# Distribution of shapes of Delaunay triangles

People haven't been rushing in to answer this question I asked on stackexchange yesterday.

The way I phrased it initially was this: Does anyone know the probability distribution of the shapes of Delaunay triangles in a constant-intensity Poisson process in the plane?

Then I added this naive comment:

Slightly later edit: One can imagine performing the experiment repeatedly and looking at the one triangle that surrounds the origin, and ask, for example, how frequently it will be obtuse; or one can imagine doing it just once and looking at all of the infinitely many triangles and asking what proportion of them are obtuse. One would (or at least I would) initially guess the two answers are the same (and similarly for other sets of shapes besides the set of all obtuse triangles). One complication in proving that would be that the shapes of the infinitely many triangles one gets by doing the experiment once are not mutually independent.
end of "slightly later edit"

Then joriki pointed out that one shouldn't expect the two to be the same. So take your pick..... (For now, I prefer joriki's interpretation: i.e. the second one is the interesting one....)

However: In view of the comments, maybe it might be even more fruitful to ask for the joint distribution of the size and shape.

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See this paper of R. E. Miles (he has plenty of related results for points on the sphere, etc, etc, mathscinet will tell you more). The results you want are in section 9 (p. 112, and thereabouts). (the paper is: On the homogeneous planar Poisson point process, R. E. Miles, Mathematical Biosciences 6 (1970).

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Cool. For any triangle with angles $\alpha$, $\beta$, and $\gamma=\pi-\alpha-\beta$, the value of the probability density is the ratio of the area of the triangle to the area of the circumscribed circle, thus is is $\text{constant}\cdot\sin\alpha\sin\beta\sin\gamma$. – Michael Hardy Dec 27 '11 at 18:21