What is a statistical test that a set of 3D vectors is uniformly distributed in all directions?

Question

A set of unit vectors in $\mathbb{R}^3$ is given. I want a statistical measurement of how uniformly their directions are distributed.

An example of a set which is uniformly distributed: generated uniform random points $p=(x,y,z): x,y,z \in [-1..1]$ and discard those having $|p|>1$, then project them onto the unit sphere, taking the set $\{\frac{p}{|p|}\}$. These points will be isotropic when viewed from the origin.

How can I measure this isotropy (or lack thereof) for a given set of unit 3D vectors?

Given a single real-valued threshold parameter, I want to answer the question "are these sufficiently uniformly-distributed (i.e. isotropic) or not?"

Answer 1

A possible way:

• transform your unit vectors into spherical coordinates
• cut the surface of the unit sphere into $N$ parts with (maybe differing) areas $A_n$ using spherical coordinates
• make a $\chi^2$-test for uniformity (watch out that each part $A_n$ on the surface has a probability $\frac{A_n}{4\pi}$)