Triangulating Simple Polygons: Pseudo-Triangulations

Triangulating a given n-vertex simple polygon means to partition the interior of the polygon into n − 2 triangles by adding n − 3 nonintersecting diagonals. Significant theoretical advances have recently been made in finding efficient polygon triangulation algorithms. However, there is substantial effort being made in finding a simple and practical triangulation algorithm. We propose the concept of pseudo-triangulation (a generalized version of triangulation in which the member triangles need not all have the same orientation), and explore some of its combinatorial and topological properties. Some of the main results of this paper are: (1) We prove the triangulation-flip-graph of a simple polygon is connected. Using this theorem we obtain a very simple linear-time algorithm to recognize whether a given triangulation of a simple polygon is its unique triangulation. (2) We prove the maximum diameter of the triangulation-flip-graph is Θ(n). (3) We prove the Spin-Number Theorem on simple polygons; an interesting topological result. (4) We propose a triangulation heuristic that uses the angular (deficit) indices, and the chord-flip operation, in a local search to transform an initial pseudo-triangulation (which is easy to construct) into a triangulation. The significant open problem with this regard is finding an effective criterion in further refinement of the heuristic regarding the selection of the chord in the chord-flip operation. key words. simple polygon, triangulation, pseudo-triangulation, chordflip operation, spin number, angular (deficit) index. AMS(MOS) subject classifications. 51M15, 68Q25.