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Exact distributions for shapes of random triangles in convex sets

Published online by Cambridge University Press:  01 July 2016

D. G. Kendall
D. G. Kendall*
Affiliation:
University of Cambridge
*
∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
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Abstract

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

Keywords

SHAPE-SPACEAUXILIARY SHAPE-DENSITY(C, W)-DIAGRAM(X, y)-DIAGRAMCOLLINEARITYQUASARS
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 2 , June 1985 , pp. 308 - 329
Copyright
Copyright © Applied Probability Trust 1985 

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