How could I calculated the intersections of two ellipses?

I have the positions of the two focus points of each ellipse,
the distance between those focus points (2e) and the diameter of the ellipse on the semimajor axis (2a).

I can calculate any point on on of the ellipses using either a angle from the center or a valid value for either x or y.
My problem is that I can't figure out to calculate an intersection.