Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T09:32:55.819Z Has data issue: false hasContentIssue false
  • English
  • Français

The Poisson-Voronoi tessellation: relationships for edges

Part of:Geometric probability and stochastic geometryGeneral convexity

Published online by Cambridge University Press:  01 July 2016

L. Muche
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Keywords

Probability density functionmomentedge lengthadjacent angledistanceneighbor

MSC classification

Primary:60D05: Geometric probability and stochastic geometry
Secondary:52A22: Random convex sets and integral geometry

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 2 , June 2005, pp. 279 - 296
Copyright
Copyright © Applied Probability Trust 2005 

References

Brakke, K. A. (1985). Statistics of random plane Voronoi tessellations. Preprint, Department of Mathematical Sciences, Susquehanna University Selinsgrove, PA.Google Scholar
Brakke, K. A. (1985). Statistics of three dimensional random Voronoi tessellations. Preprint, Department of Mathematical Sciences, Susquehanna University Selinsgrove, PA.Google Scholar
Collins, R. (1968). A geometric sum rule for two-dimensional fluid correlation functions. J. Phys. C 1, 1461–1471.Google Scholar
Collins, R. (1972). Melting and statistical geometry of simple liquids. In Phase Transitions and Critical Phenomena, eds Domb, C. and Green, M. S., Academic Press, New York, pp. 271–303.Google Scholar
Cowan, R., Quine, M. and Zuyev, S. (2003). Decomposition of gamma-distributed domains constructed from Poisson point processes. Adv. Appl. Prob. 35, 56–69.Google Scholar
Li, X.-Y., Wan, P.-J. and Frieder, O. (2003). Coverage in wireless ad-hoc sensor networks. IEEE Trans. Comput. 526, 753–763.Google Scholar
Liang, J. et al. (1998). Analytical shape computation of macromolecules. I. Molecular area and volume through alpha shape. Proteins Structure Function Genet. 33, 1–17.Google Scholar
Mecke, J. and Muche, L. (1995). The Poisson Voronoi tessellation. I. A basic identity. Math. Nachr. 176, 199–208.Google Scholar
Miles, R. E. (1970). On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127.Google Scholar
Miles, R. E. (1974). A synopsis of ‘Poisson flats in Euclidean spaces’. In Stochastic Geometry, eds Harding, E. F. and Kendall, D. G., John Wiley, New York, pp. 202–227.Google Scholar
Møller, J. (1989). Random tessellations in R d . Adv. Appl. Prob. 21, 37–73.Google Scholar
Møller, J. (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.Google Scholar
Muche, L. (1996). The Poisson–Voronoi tessellation. II. Edge length distribution functions. Math. Nachr. 178, 271–283.Google Scholar
Muche, L. (1999). Delaunay and Voronoi tessellation: Minkowski functionals and edge characteristics. Proc. S4G (Internat. Conf. Stereology, Spatial Statist. Stoch. Geom.; Prague, June 1999), eds Beneš, V., Janáček, J. and Saxl, I., Union of Czech Mathematicians and Physicists, Prague, pp. 21–30.Google Scholar
Muche, L. and Stoyan, D. (1992). Contact and chord length distributions of the Poisson Voronoi tessellation. J. Appl. Prob. 29, 467–471. (Correction: 30 (1993), 749.)Google Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.Google Scholar
Sastry, S. et al. (1998). Free volume in the hard sphere liquid. Molec. Phys. 952, 289–297.Google Scholar
Schlather, M. (2000). A formula for the edge length distribution function of the Poisson Voronoi tessellation. Math. Nachr. 214, 113–119.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar