Proclus' Ellipse Construction

This ellipse drawer would have been known to Proclus (418-485), who knew that an ellipse could be generated by tracing a point P inside a circle that rolls without slipping inside and tangent to another circle whose radius is twice as long. Such a construction brings to mind the popular spirograph game that children use to generate cycloidal curves.

Click the Play button at the lower left hand corner to generate the ellipse. What should be the lengths of OA and PC so that the curve generated will have x intercepts at (±5,0) and y intercepts at (0,±3)? Use the sliders to set OA and CP to these values and verify that the resulting curve has the correct intercepts. Enter the equations that you found in Exercise 1 into the input field to verify that Proclus's ellipse drawer does generate an ellipse.