Folding a Circle to Get an Ellipse

Date: 01/08/2001 at 09:35:01
From: Paul Rosin
Subject: Folding circles to get ellipses

I've read the following instructions:

1. Take a circle and place a point inside it.

2. Take a point on the circle and fold it to the interior point.

Repeat this process, and the envelope of folds forms an ellipse.

Can you tell me how to prove that the envelope is an ellipse? Also, 
what is its equation?

Date: 01/08/2001 at 12:05:02
From: Doctor Peterson
Subject: Re: Folding circles to get ellipses

Hi, Paul.

This is a nice construction, and is actually very closely related to 
the reflection property of an ellipse. I'll just give you a hint to 
get you started on a proof.

Consider one of the lines you make by folding:

                       *********
                 ******         ******
              ***                     ***   Q
            **                      ********
          **                A ******      /**
         *                 **+           /   *
        *                **   \ \       /     *
       *               **      \    \  /       *
      *               *         \     /\        *
      *              *           \   /    \     *
     *              *             \ /         \  *
     *             *       +-------+-------------+ B
     *             *      O       / C            *
      *           *              /              *
      *           *             /               *
       *          *            /               *
        *          *          /               *
         *         *         /               *
          **        *       /              **
            **       *     /             **
              ***     *   /           ***
                 ********/      ******
                       *********
                        P

Here I've folded the circle on line PQ, so that point B on the circle 
lies on the fixed point A. You should immediately notice that PQ is 
the perpendicular bisector of AB; that's how folding works. Draw 
segment OB from the center of the circle, and look at point C where 
this intersects PQ. This will be the point on the ellipse. 

On one hand, you can show that point C fits the definition of an 
ellipse with O and A as foci; on the other hand, if you know that 
light from one focus reflects through the other focus, you should be 
able to see that PQ is in fact tangent to this ellipse.

If you want the equation, you can use the foci and other information 
from this picture; or generate the equation from the construction by 
taking the angle of OB as a parameter, and constructing point C from 
point B. I haven't tried this, and can't tell you how hard it might 
be.

If you need more help, feel free to write.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/