Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-08T21:34:51.881Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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