High Dimensional Consistent Digital Segments
Man-Kwun Chiu, Matias Korman · SIAM Journal on Discrete Mathematics · 2018
We consider the problem of digitalizing Euclidean line segments from $\mathbb{R}^d$ to $\mathbb{Z}^d$. Christ, Pálvölgyi, and Stojaković, [ Discrete Comput. Geom., 47 (2012), pp. 691--710] showed how to construct a set of consistent digital segments (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 extend some of their results to higher dimensions. Specifically we show that any total order can be used to create a set of 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). Then we use the same approach to create a CDS in $\mathbb{Z}^d$ and observe that it only works in some cases. We fully characterize for which total orders the construction is consistent (and thus gives a CDS). In particular, this positively answers the question posed by Christ and co-authors.