Digitalizing line segments

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 [J. Chun, M. Korman, M. Nöllenburg, and T. Tokuyama. Consistent digital rays. Discrete Comput. Geom., 42(3):359–378, 2009].