Orbiting spherical coordinates around sphere

I have a sphere with radius 1 described by the spherical coordinate rotations $(s_\theta, s_\phi)$, as well as a point on that sphere described by spherical coordinates relative to (0,0) on the sphere $(p_\theta, p_\phi)$.

I want to know what the spherical coordinates of the given point are, NOT relative to the sphere.

For example with a sphere rotated at $(0,0)$ and a point $p=(p_\theta, p_\phi)$, the actual spherical coordinates of p remain $(p_\theta, p_\phi)$.

Given a sphere rotated at $(\pi,0)$ with a point $p=(p_\theta, p_\phi)$ on the sphere, the actual spherical coordinates of p are $(p_\theta+\pi, p_\phi)$.

These examples are trivial but hopefully give you an understanding of what I am looking for. When you let $s_\phi\ne0$ it becomes much more difficult to calculate.

I know it is possible to calculate this using two applications of Rodrigues' Rotations, however this involves a ridiculous amount of calculations, I hope there is a more direct way to perform this calculation. Ideally I would not have to convert to Cartesian coordinates at all.

edit: I've been working on coming up with the calculation for this problem and I came up with this plot. The graph plots rotations of points at different distances from the axis of rotation of the sphere. The equation is a square at points closest to the axis of rotation and circle as the distance reaches $\pi/2$. Therefore the equation is of the form $(x-\pi/2)^a + y^a = b$, where a and b are non-constant but I'm not sure what they are.

• Trying to do this in a spherical coordinate system seems hopeless. Why not just convert $(s_\theta,s_\phi)$ to a rotation matrix $S$, convert $(p_\theta,p_\phi)$ to rectangular coordinates $p$, compute $q=Sp$ and then convert $q$ to spherical coordinates $(q_\theta,q_\phi)$. – T L Davis Apr 24 '17 at 15:58