On some applications of generalized functionality for arithmetic discrete planes

摘要

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 (α1, α2, α3) satisfying α1v1 + α2v2 + α3v3 = w. Several applications are considered. We first study certain local configurations, that is, the (m, n)-cubes introduced in Ref. [J. Vittone, J.-M. Chassery, (n,m)-cubes and Farey Nets for Naive Planes Understanding, in: DGCI, 8th International Conference, Lecture Notes in Computer Science, vol. 1568, Springer-Verlag, 1999, pp. 76–87.]. 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: DGCI, 11th International Conference, DGCI 2003, Lecture Notes in Computer Science, vol. 2886, Springer-Verlag, 2003, pp. 144–153.].