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The proportion of triangles in a class of anisotropic Poisson line tessellations

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  07 August 2023

Nils Heerten ,
Julia Krecklenberg  and
Christoph Thäle
Nils Heerten*
Affiliation:
Ruhr University Bochum
Julia Krecklenberg*
Affiliation:
Ruhr University Bochum
Christoph Thäle*
Affiliation:
Ruhr University Bochum
*
*Postal address: Ruhr University Bochum, Universitätsstraße 150, 44801 Bochum, Germany.
*Postal address: Ruhr University Bochum, Universitätsstraße 150, 44801 Bochum, Germany.
*Postal address: Ruhr University Bochum, Universitätsstraße 150, 44801 Bochum, Germany.
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Abstract

Stationary Poisson processes of lines in the plane are studied, whose directional distributions are concentrated on $k\geq 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random polygons, which form a so-called Poisson line tessellation. The focus of this paper is to determine the proportion of triangles in such tessellations, or equivalently, the probability that the typical cell is a triangle. As a by-product, a new deviation of Miles’s classical result for the isotropic case is obtained by an approximation argument.

Keywords

Poisson line tessellationrandom trianglestochastic geometrytriangle probabilitytypical cell

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 214 - 229
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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