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Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson-Voronoi tessellation and a Poisson line process

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Pierre Calka
Pierre Calka*
Affiliation:
Université Claude Bernard Lyon 1
*
∗ Postal address: Université Claude Bernard Lyon 1, LaPCS, Bât. B, Domaine de Gerland, 50, avenue Tony-Garnier, F-69366 Lyon Cedex 07, France. Email address: [email protected]
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Abstract

In this paper, we give an explicit integral expression for the joint distribution of the number and the respective positions of the sides of the typical cell 𝒞 of a two-dimensional Poisson-Voronoi tessellation. We deduce from it precise formulae for the distributions of the principal geometric characteristics of 𝒞 (area, perimeter, area of the fundamental domain). We also adapt the method to the Crofton cell and the empirical (or typical) cell of a Poisson line process.

Keywords

Crofton cellempirical cellfundamental domainPalm distributionPoisson-Voronoi tessellationstochastic geometrytypical cell

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 551 - 562
Copyright
Copyright © Applied Probability Trust 2003 

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References

[1] Ambartzumian, R. V. (1976). A note on pseudo-metrics on the plane. Z. Wahrscheinlichkeitsth. 37, 145–155.CrossRefGoogle Scholar
[2] Ambartzumian, R. V. (1982). Combinatorial Integral Geometry, edited and with an appendix by Baddeley, A. John Wiley, New York.Google Scholar
[3] Baccelli, F. and Błaszczyszyn, B. (2001). On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation with applications to wireless communications. Adv. Appl. Prob. 33, 293–323.Google Scholar
[4] Calka, P. (2001). Mosaïques poissoniennes de l'espace euclidien. Une extension d'un résultat de R. E. Miles. C. R. Acad. Sci. Paris Sér. I Math. 332, 557–562.Google Scholar
[5] Calka, P. (2002). The explicit expression of the distribution of the number of sides of the typical Poisson–Voronoi cell. Preprint, 02-02 LaPCS, Université Claude Bernard Lyon 1.Google Scholar
[6] Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.CrossRefGoogle Scholar
[7] Goldman, A. (1996). Le spectre de certaines mosaïques poissoniennes du plan et l'enveloppe convexe du pont brownien. Prob. Theory Relat. Fields 105, 57–83.Google Scholar
[8] Goudsmit, S. (1945). Random distribution of lines in a plane. Rev. Modern Phys. 17, 321–322.Google Scholar
[9] Kumar, S. and Singh, R. N. (1995). Thermal conductivity of polycrystalline materials. J. Amer. Ceramics Soc. 78, 728–736.Google Scholar
[10] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
[11] Mecke, J. (1980). Palm methods for stationary random mosaics. In Combinatorial Principles in Stochastic Geometry, ed. Ambartzumian, R. V., Armenian Academy of Sciences Publishing House, Erevan, pp. 124–132.Google Scholar
[12] Mecke, J., Schneider, R., Stoyan, D. and Weil, W. (1990). Stochastische Geometrie. Birkhäuser, Basel.Google Scholar
[13] Meijering, J. L. (1953). Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–290.Google Scholar
[14] Miles, R. E. (1964). Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. USA 52, 901–907.Google Scholar
[15] Miles, R. E. (1964). Random polygons determined by random lines in a plane. II. Proc. Nat. Acad. Sci. USA 52, 1157–1160.Google Scholar
[16] Miles, R. E. (1973). The various aggregates of random polygons determined by random lines in a plane. Adv. Math. 10, 256–290.Google Scholar
[17] Möller, J., (1994). Lectures on Random Voronoi Tessellations. Springer, New York.CrossRefGoogle Scholar
[18] 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
[19] Pielou, E. (1977). Mathematical Ecology. Wiley-Interscience, New York.Google Scholar
[20] Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar
[21] Tanner, J. C. (1983). Polygons formed by random lines in a plane: some further results. J. Appl. Prob. 20, 778–787.Google Scholar
[22] Van de Weygaert, R. (1994). Fragmenting the Universe III. The construction and statistics of 3-D Voronoi tessellations. Astron. Astrophys. 283, 361–406.Google Scholar
[23] Zuyev, S. A. (1992). Estimates for distributions of the Voronoi polygon's geometric characteristics. Random Structures Algorithms 3, 149–162.Google Scholar