As I understand the question, you have an ellipse ‘rolling’ around a circle.

You want to express the distance between their centers as a function of the angle of rotation of the ellipse.

Under this understanding the answer is straightforward.

Suppose that is the radius of the circle, is the half the length of the major axis of the ellipse, and is the half the length of the minor axis of the ellipse.

Then

With reference to your drawing.

When then .

When then .