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Exact distributions for shapes of random triangles in convex sets
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- Journal:
- Advances in Applied Probability / Volume 17 / Issue 2 / June 1985
- Published online by Cambridge University Press:
- 01 July 2016, pp. 308-329
- Print publication:
- June 1985
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- Article
- Export citation
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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.
. A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set 
is then introduced. When 
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.