Question

Using Matlab, how to generate a net of 3^10 points that are evenly located (or distributed) on the 8-dimensional unit sphere?

Answer 1

from wikipedia n-sphere

To generate uniformly distributed
random points on the (n − 1)-sphere
(i.e., the surface of the n-ball),
Marsaglia (1972) gives the following
algorithm.

Generate an n-dimensional vector of
normal deviates (it suffices to use
N(0, 1), although in fact the choice
of the variance is arbitrary),
\mathbf{x}=(x_1,x_2,\ldots,x_n).

Now calculate the “radius” of this
point,
r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.

The vector \frac{1}{r} \mathbf{x} is
uniformly distributed over the surface
of the unit n-ball.

A matlab code fragment that accomplishes this is:

numdims = 8;
numpts = 3^10;
x = randn([numdims numpts]);
lx = repmat(sqrt(sum(x.^2,1)), [numdims 1]);
x = x./lx;
%x(:,j) is the jth point on the circle)