Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T06:57:19.747Z Has data issue: false hasContentIssue false
  • English
  • Français

Random Laguerre tessellations

Part of: Multivariate analysis General convexity Stochastic processes Designs and configurations Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Claudia Lautensack  and
Sergei Zuyev
Claudia Lautensack*
Affiliation:
University of Applied Sciences, Darmstadt
Sergei Zuyev*
Affiliation:
University of Strathclyde
*
∗ Postal address: Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany. Email address: claudia.lautensack@itwm.fraunhofer.de
∗∗ Postal address: Department of Statistics and Modelling Science, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK. Email address: sergei@stams.strath.ac.uk
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.

A systematic study of random Laguerre tessellations, weighted generalisations of the well-known Voronoi tessellations, is presented. We prove that every normal tessellation with convex cells in dimension three and higher is a Laguerre tessellation. Tessellations generated by stationary marked Poisson processes are then studied in detail. For these tessellations, we obtain integral formulae for geometric characteristics and densities of the typical k-faces. We present a formula for the linear contact distribution function and prove various limit results for convergence of Laguerre to Poisson-Voronoi tessellations. The obtained integral formulae are subsequently evaluated numerically for the planar case, demonstrating their applicability for practical purposes.

Keywords

Laguerre tessellationpower tessellationweighted Voronoi tessellationk-face densityPoisson processrandom tessellationstochastic geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 05B45: Tessellation and tiling problems 52A22: Random convex sets and integral geometry 62H11: Directional data; spatial statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 630 - 650
Copyright
Copyright © Applied Probability Trust 2008 

References

Ash, P. and Bolker, E. (1986). Generalized Dirichlet tessellations. Geometriae Dedicata 20, 230–243.Google Scholar
Aurenhammer, F. (1987). A criterion for the affine equivalence of cell complexes in R d and convex polyhedra in R d+1 . Discrete Computational Geometry 2, 49–64.Google Scholar
Aurenhammer, F. (1987). Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16, 78–96.Google Scholar
Barndorff-Nielsen, O., Kendall, W. and van Lieshout, M. (1999). Stochastic Geometry Likelihood and Computation. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
Baumstark, V. and Last, G. (2007). Some distributional results for Poisson–Voronoi tessellations. Adv. Appl. Prob. 39, 16–40.Google Scholar
Blaschke, W. (1929). Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Springer, Berlin.Google Scholar
Boissonnat, J.-D. and Yvinec, M. (1998). Algorithmic Geometry. Cambridge University Press.Google Scholar
Coxeter, H. S. M. (1969). Introduction to Geometry. John Wiley, New York.Google Scholar
Fischer, W., Koch, E. and Hellner, E. (1971). Zur Berechnung von Wirkungsbereichen in Strukturen anorganischer Verbindungen. Neues Jahrbuch für Mineralogie Monatshefte 1971, 227–237.Google Scholar
Heinrich, L. (1998). Contact and chord length distribution of a stationary Voronoi tessellation. Adv. Appl. Prob. 30, 603–618.CrossRefGoogle Scholar
Imai, H., Iri, M. and Murota, K. (1985). Voronoi diagram in the Laguerre geometry and its applications. SIAM J. Comput. 14, 93–105.Google Scholar
Lautensack, C. (2007). Random Laguerre tessellations. Lautensack, Weiler bei Bingen. Available at http://www.itwm.fhg.de/bv/theses/works/lautensack.Google Scholar
Mecke, J. (1984). Parametric representation of mean values for stationary random mosaics. Math. Operat. Statist. Ser. Statist. 3, 437–442.Google Scholar
Miles, R. E. (1971). Isotropic random simplices. Adv. Appl. Prob. 3, 353–382.CrossRefGoogle Scholar
Molchanov, I. (2005). Theory of Random Sets. Springer, Berlin.Google Scholar
Møller, J. (1989). Random tessellations in R d . Adv. Appl. Prob. 21, 37–73.Google Scholar
Møller, J. (1992). Random Johnson–Mehl tessellations. Adv. Appl. Prob. 24, 814–844.Google Scholar
Møller, J. (1994). Lectures on Random Voronoi Tessellations (Lecture Notes Statist. 87). Springer, New York.Google Scholar
Muche, L. and Stoyan, D. (1992). Contact and chord length distributions of the Poisson–Voronoi tessellation. J. Appl. Prob. 29, 467–471.CrossRefGoogle 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
Schlottmann, M. (1993). Periodic and quasi-periodic Laguerre tilings. Internat. J. Modern Phys. B 7, 1351–1363.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (1992). Integralgeometrie. Teubner, Stuttgart.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Leipzig.Google Scholar