Let $B$ be the unit sphere of an M-dimensional linear space and let $x$ be uniformly distributed on $B$. Let $A$ be an N-dimensional subspace of the said linear space (N<M). How is the projection of $x$ onto $A$, i.e. $P_Ax$, distributed? Specifically, how is the length of this projection, i.e. $\sqrt{x^{'}P_A^{'}P_Ax}$, distributed?
I can quite easily solve this problem for low dimensions (e.g. M=2, N=1), but I am looking for the general rule here. I can also simulate the distribution of $x$, and therefore that of $P_Ax$, but I would prefer an exact solution.
Special: If someone can present an exact solution, that can be rewarded by a co-authorship in a scientific publication. I am not a mathematician, but I am working on a field where mathematics is frequently encountered.