If I sample three points independently, uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their [polar sine][1]? More generally, for $k<n$ if I sample $k$ points independently uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their polar sine? An asymptotic form for large $n$ can be deduced in the case $k=2$ from Lemma 4 of Appendix A of Laarhoven's [Sieving for shortest vectors in lattices using angular locality-sensitive hashing][2], but my attempts to generalise to $k=3$ do not match empirical distributions for $n=10$. [1]: https://en.wikipedia.org/wiki/Polar_sine [2]: https://eprint.iacr.org/2014/744.pdf