Given a mean and variance, I want to be able to sample rotations that are "normally distributed" according to this mean and variance. I have a metric over rotations, the chordal metric: $\| R_1 - R_2\|$ which tells us how close two rotations are to each other. How do I do this?
One way is to consider SO(3) as a sphere $S^3$ in $R^4$, the space of quaternions, but I don't know how to setup the Gaussian distribution correctly in this case considering a single rotation in SO(3) is an antipodal pair in $S^3$.