There is a familiar scheme for choosing a random direction in n-dimensions that works well for small n.
Select successive x<sup>i</sup> from U{−1, 1} until Σx<sub>i</sub><sup>2</sup> > 1 or until we have n samples.
Discard the samples and start over when you exceed 1.
When n samples have been achieved the point whose coordinates are x<sub>i</sub> is in a random direction.
<p>This is a very useful algorithm but its efficiency degrades rapidly when n is greater than about 6.
It becomes too hard to not exceed 1.
<p>Instead we choose n random normal deviates x<sub>i</sub> and the point that they define in n-space is already is a random direction from the origin.
<p>To generate two independent random normal deviates choose random points &lt;x, y> uniformly from the square [−1, 1] × [−1, 1] until r2 = x<sup>2</sup> + y<sup>2</sup> &lt; 1.
Let q = √(−2 ln(r2))/r2.
xq and yq are independent random normal deviates.
<p>The C routine <tt>rnd</tt> in <a href=a.c>this program</a> produces random normal deviates.
<a href=../../code/Scheme/grnd.txt>This Scheme code</a> which employes some  random number generator such as <a href=../../code/mult/ff.txt>rand31 here</a> also does this algorithm.
The algorithm came from Knuth.
<p>See <a href=../PowerDist.html>this</a> for choosing a point within the nD ball.