Jump to ContentJump to Main Navigation
New Perspectives in Stochastic Geometry$
Users without a subscription are not able to see the full content.

Wilfrid S. Kendall and Ilya Molchanov

Print publication date: 2009

Print ISBN-13: 9780199232574

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199232574.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2020. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 05 June 2020

Tessellations

Tessellations

Chapter:
(p.145) 5 Tessellations
Source:
New Perspectives in Stochastic Geometry
Author(s):

Pierre Calka

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199232574.003.0005

Random tessellations and cellular structures occur in many domains of application, such as astrophysics, ecology, telecommunications, biochemistry and naturally cellular biology (see Stoyan, Kendall and Mecke 1987 or Okabe, Boots, Sugihara and Chiu 2000 for complete surveys). The theoretical study of these objects was initiated in the second half of the twentieth century by D. G. Kendall, J. L. Meijering, E. N. Gilbert and R. E. Miles, notably. Two isotropic and stationary models have emerged as the most basic and useful: the Poisson hyperplane tessellation and the Poisson–Voronoi tessellation. Since then, a large majority of questions raised about random tessellations have concerned statistics of the population of cells (‘how many cells are triangles in the plane?’, ‘how many cells have a volume greater than one?’) or properties of a specific cell (typically the one containing the origin). Two types of results are presented below: exact distributional calculations and asymptotic estimations. In the first part, we describe the two basic constructions of random tessellations (i.e. by throwing random hyperplanes or by constructing Voronoi cells around random nuclei) and we introduce the fundamental notion of typical cell of a stationary tessellation. The second part is devoted to the presentation of exact distributional results on basic geometrical characteristics (number of hyperfaces, typical k‐face, etc.). The following part concerns asymptotic properties of the cells. It concentrates in particular on the well‐known D. G. Kendall conjecture which states that large planar cells in a Poisson line tessellation are close to the circular shape. In the last part, we present some recent models of iterated tessellations which appear naturally in applied fields (study of crack structures, telecommunications). Intentionally, this chapter does not contain an exhaustive presentation of all the models of random tessellations existing in the literature (in particular, dynamical constructions such as Johnson‐Mehl tessellations will be omitted). The aim of the text below is to provide a selective view of recent selected methods and results on a few specific models.

Keywords:   random tessellations, cellular structures, astrophysics, ecology, telecommunications, biochemistry, cellular biology, Poisson hyperplane tessellation, Poisson–Voronoi tessellation, distributional calculations, asymptotic estimations

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .