Let $\mathbb{S}^2$ be the round $2-$sphere and consider $S^1_a$ and $S^2_b$ two geodesics, i.e., two great circles and suppose that they are not the same. Intuitively, I think that there is a rotation matrix by $\theta$, denotes by $R_{\theta}$, such that $R_{\theta}(S^1_a)=S^2_b$, where $\theta$ is the smallest angle between the tangent vectors at the intersection points.
Is it right? Can I consider only rotation matrix to "change" of great circles on $\mathbb{S}^2$? Or need I be carreful with orientation problems?
I appreciate any help, thanks.
The two great circles will intersect at two points. You can rotate around the axis determined by those two points in order to move one great circle to the other.