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.free.fr/metah/) shows what I am looking for. This is a slice through the sphere, showing a uniform distribution of points:

This is what I am currently getting:

You can see that there is a cluster of points at the center due to the conversion between spherical and Cartesian coordinates.

The code I am using is:

def new_positions_spherical_coordinates(self):
   radius = numpy.random.uniform(0.0,1.0, (self.number_of_particles,1)) 
   theta = numpy.random.uniform(0.,1.,(self.number_of_particles,1))*pi
   phi = numpy.arccos(1-2*numpy.random.uniform(0.0,1.,(self.number_of_particles,1)))
   x = radius * numpy.sin( theta ) * numpy.cos( phi )
   y = radius * numpy.sin( theta ) * numpy.sin( phi )
   z = radius * numpy.cos( theta )
   return (x,y,z)

Below is some MATLAB code that supposedly creates a uniform spherical sample, which is similar to the equation given by http://nojhan.free.fr/metah. I just can't seem to decipher it or understand what they did.

function X = randsphere(m,n,r)

% This function returns an m by n array, X, in which 
% each of the m rows has the n Cartesian coordinates 
% of a random point uniformly-distributed over the 
% interior of an n-dimensional hypersphere with 
% radius r and center at the origin.  The function 
% 'randn' is initially used to generate m sets of n 
% random variables with independent multivariate 
% normal distribution, with mean 0 and variance 1.
% Then the incomplete gamma function, 'gammainc', 
% is used to map these points radially to fit in the 
% hypersphere of finite radius r with a uniform % spatial distribution.
% Roger Stafford - 12/23/05

X = randn(m,n);
s2 = sum(X.^2,2);
X = X.*repmat(r*(gammainc(s2/2,n/2).^(1/n))./sqrt(s2),1,n);

I would greatly appreciate any suggestions on generating a truly uniform sample from a spherical volume in Python.

There seem to be plenty of examples showing how to sample from a uniform spherical shell, but that seems to be easier an easier problem. The issue has to do with the scaling - there should be fewer particles at a radius of 0.1 than at a radius of 1.0 to generate a uniform sample from the volume of the sphere.

Edit: Fixed and removed the fact I asked for normally and I meant uniform.

Answer 1

While I prefer the discarding method for spheres, for completeness I offer the exact solution.

In spherical coordinates, taking advantage of the sampling rule:

phi = random(0,2pi)
costheta = random(-1,1)
u = random(0,1)

theta = arccos( costheta )
r = R * cuberoot( u )

now you have a (r, theta, phi) group which can be transformed to (x, y, z) in the usual way

x = r * sin( theta) * cos( phi )
y = r * sin( theta) * sin( phi )
z = r * cos( theta )