Sampling uniformly distributed random points inside a spherical volume
geometry, matlab, python, random, uniform-distribution
Solution
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 )
Problem
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.