Sign up ×  I would like to create randomly oriented planes. This
is how I'm attempting to do that:    1. I create a 2 random unit
vectors, $\mathbf{v}_1$, and $\mathbf{v}_2$, in the $x$-$y$
plane   2. I assume that if I rotate these two vectors around a
random vector (in $x$-$y$-$z$) called $\mathbf n$, the plane on
which the two vectors lie is random and uniformly distributed
over the sphere.   3. So I randomly pick $\mathbf n$, keeping in
mind the issues of random point picking on spheres.   4. I then
use RotationMatrix[α, n].v1 (and same for v2) to get the new
vectors on the rotated and supposedly random plane. There is
some question on what to choose for the angle $\alpha$ which
gives the angle of the rotation.     • If I choose
$\alpha<\pi/2$, the vectors do not encompass the entire sphere.
• If I choose $\alpha=\pi/2$, I get a bunching of vectors at the
poles.     • If I choose $\alpha>\pi/2$, there is bunching at
various latitudes. I cannot get a uniformly random distribution
over the sphere!  Is t