A characterization of consistent digital line segments in ℤ2

Abstract

Our concern is the digitalization of line segments in ℤ2 as considered by Chun et al. [5] and Christ et al. [4]. 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 ℤd 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 ℤ2 with optimal Hausdorff distance. Christ et al. also give a characterization of CDRs in ℤ2, and they leave open the question on how to characterize CDSes in ℤ2. In this paper, we answer one of the most important open question regarding CDSes in ℤ2 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.