The Elliptic Pool Game

by Pete Horton, Laredo Junior College

The special reflection properties of the conic sections have been known for quite some time. In this article, I will focus on the reflecting properties of the ellipse. Although the ellipse can be defined in several different ways, I'll just use the following definition.

The ellipse is the set of points in the plane such that

the sum of the distances from 2 fixed points, called foci,

is a constant value. (see the figure)

For each point on the ellipse,  d1 + d2 = constant . The major axis is the segment connecting sides of the ellipse and passing through the foci. The vertices are the points where the major axis touches the ellipse.


Now for the reflecting property. Any ray (or cue ball) which leave one focus and rebounds off the side of the ellipse will be directed toward the other focus. See the figure for some example paths. For a proof, see A Pre-trigonometry Proof of th e Reflection Property of the Ellipse, The College Mathematics Journal, 418 (November 1986).
Let's play pool! The game consists of an elliptic pool table with 2 pockets located at the foci, a cue stick, and a cue ball. The rules are pretty basic. A player wins by shooting the cue ball with the cue stick in such a way so that the ball falls into one of the pockets after a single rebound off the side of the table. Of course, we'll think of the pickets and ball as actually being points, and no spins or masse' shots are permitted.
Questions: How should a player align the cue stick in order to guarantee a win if the cue ball is located
  • off of the major axis?
  • on the major axis but outside of the foci?
  • on the major axis, but between the foci?

Solutions: