An Arithmetic and Combinatorial Approach to Three-Dimensional Discrete Lines

Abstract

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 (u1, u2, u3) with coprime nonnegative integer coordinates. This generation method is based on generalized three-dimensional Euclid's algorithms acting on the triple (u1, u2, 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.

Remainder approach for the computation of digital straight line subsegment characteristics

“…As one can see on Figure 6. By removing a row from the 3D plane segment P of characteristics (3, 7, 18, 0) (0 ≤ 3x + 7y + 18z + 0 < 18) on the (x, y)-interval [1,7] × [2,5], we obtain a new plane segment P ′ of characteristics (2, 5, 11, −3) where a point of remainder 2 in P becomes leaning point for P ′ while a point of remainder 1 in P does not. The equivalent of Theorem 1 is not verified in, at least, dimension 3.…”

“…The study are regained some interest when J-P. Reveillès, proposed, among previous authors [3,5], an analytical description of a DSL 0 ≤ ax − by − c < ω in [13] (where (a, b, c) are called the characteristics or parameters of the DSL). The immediate possibilities of extensions to higher dimensions and to different scales sparked interest among arithmeticians [2] and researchers in image processing [15,19]. In this paper we are interested in a particular class of DSS recognition problems.…”

Digital Analytical Geometry: How Do I Define a Digital Analytical Object?

“…The study of Digital Analytical Lines has gained a lot of traction in the arithmetical community [57,58] It is an interesting question and it shows that direct analytical denitions for digital objects may lead to interesting topological properties.…”

Thoughts on 3D Digital Subplane Recognition and Minimum-Maximum of a Bilinear Congruence Sequence

“…See [10] for an historical perspective. The natural extension to higher dimensions has opened new venues for arithmeticians [2].…”