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We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. We may preprocess P to obtain a label and a routing table for each vertex of P . Then, we must be able to route a data packet between any two vertices p and q of P , where each step must use only the label of the target node q and the routing table of the current node.For any fixed ε > 0, we present a routing scheme that always achieves a routing path whose length exceeds the shortest path by a factor of at most 1 + ε. The labels have O(log n) bits, and the routing tables are of size O((ε −1 + h) log n). The preprocessing time is O(n 2 log n). It can be improved to O(n 2 ) for simple polygons.

We consider the problem of digitalizing Euclidean line segments from R d to Z d . Christ et al. (DCG, 2012) showed how to construct a set of consistent digital segment (CDS) for d = 2: a collection of segments connecting any two points in Z 2 that satisfies the natural extension of the Euclidean axioms to 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 consistent digital rays CDR in 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 et al..

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