Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-06-25T01:40:18.656Z Has data issue: false hasContentIssue false
  • English
  • Français

Averages for polygons formed by random lines in Euclidean and hyperbolic planes

Published online by Cambridge University Press:  14 July 2016

L. A. Santaló  and
I. Yañez
L. A. Santaló
Affiliation:
University of Buenos Aires
I. Yañez
Affiliation:
University of Madrid
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]–[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry.

Keywords

GEOMETRICAL PROBABILITYRANDOM LINESRANDOM POLYGONSPLANE OF CONSTANT CURVATUREHYPERBOLIC PLANECONVEX DOMAINSPOISSON LINE PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 1 , March 1972 , pp. 140 - 157
Copyright
Copyright © Applied Probability Trust 1972 

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

[1] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. J. Wiley, New York.Google Scholar
[2] Goudsmit, S. A. (1945) Random distributions of lines in a plane. Rev. Modern Phys. 17, 321–322.CrossRefGoogle Scholar
[3] Guggenheimer, H. (1963) Differential Geometry. McGraw Hill, New York.Google Scholar
[4] Miles, R. E. (1964) Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. 52, 901–907.CrossRefGoogle ScholarPubMed
[5] Miles, R. E. (1964) Random polygons determined by random lines in a plane (II). Proc. Nat. Acad. Sci. 52, 1157–1160.CrossRefGoogle Scholar
[6] Miles, R. E. (1970) A synopsis of Poisson flats in Euclidean spaces. Izv. Akad. Nauk Armjan. SSR Ser. Mat. V, 3, 263–285.Google Scholar
[7] Miles, R. E. (1969) Poisson flats in Euclidean spaces, Part I: A finite number of random uniform flats. Adv. Appl. Prob. 1, 211–237.CrossRefGoogle Scholar
[8] Miles, R. E. (1971) Poisson flats in Euclidean spaces, Part II: Homogeneous Poisson flats and the complementary theorem. Adv. Appl. Prob. 3, 1–43.CrossRefGoogle Scholar
[9] Miles, R. E. (1970) On the homogeneous planar Poisson point process. Math. Biosciences 6, 85–127.CrossRefGoogle Scholar
[10] Richards, P. I. (1964) Averages for polygons formed by random lines. Proc. Nat. Acad. Sci. 52, 1160–1164.CrossRefGoogle ScholarPubMed
[11] Santaló, L. A. (1953) Introduction to Integral Geometry. Hermann, Paris.Google Scholar
[12] Santaló, L. A. (1966) Valores medios para polígonos formados por rectas al azar en el plano hiperbólico. Rev. Mat. y Fis. Teor. Univ. Tucumán 16, 29–43.Google Scholar
[13] Santaló, L. A. (1967) Horocycles and convex sets in hyperbolic plane. Arch. Math. (Basel) 18, 529–533.CrossRefGoogle Scholar
[14] Santaló, L. A. (1969) Convexidad en el plano hiperbólico. Rev. Mat. y Fis. Teor. Univ. Tucuman 19, 173–183.Google Scholar