Further results on the hyperbolic Voronoi diagrams

In Euclidean geometry, it is well-known that the $k$-order Voronoi diagram in $\mathbb{R}^d$ can be computed from the vertical projection of the $k$-level of an arrangement of hyperplanes tangent to a convex potential function in $\mathbb{R}^{d+1}$: the paraboloid. Similarly, we report for the Klein ball model of hyperbolic geometry such a {\em concave} potential function: the northern hemisphere. Furthermore, we also show how to build the hyperbolic $k$-order diagrams as equivalent clipped power diagrams in $\mathbb{R}^d$. We investigate the hyperbolic Voronoi diagram in the hyperboloid model and show how it reduces to a Klein-type model using central projections.Comment: 6 pages, 2 figures (ISVD 2014

Visualizing hyperbolic Voronoi diagrams

We present an interactive software, HVD, that represents in-ternally the k-order hyperbolic Voronoi diagram of a finite set of sites as an equivalent clipped power diagram. HVD allows users to interactively browse the hyperbolic Voronoi diagrams and renders simultaneously the diagram in the five standard models of hyperbolic geometry: Namely, the Poincare ́ disk, the Poincare ́ upper plane, the Klein disk, the Beltrami hemisphere and the Weierstrass hyperboloid. 1

Voronoi Diagrams in the Hilbert Metric

The Hilbert metric is a distance function defined for points lying within a convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to any convex set, and it has numerous applications in the analysis and processing of convex bodies. In this paper, we study the geometric and combinatorial properties of the Voronoi diagram of a set of point sites under the Hilbert metric. Given any m-sided convex polygon ? in the plane, we present two randomized incremental algorithms and one deterministic algorithm. The first randomized algorithm and the deterministic algorithm compute the Voronoi diagram of a set of n point sites. The second randomized algorithm extends this to compute the Voronoi diagram of the set of n sites, each of which may be a point or a line segment. Our algorithms all run in expected time O(m n log n). The algorithms use O(m n) storage, which matches the worst-case combinatorial complexity of the Voronoi diagram in the Hilbert metric

Largest Empty Circle Centered on a Query Line

The Largest Empty Circle problem seeks the largest circle centered within the convex hull of a set $P$ of $n$ points in $\mathbb{R}^2$ and devoid of points from $P$. In this paper, we introduce a query version of this well-studied problem. In our query version, we are required to preprocess $P$ so that when given a query line $Q$, we can quickly compute the largest empty circle centered at some point on $Q$ and within the convex hull of $P$. We present solutions for two special cases and the general case; all our queries run in $O(\log n)$ time. We restrict the query line to be horizontal in the first special case, which we preprocess in $O(n \alpha(n) \log n)$ time and space, where $\alpha(n)$ is the slow growing inverse of the Ackermann's function. When the query line is restricted to pass through a fixed point, the second special case, our preprocessing takes $O(n \alpha(n)^{O(\alpha(n))} \log n)$ time and space. We use insights from the two special cases to solve the general version of the problem with preprocessing time and space in $O(n^3 \log n)$ and $O(n^3)$ respectively.Comment: 18 pages, 13 figure