Further to this question
Quaternion rotation has a nice property that you can trace any great circle you like. You specify the axis of rotation, and you will automatically follow the great circle when rotating.
However spherical coordinates only trace a great circle when $ \theta (elevation) = 0 $, and $\phi (azimuth) $ is allowed to travel $ 0.. 2\pi$.
So my question is, is there a way to formulate a $\theta, \phi$ restriction that will allow me to trace an arbitrary great circle in spherical coordinates?