Consistent digital line segments

TL;DR

This paper presents a new method for digitalizing line segments in the plane that ensures they closely approximate Euclidean segments, based on a novel approach using total orders on integers.

Contribution

It introduces a general framework for digital line segments satisfying natural Euclidean axioms, and constructs systems that are Hausdorff close to Euclidean segments.

Findings

Digital segments are optimally close to Euclidean segments in Hausdorff metric

A system of digital segments satisfying Euclidean axioms is explicitly constructed

The approach generalizes previous methods and resolves key open questions

Abstract

We introduce a novel and general approach for digitalization of line segments in the plane that satisfies a set of axioms naturally arising from Euclidean axioms. In particular, we show how to derive such a system of digital segments from any total order on the integers. As a consequence, using a well-chosen total order, we manage to define a system of digital segments such that all digital segments are, in Hausdorff metric, optimally close to their corresponding Euclidean segments, thus giving an explicit construction that resolves the main question of Chun et al.