Reflections Inside an Ellipse
- Clint Givens
In the GeoGebra applet below, you can drag the points A, B, and C to reposition the ellipse, and you can drag D and E on the circumference of the ellipse. The other segments drawn inside the ellipse are the path of a light ray which starts at D, travels to E, and then is reflected a large number of times by the interior of the ellipse. You will notice right away that the space filled up by the light ray's path, also known as its orbit, takes the shape of a hyperbola. As you move points D and E around, you will find that you can also arrange it to take the shape of a different ellipse situated inside the original ellipse. These shapes--the hyperbola or ellipse traced out by the light ray's orbit--are called envelopes.
Questions to consider: (1) What are the foci of the hyperbola / ellipse which is formed as an envelope of the light ray's orbit? (2) Under what circumstances will the envelope by a hyperbola vs. an ellipse? (3) We call the light ray's orbit closed if it eventually returns to its starting point (D) and forms an exactly repeating cycle. Some configurations of the ellipse and the segment DE seem to result in a closed, or very nearly closed, orbit. Are these orbits truly closed, or do they just appear that way because of the limited resolution of the applet and the fact that it only draws a finite number of segments? On the other hand, could it be the case that ALL orbits are closed, if only the applet could draw enough segments to show the light ray finally return to its starting point??