TEOREMA DE PICK PARA RETICULADOS BIDIMENSIONAIS E UM CRITÉRIO LINEAR PARA REGIÕES DE VORONOI

Abstract

In this article, we derive Pick’s Theorem for arbitrary two-dimensional lattices from its classical formulation on Z₂ by means of invertible linear transformations of the plane. This yields a version of the theorem valid for every two-dimensional lattice Λ ⊂ R². We then discuss the relation between this result and the area of Voronoi regions in dimension 2. We show that, in the rectangular and regular hexagonal cases, the identity Area(V_Λ) = det(Λ) can be recovered by means of Pick’s Theorem, after the construction of suitable invertible linear transformations. Moreover, we show that this linear procedure does not extend to the non-regular hexagonal case, thereby delimiting the scope of the method considered.

Keywords: Pick’s theorem; two-dimensional lattices; Voronoi regions; linear transformations; geometry of numbers.