Hostname: page-component-5c6d5d7d68-wpx84 Total loading time: 0 Render date: 2024-08-14T14:56:30.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Random tessellations in ℝd

Published online by Cambridge University Press:  01 July 2016

J. Møller
J. Møller*
Affiliation:
Aarhus University
*
∗Postal address: Department of Theoretical Statistics, Institute of Mathematics, Aarhus University, DK 8000 Aarhus, Denmark.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents a general theory of random tessellations (i.e. stochastic aggregates of disjoint and space-filling cells) in d-dimensional Euclidean space. Some particular models of random tessellations are discussed in detail.

Keywords

DELAUNEY TESSELLATIONFACET CHARACTERISTICSFLAT SECTIONSMEAN VALUE RELATIONSMOSAICSPALM MEASURESSTEREOLOGYSTOCHASTIC GEOMETRYTESSELLATIONSTESSELLATION CHARACTERISTICSVORONOI TESSELLATION
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 1 , March 1989 , pp. 37 - 73
Copyright
Copyright © Applied Probability Trust 1989 

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

Ambartzumian, R. V. (1974) Convex polygons and random tessellations. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G.. Wiley, New York.Google Scholar
Cowan, R. (1978) The use of the ergodic theorems in random geometry. Suppl. Adv. Appl. Prob. 10, 47–57.CrossRefGoogle Scholar
Cowan, R. (1980) Properties of ergodic random mosaic processes. Math. Nachr. 97, 89–102.CrossRefGoogle Scholar
Crain, I. K. and Miles, R. E. (1976) Monte Carlo estimates of the distributions of the random polygons determined by random lines in the plane. J. Statist. Comput. Simul. 4, 293–325.CrossRefGoogle Scholar
Dirichlet, G. L. (1850) Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. J. Reine Angew. Math. 40, 209–227.Google Scholar
Gilbert, E. N. (1967) Random plane networks and needle-shaped crystals. Chapter 16 of Applications of Undergraduate Mathematics in Engineering by B. Noble. Macmillan, New York.Google Scholar
Green, P. J. and Sibson, R. (1977) Computing Dirichlet tessellations in the plane. The Computer Journal 21, 168–173.CrossRefGoogle Scholar
Grünbaum, B. (1967) Convex Polytopes. Wiley, New York.Google Scholar
Hadwiger, H. (1957) Volesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Hinde, A. L. and Miles, R. E. (1980) Monte Carlo estimates of the random polygons of the Voronoi tessellation with respect to a Poisson process. J. Statist. Comput. Simul. 10, 205–223.CrossRefGoogle Scholar
Matheron, G. (1975) Random Sets and Integral Geometry, Wiley, New York.Google Scholar
Mecke, J. (1973) Invarianzeigenschaften allgemeiner. Palmscher Masse. Math. Nachr. 65, 335–344.CrossRefGoogle Scholar
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.Google Scholar
Mecke, J. (1984) Parametric representation of mean values for stationary random mosaics. Math. Operationsforsch, u. Statist., ser. statist., 15, 437–442.Google Scholar
Meijering, J. L. (1953) Inferface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–290.Google Scholar
Miles, R. E. (1964) Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. USA 52, 901–907; 1157-1160.CrossRefGoogle ScholarPubMed
Miles, R. E. (1972a) Multidimensional perspectives on stereology. J. Microsc. 95, 181–195.CrossRefGoogle Scholar
Miles, R. E. (1972b) The random division of space. Suppl. Adv. Appl. Prob., 243–266.CrossRefGoogle Scholar
Miles, R. E. (1974) A synopsis of ‘Poisson flats in Euclidean spaces’. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G.. Wiley, New York.Google Scholar
Miles, R. E. (1984) Sectional Voronoi tessellations. Rev. Unión Math. Argentina 29, 310–327.Google Scholar
Møller, J. (1986) Random tessellations in ℝd. Memoirs No. 9, Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus.Google Scholar
Radecke, W. (1980) Some mean value relations on stationary random mosaics in the space. Math. Nachr. 97, 203–210.CrossRefGoogle Scholar
Rogers, C. A. (1964) Packing and Covering. Cambridge University Press, London.Google Scholar
Schneider, R. (1979) Integralgeometrie. (Lecture Notes, in German.) Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg.Google Scholar
Stoyan, D. (1982) Stereological formulae for size distributions via marked point processes. Prob. Math. Statist. 2, 161–166.Google Scholar
Stoyan, D. (1986) On generalized random tessellations. Math. Nachr. 128, 97–101.CrossRefGoogle Scholar
Stoyan, D. and Hermann, H. (1986) Some methods for statistical analysis of planar random tessellations. Math. Operationsforsch, u. Statist., ser. statist. 17, 407–420.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Akademie-Verlag, Berlin.Google Scholar
Voronoi, G. (1908) Nouvelles applications des paramètres continus à la théorie des formes quadratiques, Deuxième Mémoire, Recherches sur les paralléloèdres primitifs. J. Reine Angew. Math. 134, 198–287.CrossRefGoogle Scholar
Watson, D. F. (1981) Computing the n-dimensional Delaunay tessellations with application to Voronoi polytopes. The Computer Journal 24, 167–172.CrossRefGoogle Scholar
Weil, W. (1986) Point processes of cylinders, particles, and flats.CrossRefGoogle Scholar
Weiss, V. (1986) Relations between mean values for stationary random hyperplane mosaics of ℝd .Google Scholar
Weiss, V. and Zähle, M. (1986) Geometric measures for random curves mosaics of ℝd .Google Scholar