Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T09:20:27.108Z Has data issue: false hasContentIssue false
  • English
  • Français

Shape distributions for planar triangles by dual construction

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

John Gates
John Gates*
Affiliation:
University of Greenwich
*
* Postal address: School of Mathematics, Statistics and Computing, The University of Greenwich, Woolwich Campus, Wellington Street, London SE18 6PF, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

A random triangle in the plane is constructed using three independent elements from a convex set of lines. Expressions are given to calculate the shape distribution from the internal width function of the line set. Two examples are given together with their maximum angle distributions; a simple inequality implies a zero collinearity constant in general. A relationship between the shape distribution and inter-line angle distribution is given.

Keywords

DUAL CONVEXITYSHAPE DISTRIBUTIONINTER-LINE ANGLE DISTRIBUTION

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 26 , Issue 2 , June 1994 , pp. 324 - 333
Copyright
Copyright © Applied Probability Trust 1994 

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. (1990) Factorization Calculus and Geometric Probability. Cambridge University Press.Google Scholar
Dryden, I. L. and Mardia, K. V. (1991) General shape distributions in a plane. Adv. Appl. Prob. 23, 259–276.Google Scholar
Gates, J. (1993) Some dual problems of geometric probability in the plane. Combinatorics, Probability and Computing, 2, 11–23.Google Scholar
Kendall, D. G. (1984) Shape manifolds, procrustean metrics and complex projective space. Bull. London Math. Soc. 16, 81–121.Google Scholar
Kendall, D. G. (1985) Exact distributions for shapes of random triangles in convex sets. Adv. Appl. Prob. 17, 308–329.Google Scholar
Kendall, D. G. and Kendall, W. S. (1980) Alignment in two-dimensional random sets of points. Adv. Appl. Prob. 12, 380–424.Google Scholar
Kendall, D. G. and Le, H.-L. (1986) Exact shape-densities for random triangles in convex polygons. In Analytic and Geometric Stochastics, Suppl. Adv. Prob., 59–72.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
Small, C. G. (1982) Random uniform triangles and the alignment problem. Math. Proc. Camb. Phil. Soc. 91, 315–322.Google Scholar