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

The oscillating behavior of the pair correlation function in galaxies

The pair correlation function (PCF) for galaxies presents typical oscillations in the range 20-200 Mpc/h which are named baryon acoustic oscillation (BAO). We first review and test the oscillations of the PCF when the 2D/3D vertexes of the Poissonian Voronoi Tessellation (PVT) are considered. We then model the behavior of the PCF at a small scale in the presence of an auto gravitating medium having a line/plane of symmetry in 2D/3D. The analysis of the PCF in an astrophysical context was split into two, adopting a non-Poissonian Voronoi Tessellation (NPVT). We first analyzed the case of a 2D cut which covers few voids and a 2D cut which covers approximately 50 voids. The obtained PCF in the case of many voids was then discussed in comparison to the bootstrap predictions for a PVT process and the observed PCF for an astronomical catalog. An approximated formula which connects the averaged radius of the cosmic voids to the first minimum of the PCF is given.Comment: 19 pages 14 figure