Related papers: On the Structure of Higher Order Voronoi Cells
We study Voronoi diagrams for distance functions that add together two convex functions, each taking as its argument the difference between Cartesian coordinates of two planar points. When the functions do not grow too quickly, then the…
Multicellular rosettes are observed in different situations such as morphogenesis, wound healing, and cancer progression. While some molecular insights have been gained to explain the presence of these assemblies of five or more cells…
We present a new particle-merging algorithm for the particle-in-cell method. Based on the concept of the Voronoi diagram, the algorithm partitions the phase space into smaller subsets, which consist of only particles that are in close…
Since the Voronoi diagram appears in many applications, the topic of improving its computational efficiency remains attractive. We propose a novel yet efficient method to compute Voronoi diagrams bounded by a given domain, i.e., the clipped…
We investigate the statistical properties of two dimensional random cellular systems (froths) in term of their shell structure. The froth is analyzed as a system of concentric layers of cells around a given central cell. We derive exact…
We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein's non-conformal disk model are hyperplanes that can be interpreted as power bisectors of…
The intersections of beta-Voronoi, beta-prime-Voronoi and Gaussian-Voronoi tessellations in $\mathbb{R}^d$ with $\ell$-dimensional affine subspaces, $1\leq \ell\leq d-1$, are shown to be random tessellations of the same type but with…
This paper introduces nucleus clustering in Voronoi tessellations of plane surfaces with applications in the geometry of digital images. A \emph{nucleus cluster} is a collection of Voronoi regions that are adjacent to a Voronoi region…
We introduce a dynamical system based on the vertices of Voronoi tessellations. This dynamical system acts on finite or discrete point sets in the plane, taking a point set to the vertex set of its Voronoi tessellation. We explore the…
In various types of many-particle systems, bidispersity is frequently used to avoid spontaneous ordering in particle configuration. In this study, the relation between bidispersity and disorder degree of particle configuration is…
We study the geometry and complexity of Voronoi cells of lattices with respect to arbitrary norms. On the positive side, we show for strictly convex and smooth norms that the geometry of Voronoi cells of lattices in any dimension is similar…
In this note, a simple description of zone theorem in three dimensions is given.Arrangements in three dimensions are useful for constructing higher-order Voronoi diagrams in plane. An elementary and very intuitive treatment of this result…
Since the time of Minkowski a basic problem in number theory has been to find lower bounds for the absolute value $\Delta(K)$ of the discriminant of a number field $K$ in terms of the degree $n(K)$ of $K$. In this paper we study another…
We analyze the structure of two dimensional disordered cellular systems generated by extensive computer simulations. These cellular structures are studied as topological trees rooted on a central cell or as closed shells arranged…
The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define…
We address the problem of replicating a Voronoi diagram $V(S)$ of a planar point set $S$ by making proximity queries, which are of three possible (in decreasing order of information content): 1. the exact location of the nearest site(s) in…
Given a network, the statistical ensemble of its graph-Voronoi diagrams with randomly chosen cell centers exhibits properties convertible into information on the network's large scale structures. We define a node-pair level measure called…
An important open question in the modeling of biological tissues is how to identify the right scale for coarse-graining, or equivalently, the right number of degrees of freedom. For confluent biological tissues, both vertex and Voronoi…
$n$ independent random points drawn from a density $f$ in $R^d$ define a random Voronoi partition. We study the measure of a typical cell of the partition. We prove that the asymptotic distribution of the probability measure of the cell…
Given a set S of n points in general position, we consider all k-th order Voronoi diagrams on S, for k=1,...,n, simultaneously. We deduce symmetry relations for the number of faces, number of vertices and number of circles of certain…