Consider the natural group action of $O(2)$ on the set of pairs of vectors of the $\mathbb{S}^{1}$ unit circle.
What is it's orbit for the group $O(2)$, give a total invariant.?
Same questions of $SO(2)$, and give the quotient $X/SO(2)$ (probably the space of all classes of orbits)
Thanks for any advice.
It seems as though $O(2)$ yields all possible pairs of vectors of $\mathbb{S}^1$ $\ldots$