# Random Gaussian Tetrahedra

Steven Finch

Arxiv ID: 1005.1033•Last updated: 3/22/2022

Given independent normally distributed points A,B,C,D in Euclidean 3-space,
let Q denote the plane determined by A,B,C and D^ denote the orthogonal
projection of D onto Q. The probability that the tetrahedron ABCD is acute
remains intractable. We make some small progress in resolving this issue. Let
Gamma denote the convex cone in Q containing all linear combinations
A+r*(B-A)+s*(C-A) for nonnegative r, s. We compute the probability that D^
falls in (B+C)-Gamma to be 0.681..., but the probability that D^ falls in Gamma
to be 0.683.... The intersection of these two cones is a parallelogram in Q
twice the area of the triangle ABC. Among other issues, we mention the
distribution of random solid angles and sums of these.

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