Generalized Functionality for Arithmetic Discrete Planes

Abstract

Abstract. The discrete plane P (a, b, c, µ, ω) is the set of points (x, y, z) ∈ Z 3 satisfying 0 ≤ ax+by+cz+µ < ω. In the case ω = max (|a|, |b|, |c|), the discrete plane is said naive and is well-known to be functional on a coordinate plane. The aim of our paper is to extend the notion of functionality to a larger family of arithmetic discrete planes by introducing a suitable orthogonal projection direction (α, β, γ) satisfying αa + βb + γc = ω. We then apply this functionality property to the enumeration of some local configurations, that is, the (m, n)-cubes such as introduced in [VC99].Keywords: digital planes; arithmetic planes; local configurations; functionality of discrete planes.The discrete plane P (a, b, c, µ, ω) is the set of integer points (x, y, z) ∈ Z 3 satisfying 0 ≤ ax + by + cz + µ < ω. In the case ω = max (|a|, |b|, |c|), the discrete plane is said naive and is well-known to be functional on one of the coordinate planes, that is, for any point of P of this coordinate plane, there exists a unique point in the discrete plane obtained by adding to P a third coordinate. Naive planes have been widely studied, see for instance [Rev91, DRR94, DR95, AAS97, VC97, Col02, BB02].The present paper extends the notion of functionality for naive discrete planes to a larger family of arithmetic discrete planes. For that purpose, instead of projecting on a coordinate space, we introduce a suitable orthogonal projection on a plane along a direction (α, β, γ), in some sense dual to the normal vector of the discrete plane P (a, b, c, µ, ω), that is, αa + βb + γc = ω, so that the projection of Z 3 and the points of the discrete plane are in one-to-one correspondence. One interest of the notion of functionality is that it reduces a threedimensional problem to a two-dimensional one, allowing a better understanding of the combinatorial and geometric properties of discrete planes. We thus apply this functionality property to the enumeration of some local configurations, the (m, n)-cubes, for a large family of arithmetic discrete planes, following the approach of [Vui99,BV01].For clarity issues, we have chosen to work here in a three-dimensional space but all the results and methods presented extend in a natural way to R n , with n ≥ 2, as well as to arithmetic discrete lines.

On some applications of generalized functionality for arithmetic discrete planes

“…Let us note that we have tried to make this paper essentially self-contained. This paper is an extended version of [BFJ05], but also contains some new applications of functionality.…”

Characterization of the Closest Discrete Approximation of a Line in the 3-Dimensional Space

“…In the present paper, our purpose is to introduce a modeling of discrete lines in the 3-dimensional space such that topological properties and the relationship with the closest integer points to the Euclidean line with same parameter are easily determined. We propose a representation of the 1-dimensional linear discrete object inspired by the notion of functionality [10,11]. Indeed, a connected discrete line in the 3-dimensional space should verify some conditions similar to this notion: we can define subsets of Z 3 such that the discrete line is connected only if it contains at least a point of each of them.…”