  We consider the problem of digitalizing Euclidean line segments from
$\mathbb{R}^d$ to $\mathbb{Z}^d$. Christ {\em et al.} (DCG, 2012) showed how to
construct a set of {\em consistent digital segment} (CDS) for $d=2$: a
collection of segments connecting any two points in $\mathbb{Z}^2$ that
satisfies the natural extension of the Euclidean axioms to $\mathbb{Z}^d$. In
this paper we study the construction of CDSs in higher dimensions.
  We show that any total order can be used to create a set of {\em consistent
digital rays} CDR in $\mathbb{Z}^d$ (a set of rays emanating from a fixed point
$p$ that satisfies the extension of the Euclidean axioms). We fully
characterize for which total orders the construction holds and study their
Hausdorff distance, which in particular positively answers the question posed
by Christ {\em et al.}.
