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Representation of Euclidean objects in a digital space has been a focus of research for over 30 years. Digital line segments are particularly important as other digital objects depend on their definition (e.g., digital convex objects or digital star-shaped objects). It may be desirable for the digital line segment systems to satisfy some nice properties that their Euclidean counterparts also satisfy. The system is a consistent digital line segment system (CDS) if it satisfies five properties, most notably the subsegment property (the intersection of any two digital line segments should be connected) and the prolongation property (any digital line segment should be able to be extended into a digital line). It is known that any CDS must have Ω(log n) Hausdorff distance to their Euclidean counterparts, where n is the number of grid points on a segment. In fact this lower bound even applies to consistent digital rays (CDR) where for a fixed p ∈ ℤ², we consider the digital segments from p to q for each q ∈ ℤ². In this paper, we consider families of weak consistent digital rays (WCDR) where we maintain four of the CDR properties but exclude the prolongation property. In this paper, we give a WCDR construction that has optimal Hausdorff distance to the exact constant. That is, we give a construction whose Hausdorff distance is 1.5 under the L_∞ metric, and we show that for every ε > 0, it is not possible to have a WCDR with Hausdorff distance at most 1.5 - ε. 

Visibility problems are fundamental to computational geometry, and many versions of geometric set cover where coverage is based on visibility have been considered. In most settings, points can see "infinitely far" so long as visibility is not "blocked" by some obstacle. In many applications, this may be an unreasonable assumption. In this paper, we consider a new model of visibility where no point can see any other point beyond a sight radius ρ. In particular, we consider this visibility model in the context of terrains. We show that the VC-dimension of limited visibility terrains is exactly 7. We give lower bound construction that shatters a set of 7 points, and we prove that shattering 8 points is not possible. 

We give polynomial-time algorithms that solve the pseudo-polygon visibility graph recognition and reconstruction problems. Our algorithms are based on a new characterization of the visibility graphs of pseudo-polygons. 

In this paper, we consider the Visibility Graph Recognition and Reconstruction problems in the context of terrains. Here, we are given a graph G with labeled vertices v₀, v₁, …, v_{n-1} such that the labeling corresponds with a Hamiltonian path H. G also may contain other edges. We are interested in determining if there is a terrain T with vertices p₀, p₁, …, p_{n-1} such that G is the visibility graph of T and the boundary of T corresponds with H. G is said to be persistent if and only if it satisfies the so-called X-property and Bar-property. It is known that every "pseudo-terrain" has a persistent visibility graph and that every persistent graph is the visibility graph for some pseudo-terrain. The connection is not as clear for (geometric) terrains. It is known that the visibility graph of any terrain T is persistent, but it has been unclear whether every persistent graph G has a terrain T such that G is the visibility graph of T. There actually have been several papers that claim this to be the case (although no formal proof has ever been published), and recent works made steps towards building a terrain reconstruction algorithm for any persistent graph. In this paper, we show that there exists a persistent graph G that is not the visibility graph for any terrain T. This means persistence is not enough by itself to characterize the visibility graphs of terrains, and implies that pseudo-terrains are not stretchable.