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The shapes of a random sequence of triangles

Published online by Cambridge University Press:  01 July 2016

G. S. Watson
G. S. Watson*
Affiliation:
Princeton University
*
∗Postal address: Fine Hall, Princeton University, Washington Rd, Princeton, NJ 08544, USA.
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Abstract

A triangle with vertices z 1, z 2 , z 3 in the complex plane may be denoted by a vector Z , Z = [z 1, z 2, z 3]t . From a sequence of independent and identically distributed 3×3 circulants {C j }∞ 1, we may generate from Z 1 the sequence of vectors or triangles {Z j }∞ 1, by the rule Z j = C j Z j–1 (j> 1), Z 1=Z . The ‘shape’ of a set of points, the simplest case being three points in the plane has been defined by Kendall (1984). We give several alternative, ab initio discussions of the shape of a triangle, and proofs of a limit theorem for shape of the triangles in the sequence {Z j }∞ 1. In Appendix A, the shape concept is applied to the zeros of a cubic polynomial. Appendix B contains some further remarks about shape. Appendix C uses the methods of this paper to give proofs of generalizations of two old theorems on triangles.

Keywords

CIRCULANTMOEBIUS TRANSFORMATIONSNAPOLEON's THEOREM

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 1 , March 1986, pp. 156 - 169
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

This paper is dedicated to the memory of two old friends, Marc Kac and Elliot W. Montroll. Kac was drawn into mathematics by a dissatisfaction with Vieta's substitution. Montroll used circulants extensively. Both loved dealing with the simplest case.

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