Given two points $x_0$ and $x_1$ lying on the hypersphere $S^n$, how can I derive a formula for a orthogonal transformation that will move $x_0$ to $x_1$?
I could decompose $x_0$ and $x_1$ into hyper-spherical coordinates and figure out a rotation between them, but I am sure there is a more elegant way of solving this problem.