Generate a random sample of points distributed on the surface of a unit sphere

Question

I am trying to generate random points on the surface of the sphere using numpy. I have reviewed the post that explains uniform distribution here. However, need ideas on how to generate the points only on the surface of the sphere. I have coordinates (x, y, z) and the radius of each of these spheres.

I am not very well-versed with Mathematics at this level and trying to make sense of the Monte Carlo simulation.

Any help will be much appreciated.

Thanks, Parin

Answer 1

Based on the last approach on this page, you can simply generate a vector consisting of independent samples from three standard normal distributions, then normalize the vector such that its magnitude is 1:

import numpy as np

def sample_spherical(npoints, ndim=3):
    vec = np.random.randn(ndim, npoints)
    vec /= np.linalg.norm(vec, axis=0)
    return vec

For example:

from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import axes3d

phi = np.linspace(0, np.pi, 20)
theta = np.linspace(0, 2 * np.pi, 40)
x = np.outer(np.sin(theta), np.cos(phi))
y = np.outer(np.sin(theta), np.sin(phi))
z = np.outer(np.cos(theta), np.ones_like(phi))

xi, yi, zi = sample_spherical(100)

fig, ax = plt.subplots(1, 1, subplot_kw={'projection':'3d', 'aspect':'equal'})
ax.plot_wireframe(x, y, z, color='k', rstride=1, cstride=1)
ax.scatter(xi, yi, zi, s=100, c='r', zorder=10)

The same method also generalizes to picking uniformly distributed points on the unit circle (ndim=2) or on the surfaces of higher-dimensional unit hyperspheres.