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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Statistics and Probability
Cited by
6 articles.
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