Pencil of Lines on the $2-D$ Torus

Abstract

We consider the problem of finding the intersection points of a pencil of lines with rational slope on the $2$-dimensional torus. We show that the intersection points belonging to all the lines in the pencil form a finite cyclic group. We also exhibit a generator for this group in terms of the coefficients of the lines. The need for the results presented in this paper arose in dealing with a discrete limited angle model for computerized tomography $(Cf.[3],[5])$.