On geodesic properties of polygons relevant to linear time triangulation

Triangulating a simple polygon of n vertices in O(n) time is one of the main open problems in computational geometry. The fastest algorithm to date, due to Tarjan and van Wyk, runs in O(n log log n), but several classes of simple polygons have been shown to admit linear time traingulation. Famous examples of such classes are: star-shaped, monotone, spiral, edge visible, and weakly externally visible polygons. The notion of geodesic paths is used here to characterize all classes of polygons for which linear time triangulation algorithms are known. First we introduce a new class of polygons, palm polygons, which subsumes many known classes of polygons for which linear time triangulation algorithms exist, and present a linear time algorithm for triangulating polygons in this class. Then a class of polygons, crab polygons, is defined and shown to contain all classes of existing polygons for which linear time triangulation algorithms are known. As a byproduct of this characterization, a new, very simple linear time algorithm for triangulating star-shaped polygons is obtained.