The animation illustrates a technique for drawing an ellipse that does not use the focal points of the ellipse. The radius of the inner circle is length of the vertical semiminor axis and the radius of the outer circle is the length of the horizontal semimajor axis. The blue points shown are on the elliptical curve that is being drawn in the animation. The purple radial lines are used to find the blue ellipse points. Can you figure by studying the animation how the locations of the blue points are being determined?

animation

A detailed explanation of how to obtain the ellipse points and verification that these points really are on the ellipse being drawn is available in the pdf file ImageDetail.

These images are created with Mathematica using a command I developed called DrawEllipse. You can download a Mathematica notebook containing the command and examples of how to use DrawEllipse by following the Notebooks link.