Ball triangle picking is the selection of triples of points (corresponding to vertices of a general triangle) randomly placed inside a ball. random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].
The distribution of areas of a triangle with vertices picked at random in a unit ball is illustrated above. The mean triangle area is
(1)
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(Buchta and Müller 1984, Finch 2010).
random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].
The determination of the probability for obtaining an acute triangle by picking three points at random in the unit disk was generalized by Hall (1982) to the -dimensional ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals. Let be the probability that three points chosen independently and uniformly from the -ball form an acute triangle, then
(2)
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(3)
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These can be combined and written in the slightly messy closed form
(4)
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where is a regularized hypergeometric function.
The first few are
(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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(12)
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(OEIS A093756 and A093757, OEIS A093758 and A093759, and OEIS A093760 and A093761), plotted above.
The case corresponds to disk triangle picking.