I’d like to generate N random 3-dimensional vectors (uniformly) on the unit sphere but with the condition, that their sum is equal to 0. My attempt was to generate N/2 random unit vectors, while the other are just the same vectors with a minus sign. The problem is, as I’m trying to achieve as little correlation as possible, and my idea is obviously not ideal, since half of my vectors are perfectly anti-correlated with their corresponding pair.
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Answer
Your problem does not really have a solution, but you can generate a set of vectors that are going to be slightly less visibly correlated than your original solution of negating them. To be precise, if you generate N / 2 vectors and negate them, then rotate the negated vectors about their sum by any angle, you can guarantee that the sum will be zero and the correlation will be a more complicated rotation than a negative identity matrix.
import numpy as np from scipy.spatial.transform import Rotation N = 10 v1 = np.random.normal(size=(N / 2, 3)) v1 /= np.linalg.norm(v1, axis=1, keepdims=True) axis = v1.sum(0) rot = Rotation.from_rotvec(np.random.uniform(2.0 * np.pi) * axis / np.linalg.norm(axis)) v2 = rot.apply(-v1) result = np.concatenate((v1, v2), axis=0)
This assumes that N is even in all cases. The normal distribution is a fairly standard method to generate points uniformly on a sphere: https://mathworld.wolfram.com/SpherePointPicking.html.
If you had some leeway from the sum being exactly zero, you could align two random sets of N / 2 vectors so that their sums point opposite each other.