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?
The equations of the cut at some $x_p$ are given by $$x^2+y^2+z^2=r^2$$ and $$x=x_p$$ We can the write $$y^2+z^2=r^2-x_p^2$$ Now plug in the expression for $y$ and $z$ in polar coordinates:$$\begin{align}z&=r\cos\theta\\y&=r\sin\theta\sin\phi\end{align}$$ You get $$\cos^2\theta+\sin^2\theta\sin^2\phi=1-\frac{x_p^2}{r^2}$$ To get the limits, note that $\cos\theta$ varies between $\pm\sqrt{1-\frac{x_p^2}{r^2}}$. Then you can write the expression for $\sin\phi$ in terms of $\theta$.
If you do a cut at $y_p$ instead, just replace $x_p$ with $y_p$ and use $\cos\phi$ instead of $\sin\phi$.