On some applications of generalized functionality for arithmetic discrete planes

Abstract

Naive discrete planes are well known to be functional on a coordinate plane. The aim of our paper is to extend the functionality concept to a larger family of arithmetic discrete planes, by introducing suitable projection directions (a 1 , a 2 , a 3 ) satisfying a 1 v 1 + a 2 v 2 + a 3 v 3 = w. Several applications are considered. We first study certain local configurations, that is, the (m, n)-cubes introduced in We compute their number for a given (m, n) and study their statistical behaviour. We then apply functionality to formulate an algorithm for generating arithmetic discrete planes, inspired by Debled-Renesson [I. Debled-Renesson, Reconnaissance des Droites et Plans Discrets, Thèse de doctorat, Université Louis Pasteur, Strasbourg, France, 1995.]. We also prove that an arithmetic discrete plane may be endowed with a combinatorial surface structure, in the spirit of Ref.[Y. Kenmochi, A. Imiyam Combinatorial topologies for discrete planes, in

About thin arithmetic discrete planes

“…An arithmetic discrete plane P( v, µ, ω) with dim Q v = 1 is called rational, otherwise it is called irrational, according to [AAS97,BFJP07]. From now on, we shall agree that any representation P( v, µ, ω) of a rational arithmetic discrete plane satisfies:…”

“…Indeed, naive planes are well known to be functional, that is, in a one-to-one correspondence with the integer points of one of the coordinate planes by an orthogonal projection map. The notion of functionality for naive arithmetic discrete planes can be extended to a larger family of arithmetic discrete planes, such as described in [BFJP07], by introducing a suitable projection mapping. Functionality allows the reduction of a three-dimensional problem to a twodimensional one, and thus leads to a better understanding of the combinatorial and geometric properties of arithmetic discrete planes.…”

“…Functionality allows the reduction of a three-dimensional problem to a twodimensional one, and thus leads to a better understanding of the combinatorial and geometric properties of arithmetic discrete planes. We propose here an alternative strategy to the functional one developed in [BFJP07] for the study of arithmetic discrete planes that are not necessarily naive or standard. Instead of taking a suitable projection mapping, we continue to work with the standard selection window of size || v|| 1 , but we compare our selection window [0, ω), for ω < || v|| 1 , with the standard one [0, || v|| 1 ).…”