Euler's characteristics and Pick's theorem

Abstract

Pick's Theorem is used to compute the area of lattice polygons. In this paper we present a very general definition of a polygon to obtain the definition of faces and holes of a polygon. The decomposition of a polygon into triangles is briefly discussed. Then lattices are introduced with their corresponding elementary triangles and triangulations. We recall and prove Pick's Theorem for simple polygons. Then a generalization of Pick's Theorem is proved for very general lattice polygons, generalized lattice polygons and union of generalized lattice polygons using original and modified Euler's characteristics.

A formal proof of Pick's Theorem

“…It would also be natural to consider formalising extensions and generalisations of Pick's Theorem, and other related results. One possibility is to extend it to non-simple polygons using Euler characteristics (Dubeau and Labbé 2007). Another is to consider higher-dimensional generalisations using Ehrhart polynomials (Ehrhart 1967).…”