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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 z1, z2, z3 in the complex plane may be denoted by a vector Z, Z = [z1, z2, z3]t. From a sequence of independent and identically distributed 3×3 circulants {Cj}∞1, we may generate from Z1 the sequence of vectors or triangles {Zj}∞1, by the rule Zj = CjZj–1 (j> 1), Z1=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 {Zj}∞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
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.

References

Berlekamp, E. R., Gilbert, E. N. and Sinders, F. W. (1965) A polygon problem. Amer. Math. Monthly 72, 233–241.Google Scholar
Davis, P. J. (1979) Circulant Matrices. Wiley, New York.Google Scholar
Douglas, I. (1940a) On linear polygon transformations. Bull. Amer. Math. Soc. 146, 551–560.Google Scholar
Douglas, J. (1940b) Geometry of polygons in the complex plane. J. Math. Phys. 19, 93–130.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. 2. Wiley, New York.Google Scholar
Ford, L. R. (1929) Automorphic Functions. McGraw-Hill, New York.Google Scholar
Hall, G. G. (1967). Applied Group Theory. American Elsevier, New York.Google Scholar
Henrici, P. (1974) Applied and Computational Complex Analysis , Vol. 1. Wiley-Interscience, New York.Google Scholar
Kendall, D. G. (1983) The shape of Poisson and Delaunay triangles. In Studies in Probability and Related Topics , ed. Demetrescu, M. C. and Iosifescu, M. Nagard, Sofia.Google Scholar
Kendall, D. G. (1984) Shape manifolds, procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
Neumann, B. H. (1941) Some remarks on polygons. J. London Math. Soc. 16, 230–245.Google Scholar
Neumann, B. H. (1942) A remark on polygons. J. London Math. Soc. 17, 165–166.CrossRefGoogle Scholar
Neumann, B. H. (1982) Planar polygons revisited. J. Appl. Prob. 19A, 113–122.Google Scholar
Veitch, J. and Watson, G. S. (1984) Cyclic transformations of k points in m dimensions.Google Scholar
Watson, G. S. (1983) Random triangles. In Proc. Conf. Stochastic Geometry, Aarhus ed. Barndorff-Nielsen, O. Google Scholar