Is there a generalization of Euler angles that handles the case where inversions are allowed?
I'm trying to figure out how to parameterize elements of $O(3)$ for a computer application; if I was only concerned with rotations, I'd use Euler angles, but my problem also allows for inversions.
Since $O(3)$ is the disjoint union of $SO(3)$ and $A$, where $A$ consists of all matrices in $SO(3)$ multiplied by the negative identity, the best hope you have is to take a parameterization of $SO(3)$ and make two copies of it, i.e., if you have a space $X$ and a parameterization $$ P: X \to SO(3), $$ define $$ Q: X \times \{-1, 1 \} \to O(3) : (x, t) \mapsto t P(X). $$ $Q$ is then a parameterization of $O(3)$.
