High Dimensional Consistent Digital Segments

TL;DR

This paper extends the concept of consistent digital segments from 2D to higher dimensions, characterizing conditions for their construction and analyzing their geometric properties.

Contribution

It introduces a method to construct consistent digital rays in higher dimensions based on total orders and characterizes when this construction is valid.

Findings

Any total order can generate consistent digital rays in $\

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,

Abstract

We consider the problem of digitalizing Euclidean line segments from $\mathbb{R}^d$ to $\mathbb{Z}^d$. Christ {\em et al.} (DCG, 2012) showed how to construct a set of {\em consistent digital segment} (CDS) for $d=2$: a collection of segments connecting any two points in $\mathbb{Z}^2$ that satisfies the natural extension of the Euclidean axioms to $\mathbb{Z}^d$. In this paper we study the construction of CDSs in higher dimensions. We show that any total order can be used to create a set of {\em consistent digital rays} CDR in $\mathbb{Z}^d$ (a set of rays emanating from a fixed point $p$ that satisfies the extension of the Euclidean axioms). We fully characterize for which total orders the construction holds and study their Hausdorff distance, which in particular positively answers the question posed by Christ {\em et al.}.