The sectional Poisson Voronoi tessellation is not a Voronoi tessellation

Advances in Applied Probability

Volumn 28, Issue 2, 1996, Pages 356-376

  Chiu, S N ^a,c   Van De Weygaert, R ^b,d   Stoyan, D ^a  

^a TU BERGAKADEMIE FREIBERG   (Germany)

^b Sterrewacht Leiden Max Planck I

^c HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

^d UNIVERSITY OF GRONINGEN   (Netherlands)

Author keywords

Poisson process; Sectional Voronoi tessellation; Stochastic geometry; Voronoi tessellation

Indexed keywords

EID: 0002548910     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0001867800048515     Document Type: Article

References (27)

1. ASH, P. F. AND BOLKER, E. D. (1986) Generalized Dirichlet tessellations. Geom. Dedicat. 20, 209-243.
2. AURENHAMMER, F. (1987) Power diagrams: properties, algorithms and applications. SIAM J. Comput. 16, 78-96.
3. COWAN, R. (1984) A model for random packing of disks in the neighbourhood of one disk. SIAM J. Appl. Math. 44, 839-853.
4. GOLDWIRTH, D. S., DA COSTA, L. N. AND VAN DE WEYGAERT, R. (1994) The two-point correlation function and the size of voids. Month. Not. R. Astron. Soc., submitted.
5. IMAI, H., IRI, M. AND MUROTA, K. (1985) Voronoi diagram in the Laguerre geometry and its applications. SIAM J. Comput. 14, 93-105.
6. LORZ, U. (1991) Distribution of cell characteristics of the spatial Poisson-Voronoi tessellation and plane sections. In Geometrical Problems of Image. Ser. Research in Informatics, 4. ed. U. Eckhardt, A. Hübler, W. Nagel and G. Werner. pp. 171-178. Akademie, Berlin.
7. MECKE, J. (1984) Parametric representation of mean values for stationary random mosaics. Math. Operat. Statist. 15, 437-442.
8. MECKE, J. AND MUCHE, L. (1995) The Poisson Voronoi tessellation. I. A basic identity. Math. Nachr., to appear.
9. MILES, R. E. (1974) A synopsis of 'Poisson flats in euclidean spaces'. In Stochastic Geometry, ed. E. F. Harding and D. G. Kendall. pp. 202-227. Wiley, New York.
10. MILES, R. E. (1984) Sectional Voronoi tessellations. Rev. Union Math. Argentina 29, 310-327.
11. d. Adv. Appl. Prob. 21, 37-73.
12. MØLLER, J. (1994) Lectures on Random Voronoi Tessellations. (Lecture Notes in Statistics 87) Springer, New York.
13. OKABE, A., BOOTS, B. N. AND SUGIHARA, K. (1992) Spatial Tessellations. Concepts and Applications of Voronoi Diagrams. Wiley, New York.
14. PREPARATA, F. P. AND SHAMOS, M. I. (1985). Computational Geometry. An Introduction. Springer, New York.
15. RIVIER, N. (1993) Order and disorder in packing and froths. In Disorder and Granular Media. ed. D. Bideau and A. Hansen, pp. 55-102. Elsevier, Amsterdam.
16. SHAMOS, M. I. (1978) Computational Geometry. Ph.D. Thesis. Yale University.
17. SHAMOS, M. I. AND HOEY, D. (1975) Closest-point problems. Proc. 16th Ann. IEEE Symposium on Foundations of Computer Science. Berkeley, CA. pp. 151-162.
18. STOYAN, D., KENDALL, W. S. AND MECKE, J. (1987) Stochastic Geometry and Its Applications. Wiley, Chichester.
19. STOYAN, D. AND STOYAN, H. (1980) Gedanken zur Entstehung der Säulenformen bei Basalten. Z. Geol. Wissensch. 8, 1529-1537.
20. TELLEY, H. (1989) Modélisation er Simulation Bidimensionelle de la Croissance des Polycristaux. Ph.D. Thesis, Ecole Polytechnique Fédérale de Lausanne.
21. TELLEY, H., LIEBLING, TH. AND MOCELLIN, A. (1993) The Laguerre model of grain growth. Technical Report RO930124. Ecole Polytechnique Fédérale de Lausanne.
22. VAN DE WEYGAERT, R. (1991a) Voids and the Geometry of Large Scale Structure. Ph.D. Thesis, Leiden University.
23. VAN DE WEYGAERT, R. (1991b) Quasi-periodicity in deep redshift surveys. Month. Not. R. Astron. Soc. 249, 159-163.
24. VAN DE WEYGAERT, R. (1994) Fragmenting the Universe. III. The construction and statistics of 3-D Voronoi tessellation. Astron. Astrophys. 283, 361-406.
25. VORONOI, G. F. (1908) Nouvelles applications des paramètres continus à la théorie des formes quadratiques, Deuxième Mémoire: Recherches sur les paralléloèdres primitifis. J. Reine Angew. Math. 134, 198-287.
26. XUE, X., TELLEY, H. AND LIEBLING, TH. (1993) Polycrystal grain growth 3-D simulation by Laguerre diagrams: equation of movement. Technical Report RO931206. Ecole Polytechnique Fédérale de Lausanne.
27. XUE, X., TELLEY, H. AND LIEBLING, TH. (1994) Polycrystal grain growth 3-D simulation by Laguerre diagrams: simulation result. Technical Report RO940222. Ecole Polytechnique Fédérale de Lausanne.