Triangulations and Pick's Theorem

Abstract

where Vi and Vb, respectively, denote the number of lattice points in the interior and on the boundary of P. Observe that Vb includes, in addition to the vertices, any lattice points which occur on the boundary between the vertices. An interesting proof of Pick's theorem is contained in [3]. The proof centers around showing that the area of a so-called triangle is 1/2; a primitive triangle has no lattice points inside or on the boundary except for the (non-collinear) vertices themselves. It is not difficult to convince oneself that any simple polygon P can be decomposed into primitive triangles by appropriately joining up its lattice points with non-intersecting segments. For such a triangulation, Pick's theorem merely gives