A First Look into a Formal and Constructive Approach for Discrete Geometry Using Nonstandard Analysis

Abstract

In this paper, we recall the origins of discrete analytical geometry developed by J-P. Reveillès [1] in the nonstandard model of the continuum based on integers proposed by Harthong and Reeb [2,3]. We present some basis on constructive mathematics [4] and its link with programming [5,6]. We show that a suitable version of this new model of the continuum partly fits with the constructive axiomatic of R proposed by Bridges [7]. The aim of this paper is to take a first look at a possible formal and constructive approach to discrete geometry. This would open the way to better algorithmic definition of discrete differential concepts.

Arithmetization of a Circular Arc

“…Using these mappings, it can be shown that the digital curve C d (0, R) is isomorphic to the initial Euclidean circle C(0, R). However, in this article we are concentrating on the properties of the digital curve C d (0, R) (see [13,14] for more details on such a proof).…”

“…The second step consists in computing an upper bound for ||z n − z(t n )|| of equation (13). To do so, we consider ||z n+1 − z(t n+1 )||.…”

“…The approximation that appears in the process of drawing a discrete curve on a screen is not described anymore as a sampling error but as the result of a strongly contracting rescaling in the discrete world. For more details, readers may refer to [13,14].…”

“…At the last DGCI (and in an extended version in the Pattern Recognition journal) [13,14], we took a new look at the ideas that led to Reveillès' discrete straight line. In this paper we decided to reconsider Holin's work [15][16][17] on discrete circles generated by the preceding approach.…”

Ω-Arithmetization of Ellipses

“…We consider a new unit which is an infinitely large integer named ω = (ω n ) n∈N , which can be Ω himself. This scaling strongly contracts Z so that the result looks like R [7]. More formally, we defined the Harthong-Reeb with Ω-numbers as follows.…”