What would be the equation of an arbitrary circle rotated along some angle theta around the X-axis in spherical coordinates? For simplicity we may assume that it is a circle with constant radius r.
It seems like it has something to with having polar coordinates within the latitudinal and longitudinal spherical map. However, I'm not entirely certain.
Hint:
You can start from a circle in the $x-y$ plane centered at the origin that is represented by the parametric equation: $$ \begin{bmatrix} x\\y\\z \end{bmatrix} =\begin{bmatrix} r\cos t\\r\sin t\\0 \end{bmatrix} \qquad 0\le t<2\pi $$ Now using a matrix that represents an isometry you can transform this circle to another one rotated in any plane passing thorough the origin, and if you use translations, you can also change the center of the circle.
As an example, the matrix: $$ \begin{bmatrix}1&0&0\\ 0&\cos \theta&\sin \theta\\ 0&-\sin \theta&\cos \theta \end {bmatrix} $$ represents a rotation of angle $\theta$ around the $x-$axis, and the circle rotated in such a way has equation: $$ \begin{bmatrix} x\\y\\z \end{bmatrix} =\begin{bmatrix} r\cos t\\r\sin t \cos \theta\\-r \sin t \sin \theta \end{bmatrix} \qquad 0\le t<2\pi $$
in the same way you can rotate the circle around any axis of rotation, using the matrix that represents such rotation (see here).