Key words Voronoi tessellation, stationary Poisson process, point process of nodes, second moment measure, pair correlation function, asymptotic variance, central limit theorem, vertices of the typical cell MSC (2000) Primary: 60D05, 60G55; Secondary: 62F12, 60G60In this paper we derive representation formulae for the second factorial moment measure of the point process of nodes and the second moment of the number of vertices of the typical cell associated with a stationary normal Voronoi tessellation in R d . In case the Voronoi tessellation is generated by a stationary Poisson process with intensity λ > 0 the corresponding pair correlation function g V,λ (r) can be expressed by a weighted sum of d+2 (numerically tractable) multiple parameter integrals. The asymptotic variance of the number of nodes in an increasing cubic domain as well as the second moment of the number of vertices of the typical Poisson Voronoi cell are calculated exactly by means of these parameter integrals. The existence of a (d − 1)st-order pole of g V,λ (r) at r = 0 is proved and the exact value of limr→0 r d−1 g V,λ (r) is determined. In the particular cases d = 2 and d = 3 the graph of gV,1(r) including its local extreme points, the points of level 1 of gV,1(r) and other characteristics are computed by numerical integration. Furthermore, an asymptotically exact confidence interval for the intensity of nodes is obtained.A tessellation of the d-dimensional Euclidean space R d is a subdivision of this space into countably many compact sets {C i , i ∈ N} (called cells or crystals), where two such cells have no common interior points and the number of cells intersecting any bounded subset of R d is finite. Usually the cells are assumed to be convex so that each C i becomes a convex d-polytope. In this case the boundary ∂C i consists of s-polytopes with s ∈ {0, 1, . . . , d−1} called s-faces in what follows. As usual 0-faces, 1-faces, and (d − 1)-faces are called vertices, edges, and facets, respectively. The union of all vertices forms the countable set of nodes of the tessellation.Many real-life tessellations are random, see e.g. Stoyan et al. (1995), Okabe et al. (2000. There are various stochastic-geometric mechanisms to generate a random tessellation {C i , i ∈ N} which can be defined over a hypothetical probability space [Ω, A, P] such that the above properties are satisfied P-almost surely. The random tessellation is said to be stationary if its distribution is invariant under translation of the cells.Throughout the present paper we are concerned with stationary Voronoi (or Dirichlet, or Thiessen) tessellations (briefly VT's) generated by stationary point processes. It should be mentioned that some authors use the term VT merely relative to a homogeneous Poisson point process (henceforth such a VT will be called Poisson Voronoi tessellation (briefly PVT)). A mathematically rigorous and self-contained presentation of the basic material (mean-value relationships, PVT's, stochastic-and integral-geometric tools) on random VT's can be found...
This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.
In a unified approach, this paper presents distributional properties of a Voronoi tessellation generated by a homogeneous Poisson point process in the Euclidean space of arbitrary dimension. Probability density functions and moments are given for characteristics of the ‘typical’ edge in lower-dimensional section hyperplanes (edge lengths, adjacent angles). We investigate relationships between edges and their neighbours, called Poisson points or centres; namely angular distributions, distances, and positions of neighbours relative to the edge. The approach is analytical, and the results are given partly explicitly and partly as integral expressions, which are suitable for the numerical calculations also presented.
This paper gives basic relations between the stationary Poisson point process and the point process of vertices of the corresponding Voronoi tessellation in Rd and of planar sections through it. The results are based on a study of the Palm distribution of the point process of vertices. An identity is given connecting the distribution of a Poisson point process and the Palm distribution with respect to the vertices of the corresponding Voronoi tessellation. Distributional properties for the edges are discussed. Finally, identities are given for characteristics of the "typical" edge and an edge chosen at random emanating from the "typical" vertex.
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