Guided Center Approximation
The Guided Center Approximation. Here is a guided center for an ellipse with eccentrity of 0.01. We wish to approximate the green ellipse's orbit by using a circle (dashed blue) and a 2:1 ellipse (dashed magenta). The yellow dot is the center of the circle, and focus of the green ellipse. The magneta ellipse is the 2:1 ellipse, which is rides around the circumferance of the circle. The red dot is the approximated orbit, which has retrograde orbit on the magneta ellipse relative to the magneta direction of motion on the blue circle. The red dot has a light pink trail following it's orbit. Thus the approximation can be visiually evaluated from the overlap of the pink trail on the green ellipse.
Attached is the Mathematica script for this visualization.