An arithmetic and combinatorial approach to three-dimensional discrete lines

Abstract

The aim of this paper is to discuss from an arithmetic and combinatorial viewpoint a simple algorithmic method of generation of discrete segments in the three-dimensional space. We consider discrete segments that connect the origin to a given point (u 1,u 2,u 3) with coprime nonnegative integer coordinates. This generation method is based on generalized three-dimensional Euclid's algorithms acting on the triple (u1,u 2,u3). We associate with the steps of the algorithm substitutions, that is, rules that replace letters by words, which allow us to generate the Freeman coding of a discrete segment. We introduce a dual viewpoint on these objects in order to measure the quality of approximation of these discrete segments with respect to the corresponding Euclidean segment. This viewpoint allows us to relate our discrete segments to finite patches that generate arithmetic discrete planes in a periodic way. © 2011 Springer-Verlag.