Best way to get random spherical distribution using geometry nodes?

Question

I want to create a spherical distribution of points. At the moment I use this node setup.

But this creates noticeable higher densities on diagonals

It should be more uniform like this

What's a better method for creating a spherical distribution?

Answer 1

Discarding Approach

Spawn twice as many points as you need (I do an exact sphere radius vs cube radius ratio but you're not guaranteed to get enough points regardless how many spare points you generated due to the nature of randomness) inside a cube, and delete the excess that wasn't inside the sphere:

360-430 ms for 10 million points

Proper Maths Approach

Taken from here: Karthik Karanth: Generating Random Points in a Sphere

function getPoint() {
    var u = Math.random();
    var v = Math.random();
    var theta = u * 2.0 * Math.PI;
    var phi = Math.acos(2.0 * v - 1.0);
    var r = Math.cbrt(Math.random());
    var sinTheta = Math.sin(theta);
    var cosTheta = Math.cos(theta);
    var sinPhi = Math.sin(phi);
    var cosPhi = Math.cos(phi);
    var x = r * sinPhi * cosTheta;
    var y = r * sinPhi * sinTheta;
    var z = r * cosPhi;
    return {x: x, y: y, z: z};
}

300-340 ms for 10 million points

Answer 2

In this particular case, you find that the Random Value node is not as random as its name suggests....

If you want to solve this mathematically, use the wonderful answer from @Markus von Broady.

However, if you want to use fewer nodes, and stick to using Random Value, you could solve it as follows:

• Here I use many points created at the same position.
• I rotate them with random values between $-\pi$ and $\pi$, so that they are positioned in a circle.
• Then I rotate these positions again on a random axis so that the previously circularly positioned points form a sphere.
• Finally I scale the positions with a value between $0$ and $1$ with another Random Value node, but a different value for Seed (!)

You can also replace the first two operations with one, but it is important that at least one of your Random Value has a different seed value.

_{(Blender 3.6+)}

Answer 3

No way intended to compete with Markus' answer, which is proved, where this is not.

Following the Math Exchange answer here, though, with its commentary, and told that (Blender) Perlin Noise's distribution is Gaussian-like, this is another shot at it:

.. it's just interesting that there's no obvious visible bias? But I wouldn't trust it to do strict statistical sampling.