Sampling uniformly distributed random points inside a spherical volume

Question

I am looking to be able to generate a random uniform sample of particle locations that fall within a spherical volume.

The image below (courtesy of http://nojhan.fre

Answer 1

You can just generate random points in spherical coordinates (assuming that you are working in 3D): S(r, θ, φ ), where r ∈ [0, R), θ ∈ [0, π ], φ ∈ [0, 2π ), where R is the radius of your sphere. This would also allow you directly control how many points are generated (i.e. you don't need to discard any points).

To compensate for the loss of density with the radius, you would generate the radial coordinate following a power law distribution (see dmckee's answer for an explanation on how to do this).

If your code needs (x,y,z) (i.e. cartesian) coordinates, you would then just convert the randomly generated points in spherical to cartesian coordinates as explained here.