高维一致数字段
High Dimensional Consistent Digital Segments
摘要：
我们考虑从$\mathbb{R}^d$到$\mathbb{Z}^d$的欧氏线段的数字化问题。基督{ \em等} (DCG，2012）示出了如何为$d=2$构建{ \em一致数字段}（CDS）的集合：连接$\mathbb{Z}^2$中的任意两个点的段的集合，其满足欧几里德公理到$\mathbb{Z}^d$的自然扩展。在本文中，我们研究了在更高的维度的CDS的建设。我们表明，任何总的顺序可以用来创建一组{ \em一致的数字射线} CDR在$\mathbb{Z}^d$（一组射线从一个固定的点$p$，满足扩展的欧几里得公理）。我们充分刻画的总订单的建设持有和研究他们的豪斯多夫距离，特别是积极回答了基督{ \em等人}提出的问题。
英文摘要：
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.}.
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