A Characterization of Consistent Digital Line Segments in Two Dimensions

TL;DR

This paper characterizes the conditions that define consistent digital line segments in two dimensions, answering a key open question and providing a complete understanding of their structure.

Contribution

It provides the first necessary and sufficient conditions to characterize two-dimensional consistent digital line segments.

Findings

Established necessary and sufficient conditions for CDS in 2D.

Resolved the open problem posed by Christ et al. in 2012.

Enhanced understanding of digital line segment systems.

Abstract

Our concern is the digitalization of line segments in two dimensions as considered by Chun et al.[Discrete Comput. Geom., 2009] and Christ et al.[Discrete Comput. Geom., 2012]. The key property that differentiates the research of Chun et al. and Christ et al. from other research in digital line segment construction is that the intersection of any two segments must be connected. Such a system of segments is called a consistent digital line segments system (CDS). Chun et al. give a construction for all segments in higher dimensions that share a common endpoint (called consistent digital rays (CDR)) that has asymptotically optimal Hausdorff distance, and Christ et al. give a complete CDS in two dimensions with optimal Hausdorff distance. Christ et al. also give a characterization of CDRs in two dimensions, and they leave open the question on how to characterize CDSes in two dimensions. In this paper, we answer the most important open question regarding CDSes in two dimensions by giving the characterization asked for by Christ et al. We obtain the characterization by giving a set of necessary and sufficient conditions that a CDS must satisfy.