generate 3 random variables uniformly that correspond to a hyper plane.

Question

I am doing simulation that I want a point םמ a sphere to be picked at random. I used spherical coordinates, to uniformly generate $\theta$ ,$\phi$, but I found it that it does not really uniformly generates the values along x,y,z. x and y histogram peaked at 0, while z histogram peaked at r. Is there a way to make it more random?

Answer 1

Why does your title say "hyper plane" but your question talks about spheres?

If you want point on the sphere, read this.

If you want points within the sphere, a simple solution is to generate $x$, $y$, and $z$ uniformly in $[-r,r]$ and simply reject all points that are outside $x^2+y^2+z^2>r^2$. Also see this related (identical?) question.

Answer 2

From the rest of the question, I gather that you're using "within a sphere" in the precise technical sense in which it's synonymous with "on a sphere".

The uniform distribution over a sphere is uniform in $z$ (equivalently in $\cos\theta$), so you can choose $z$ uniformly from $[-1,1]$ and $\phi$ uniformly from $[0,2\pi]$ and use

$$ x=\sqrt{1-z^2}\cos\phi\;,\\ y=\sqrt{1-z^2}\sin\phi\;,\\ z=z\;. $$

Then by symmetry $x$ and $y$ will also be uniform over $[-1,1]$.

Alternatively, you can draw three coordinates from the normal distribution and normalise them; or you can draw three coordinates in the cube $[-1,1]^3$, reject them if they're outside the unit sphere and otherwise normalise them. That last variant requires the fewest non-elementary operations, just one square root for the normalisation (since the distance from the origin can be checked without drawing the root).