A Census of Planar Triangulations

Let P be a closed region in the plane bounded by a simple closed curve, and let S be a simplicial dissection of P . We may say that S is a dissection of P into a finite number α of triangles so that no vertex of any one triangle is an interior point of an edge of another. The triangles are ‘'topological” triangles and their edges are closed arcs which need not be straight segments. No two distinct edges of the dissection join the same two vertices, and no two triangles have more than two vertices in common.