Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T20:02:56.080Z Has data issue: false hasContentIssue false
  • English
  • Français

The proportion of triangles in a Poisson-Voronoi tessellation of the plane

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Andrew Hayen  and
Malcolm Quine
Andrew Hayen*
Affiliation:
University of Sydney
Malcolm Quine*
Affiliation:
University of Sydney
*
∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia.
∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

By using an adaptation of the radial generation method, we give an integral formula for the proportion of triangles in a Poisson-Voronoi tessellation, which gives a value of 0.0112354 to 7 decimal places. We also obtain the first four moments of some characteristics of triangles.

Keywords

PoissonVoronoitessellationPalm probabilitytrianglemoments

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 32 , Issue 1 , March 2000 , pp. 67 - 74
Copyright
Copyright © Applied Probability Trust 2000 

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] Berntsen, J., Espelid, T. O. and Genz, A. (1991a). An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Software 17, 437–451.Google Scholar
[2] Berntsen, J., Espelid, T. O. and Genz, A. (1991b). Algorithm 698: DCUHRE: An adaptive multidimensional integration routine for a vector of integrals. ACM Trans. Math. Software 17, 452–456.Google Scholar
[3] Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.Google Scholar
[4] Heinrich, L. and Schüle, E. (1995). Generation of the typical cell of a non-Poissonian Johnson–Mehl tesselation. Comm. Statist. Stochastic Models 11, 541–560.CrossRefGoogle Scholar
[5] Hinde, A. L. and Miles, R. E. (1980). Monte-Carlo estimates of the distributions of random polygons of the Voronoi tessellation with respect to a Poisson process. J. Statist. Comput. Simul. 10, 205–223.Google Scholar
[6] Meijering, J. L. (1953). Interface area, edge length and number of vertices in crystal aggregates with random nucleation. Philips Res. Rept. 8, 270–290.Google Scholar
[7] Miles, R. E. (1964). Random polygons determined by random lines in a plane. Proc. Natl. Acad. Sci. USA 52, 117–136.Google Scholar
[8] Miles, R. E. and Maillardet, R. J. (1982). The basic structures of Voronoi and generalized Voronoi polygons. In Essays in Statistical Science, eds. Gani, J. and Hannan, E. J. ({J. Appl. Prob.} 19A). Applied Probability Trust, Sheffield, pp. 97–111.Google Scholar
[9] Møller, J., (1994). Lectures on Random Voronoi Tessellations (Lecture Notes in Statist. 87). Springer, New York.Google Scholar
[10] Okabe, A., Boots, B. and Sugihara, K. (1992). Spatial Tessellations. Concepts and Applications of Voronoi Diagrams. John Wiley, Chichester.Google Scholar
[11] Quine, M. P. and Watson, D. F. (1984). Radial generation of n-dimensional Poisson processes. J. Appl. Prob. 21, 548–557.Google Scholar
[12] Stoyan, D. and Stoyan, H. (1994). Fractals, Random Shapes and Point Fields. John Wiley, Chichester.Google Scholar
[13] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications. 2nd edn. John Wiley, New York.Google Scholar
[14] Tanner, J. C. (1983). The proportion of quadrilaterals formed by random lines in a plane. J. Appl. Prob. 20, 400–404.Google Scholar