Expression for a Distance in Ptolemaic Epicycles

I’m studying the geocentric Ptolemaic model of an upper Planet. My goal is an expression of the (squared) distance of $h$ from the planet to Earth, depending on the angle $beta$.

Let me explain you how the model works. $E$ is the earth. The deferent $c$ is a circle around the middle between the earth and the equant. $H$ moves on $c$ and is the center of the epicycle. I assume the ratio of both radii is known. While $M$ is the center of the circle, the angular speed of $H$ is uniform w.r.t the equant, not uniform w.r.t $M$ or $H$.

My goal is an expression $h^2=f(beta)$, not necessarily a time-dependent expression of the Planet’s coordinates. Do I need a full parameterization of the Planet’s position anyway? I suppose it might be possible with some trigonometry. However I could’t find it.