The topic of generating random points on a unit sphere has come up several times here. The standard method using normal random samples generates points (x, y, z) in Cartesian coordinates.
If you wanted points in spherical coordinates, you could first generate points in Cartesian coordinates, then convert the points to spherical coordinates. But it would seem more natural, and possibly more efficient, to directly generate spherical coordinates (ρ, θ, ϕ). That’s what we’ll do in this post.
Unfortunately there are several conventions for spherical coordinates, as described here, so we should start by saying we’re using the math convention that (ρ, θ, ϕ) means (radius, longitude, polar angle). The polar angle ϕ is 0 at the north pole and π at the south pole.
Generating the first two spherical coordinate components is trivial. On a unit sphere, ρ = 1, and θ we generate uniformly on [0, 2φ]. The polar angle ϕ requires more thought. If we simply generated φ uniformly from [0, π] we’d put too many points near the poles and too few points near the equator.
As I wrote about a few days ago, the x, y, and z coordinates of points on a sphere are uniformly distributed. In particular, the z coordinate is uniformly distributed, and so we need cos ϕ to be uniformly distributed, not ϕ itself. We can accomplish this by generating uniform points from [−1, 1] and taking the inverse cosine.
Here’s Python code summarizing the algorithm for generating random points on a sphere in spherical coordinates.
from numpy import random, pi, arccos
def random_pt() # in spherical coordinates
rho = 1
theta = 2*pi*random.random()
phi = arccos(1 - 2*random.random())
return (rho, theta, phi)
See the next post for a demonstration.