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Explicit formulae for polygonally generated shape-densities in the basic tile

Published online by Cambridge University Press:  24 October 2008

Hui-Lin Le
Hui-Lin Le
Affiliation:
Statistical Laboratory, University of Cambridge
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Abstract

Kendall and Le [4] gave an exact algorithm for the computation of a polygonally generated (auxiliary) shape-density m̃(x, y) which we used to obtain ‘exact’ numerical densities for given (x, y). Here I derive an explicit formula for m̃(x, y), for an arbitrary convex plane polygon K, valid when the shape-point (x, y) lies in what will be called the ‘(upper) basic tile’ of the associated singular tessellation. In most circumstances this is all that is needed for the statistical analysis of collinearities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 2 , March 1987 , pp. 313 - 321
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Kendall, D. G.. Shape-manifolds, procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16 (1984), 81–121.CrossRefGoogle Scholar
[2]Kendall, D. G.. Exact distributions for shapes of random triangles in convex sets. Adv. Appl. Prob. 17 (1985), 308–329.CrossRefGoogle Scholar
[3]Kendall, D. G. and Kendall, W. S.. Alignments in two-dimensional random sets of points. Adv. Appl. Prob. 12 (1980), 380–424.CrossRefGoogle Scholar
[4]Kendall, D. G. and Le, H.-L.. Exact shape-densities for random triangles in convex polygons. Adv. Appl. Prob. (in the Press).Google Scholar
[5]Small, C. G.. Random uniform triangles and the alignment problem. Math. Proc. Cambridge Philos. Soc. 91 (1982), 315–322.CrossRefGoogle Scholar