  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.
