Random Points on Shpere

To generate a random unit vector in $R^n$ , the most trivial approach is to use spherical coordinates and generate the angles uniformly at random. Here we introduce another way : sample $n$ independent identical random variables from Gaussian distribution $N(0, 1)$ , denoted as $x_1, x_2, ..., x_n$ . Then a unit vector $v$ is obtained by

v = \frac{1}{\sum_{i = 1}^n x_i^2 }(x_1, x_2, ..., x_n)

The key observation is that all points at a fixed distance from the origin is equally generated. To see this, the joints distribution of $x_1, x_2, ..., x_n$ is given by

p = \frac{1}{(2 \pi)^{n / 2} } \exp \big( - \frac{\sum_{i=1}^n x_i^2 }{ 2 } \big)

$p$ is fixed when $\sum_{i = 1}^n x_i^2$ is fixed.