A Formula for the Edge Length Distribution Function of the Poisson Voronoi Tessellation

2000 ◽  
Vol 214 (1) ◽  
pp. 113-119 ◽  
Author(s):  
Martin Schlather
Keyword(s):  
Distribution Function ◽  
Voronoi Tessellation ◽  
Length Distribution ◽  
Edge Length

scholarly journals Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

2010 ◽  
Vol 42 (1) ◽  
pp. 48-68 ◽  
Author(s):  
L. Muche
Keyword(s):  
Voronoi Tessellation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Dimensional Euclidean Space ◽  
Spherical Contact ◽  
Chord Length Distribution ◽  
Special Cases ◽  
Contact Distribution ◽  
Contact Distribution Function

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

scholarly journals Volume degeneracy of the typical cell and the chord length distribution for Poisson-Voronoi tessellations in high dimensions

2008 ◽  
Vol 40 (04) ◽  
pp. 919-938 ◽  
Author(s):  
Kasra Alishahi ◽  
Mohsen Sharifitabar
Keyword(s):  
Space Dimension ◽  
Voronoi Tessellation ◽  
Length Distribution ◽  
Chord Length ◽  
Voronoi Tessellations ◽  
Constant Intensity ◽  
Chord Length Distribution ◽  
Typical Cell ◽  
Centered Ball ◽  
Contact Distribution

This paper is devoted to the study of some asymptotic behaviors of Poisson-Voronoi tessellation in the Euclidean space as the space dimension tends to ∞. We consider a family of homogeneous Poisson-Voronoi tessellations with constant intensity λ in Euclidean spaces of dimensions n = 1, 2, 3, …. First we use the Blaschke-Petkantschin formula to prove that the variance of the volume of the typical cell tends to 0 exponentially in dimension. It is also shown that the volume of intersection of the typical cell with the co-centered ball of volume u converges in distribution to the constant λ−1(1 − e−λu ). Next we consider the linear contact distribution function of the Poisson-Voronoi tessellation and compute the limit when the space dimension goes to ∞. As a by-product, the chord length distribution and the geometric covariogram of the typical cell are obtained in the limit.

scholarly journals The chord-length distribution of a polyhedron

2020 ◽  
Vol 76 (4) ◽  
pp. 474-488
Author(s):  
Salvino Ciccariello
Keyword(s):  
Distribution Function ◽  
Length Distribution ◽  
Chord Length ◽  
Square Root ◽  
Trigonometric Functions ◽  
Chord Length Distribution ◽  
Analytic Expressions ◽  
Analytic Expression ◽  
Positive Integers ◽  
Inverse Trigonometric Functions

The chord-length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

Contact and chord length distribution of a stationary Voronoi tessellation

1998 ◽  
Vol 30 (03) ◽  
pp. 603-618 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Point Process ◽  
Voronoi Tessellation ◽  
Pair Correlation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Closed Form Expression ◽  
Voronoi Tessellations ◽  
Chord Length Distribution ◽  
Contact Distribution

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝ d in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

Contact and chord length distribution of a stationary Voronoi tessellation

1998 ◽  
Vol 30 (3) ◽  
pp. 603-618 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Point Process ◽  
Voronoi Tessellation ◽  
Pair Correlation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Closed Form Expression ◽  
Voronoi Tessellations ◽  
Chord Length Distribution ◽  
Contact Distribution

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝd in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell

2003 ◽  
Vol 35 (4) ◽  
pp. 863-870 ◽  
Author(s):  
Pierre Calka
Keyword(s):  
Distribution Function ◽  
Explicit Expression ◽  
Voronoi Tessellation ◽  
Voronoi Cell ◽  
Two Dimensional ◽  
Typical Cell

In this paper, we give an explicit expression for the distribution of the number of sides (or equivalently vertices) of the typical cell of a two-dimensional Poisson-Voronoi tessellation. We use this formula to give a table of numerical values of the distribution function.

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