Rotation matrix in spherical coordinates When given arbitrary point on a unit sphere $a = (\theta, \phi)$ and an arbitrary axis $\vec{A}=(\Theta, \Phi)$, can we have an algebraic expression for $a_1=(\theta_1, \phi_1)$ which is a rotation of $a$ around $\vec{A}$ to the angle $\beta$? 
Points and axes are not on the coordinate planes, values are not trivial: $\theta \neq 0$, $\phi \neq 0$, $\Theta \neq 0$, $\Phi \neq 0$, $\beta \neq 0$.
Can this be done without transformation through Cartesian? Otherwise the analytic form becomes too complicated. If there is a particular case for $\Phi \rightarrow 0$ ($\sin{\Phi} \approx \Phi$, but not for the other values) it is also fine. 
Thanks

In this post I explain how to deal when the rotation axis is one of the Cartesian axes. 
For an abritrary rotation axis $\vec{A}=(\Theta,\Phi)$, using the same notations, the formula is
$$
(\theta', \phi') = (Q^{-1} \circ R_{\vec{A}}(\beta) \circ Q)(\theta,\phi)
$$
where the rotation operator $R_{\vec{A}}(\beta)$ is
$$
R_{\vec{A}}(\beta) = R_z(\Phi)R_y(\Theta)R_z(\beta){R_y(\Theta)}^\dagger{R_z(\Phi)}^\dagger
$$
("${}^\dagger$" is the notation for the conjugate transpose).
