I am unclear about how to parameterize for x,y on the ellipse shown in the figure based on lines drawn from the focus, point "P", (not the center, point "O") and for the angle $\phi$. I believe the drawing shows what I am looking for. For the standard form of this ellipse, $a=\frac{R}{\sin\beta}$ and b = R.
Edit #1: Removed confusing statement regarding q.
Edit #2: For a standard ellipse the parameterization would be $x = a \cos \alpha$ and $y = b\sin \alpha$. If it is offset to the focus as shown in the diagram, then I think the parameterization is $x = a \cos \alpha + c$ and $y = b\sin \alpha$. The problem seems to be getting the parameterization in terms of $\phi$. Note that the diagram has been revised to show the angle $\alpha$.
