  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.
