My question is related to here: Given two orientation $A,B\in SO(3)$, and let $P,Q \in S^{3}$ (the 3-sphere in 4-space) such that $f(P) = A$ and $f(Q) = B$. I want to find the "shortest distance" between the given two orientations. My questions are:
1) Does the axis-angle representation give the "shortest distance" between two orientations? I'm not sure if I fully understand Dr. Hughes' answer (here), it seem to be that he only considers the case when $P$ and $Q$ are close, what if they are not close?
2) Since $P,Q \in S^3$, is the "shortest distance" the same as the distance between $P$ and $Q$ on $S^{3}$ (calculated via integration on manifold)?
3) If my reasoning is correct, is there anyway I can find the close form expression or the interpolation of the geodesic between $P$ and $Q$?
Really appreciate your help.