Rules of Elliptical Billiards Table

You would only want to play actual billiards on a rectangular table, but playing imaginary billiards on an elliptical table yields some interesting mathematical insights and puzzles. Wherever you shoot the ball, the results are predictable and fall into a only few simple patterns. Proving some of these results would require calculus, but the rules themselves are intuitive and arise from the simple rule of reflection: that the angle of incidence equals the angle of reflection.

Angle of Reflection

The rule of reflection is that the angle of incidence equals the angle of reflection. On a normal rectangular billiards table, a ball hitting the wall bounces away at the same angle reflected across a line perpendicular to the wall and intersecting it at the point of reflection. The same holds true on an elliptical billiards table.

On an elliptical billiards table, the ball hitting the wall at point P is reflected as if the wall is a straight line tangent to the ellipse at point P. A tangent line is a line whose slope is the first derivative of the curve at point P, and intersects the curve at point P. There is a unique tangent line for every point on an ellipse.

Reflection and Foci

An ellipse has two foci, F1 and F2. They lie on the major axis of the ellipse, the line which bisects it horizontally, and are equidistant from the origin. Line segments originating at the foci and intersecting the ellipse at point P form equal angles with the tangent line at point P. Therefore, a ball that passes through a focus will pass through the other focus after the first bounce. After the second bounce the ball will pass through the first focus again, and so on. If the ball continues bouncing, its path will come closer and closer to lying along the major axis.

If the first shot does not pass through a focus, the ball will never pass through a focus no matter how many times it bounces. In this case, if you trace the path of the ball the negative space of the resulting pattern will be a smaller ellipse, with the same foci as the original ellipse. The more times the ball bounces, the closer the resulting shape looks like an ellipse, and as the number of bounces approaches infinity the shape approaches being a true ellipse.

Periodical Paths

This is an exception to the rule stated above about a ball that does not pass through the foci describing an interior ellipse. In some special cases, such as where the first bounce is a line segment from a point where the major axis intercepts the ellipse to a point where the minor axis (the line of vertical symmetry) intercepts the ellipse, the path of the ball will describe a regular polygon. The ball will never deviate from this path no matter how many times it bounces.

When the Ball Passes Between the Foci

If the initial shot passes between the foci of the ellipse, every subsequent bounce will pass between the foci of the ellipse. After many bounces, the traced path of the ball will approximate a hyperbola, an hourglass-shaped figure, whose foci are the foci of the ellipse. At the limit where the number of bounces approaches infinity, the figure is infinitely close to being a hyperbola.

Circular Tables

A circle is an ellipse with foci at the same position. As in an ellipse, the path of a ball that does not initially pass through the focus will never pass through the focus, and if the path is not periodical it will describe an interior circle with the same center as the original circle. If the path of the ball is periodical, it will describe a polygon.