁,…, P_k in time O(n + s log s). s may be viewed as a measure of non-convexity. In particular, s is always bounded by the number of concave angles + 1, and is usually much smaller. We also describe two new applications of triangulation. Given a triangulation of the plane with respect to a set of k pairwise non-intersecting simple polygons, then the intersection of this set with a convex polygon Q can be computed in time linear with respect to the combined number of vertices of the k + 1 polygons. Such a result had only be known for two convex polygons. The other application improves the bound on the number of convex parts into which a polygon can be decomposed.