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Ball Triangle Picking

The determination of the probability for obtaining an Obtuse Triangle by picking 3 points at random in the unit Disk was generalized by Hall (1982) to the $n$-D Ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals, with the first few solutions being

$\displaystyle P_2$ $\textstyle =$ $\displaystyle {9\over 8}-{4\over\pi^2}\approx 0.72$  
$\displaystyle P_3$ $\textstyle =$ $\displaystyle {\textstyle{37\over 70}}\approx0.53$  
$\displaystyle P_4$ $\textstyle \approx$ $\displaystyle 0.39$  
$\displaystyle P_5$ $\textstyle \approx$ $\displaystyle 0.29.$  

The case $P_2$ corresponds to the usual Disk case.

See also Cube Triangle Picking, Obtuse Triangle


Buchta, C. ``A Note on the Volume of a Random Polytope in a Tetrahedron.'' Ill. J. Math. 30, 653-659, 1986.

Hall, G. R. ``Acute Triangles in the $n$-Ball.'' J. Appl. Prob. 19, 712-715, 1982.

© 1996-9 Eric W. Weisstein