I am searching the spherical coordinates for the circular edge that are obtained when a sphere is cut at a certain position with a plane. The sphere has herby a radius $r$ and is focused at the center of a coordinate system. The plane cut is performed at a certain $x, y,$ or $z$ position (see an exemplary cut in the linked image).
What I am now interested in is finding the parametrization of the cutting edge, however not as parametrization of a circle, but instead in spherical coordinates of the sphere. This means I want to find the coordinates of every point on the cut, expressed in the spherical coordinate system. For a cut through the z-plane the solutions looks trivial with a azimuth angle changing between $0$ and $2\pi$ and a fixed elevation angle, as seen in the exemplary image. However the solution is not trivial for a cut through the $x$, or $y$ plane.
Does anyone know the solution for it?