TITLE: How to generate random points on a sphere?
QUESTION [61 upvotes]: How do I generate $1000$ points $\left(x, y, z\right)$ and make sure they land on a sphere whose center is
$\left(0, 0, 0\right)$ and its diameter is $20$ ?. 
Simply, how do I manipulate a point's coordinates so that the point lies on the sphere's "surface" ?.

REPLY [3 votes]: Wolfram Mathworld provides a methodology for randomly picking a point on a sphere:

To obtain points such that any small area on the sphere is expected to contain the same number of points, choose $u$ and $ν$ to be random variates on $[0,1]$. Then: $$\begin{array}{ll}\theta=2\pi u\\
\varphi= arccos(2v - 1)\end{array}$$ gives the spherical coordinates for a set of points which are uniformly distributed over $\mathbb{S}^2$.