**ELLIPTICAL BILLIARD TABLES**

**Introduction**

Normally a billiards game is played on a rectangular table. On a *"mathematical billiard table"* we consider only one ball.
Moreover, this ball is reduced to a point that moves in a straight line until it hits an edge and bounces back in such a way that the *angle of
reflection equals the angle of incidence.* Both angles can be considered as the angles formed by the side of the table involved
and the incoming and outgoing lines.
The mathematician Charles L. Dodgson, better known as Lewis Carroll, the author of *"Alice in Wonderland"* thought and wrote about *circular billiard
tables*.

We'll consider a billiard table that takes the shape of an *ellipse*.
This page exists since 2002 but only very recently (July 2007) I discovered the book *Geometry and Billiards* by *Serge Tabachnikov*. In this book (186p) the author threatens different billiard shapes and the (often not so elementary) mathematics behind them. The bouncing law is linked to the reflection law in optics and to the variational principle behind it: minimizing the time a light ray needs to travel from the original to the reflected point. As to elliptical billiards this leads in a short and elegant way to the property that *a shot passing through one focus is reflected through the other focus*. A nice geometrical proof of this property is mentioned further. In the same publication one can find a proof of the property that *a billiard trajectory inside an ellipse remains tangent to a fixed confocal conic.*
The book also contains an extensive bibliography.

The animations on this page have been made using *Maple*.

First we remind some facts about ellipses.

Given two fixed point F1 and F2 in a plane, an ellipse is the locus of all points P in that plane for which the sum of
the distances to F1 and F2 equals a given value *2a*.

The given points F1 and F2 are called the *foci* of the ellipse, PF1 and PF2 are called the *focal radii* of the point P .

Given the value

The property that will be most useful in the further treatment of our subject is illustrated on the next picture. For an elegant geometrical proof click HERE.

This property can be formulated in an other way, using the

The ball is positioned in P1 and shot in the direction of the focus F1. If the ball hits the border of the table in P2 it will be be bounced according to the law of reflection: the angle between the
incoming path and the normal in P2 equals the angle between this normal and the outgoing path.
As a consequence of the property last mentioned the ball will bounce back along P2F2 !

After a new hit the ball will bounce through the other focus F1. In the following illustration we consider a sequence of
consecutive bounces.

After a rather small number of bounces the ball seems to travel along the major axis!

If the first shot doesn't pass over a focus none of all subsequent reflected paths will pass over a focus. But also in this
case the mathematical elliptical billiard game leads to interesting patterns!

** General situation**

In the following animated picture 50 reflections are shown.

** Some exceptions: periodical paths.**

One easily can see that there exist particular situations: in some cases the paths are *periodical* and don't
lead to a new ellipse inside the given one. We give two examples: a *quadrangular* path and a *hexagonal* path.

** The circle as a special ellipse.**

A circle can be considered as a special ellipse: the foci F1 and F2 coincide, *a = b = radius circle ; c = 0*.
If a path is non periodical the evolute of the reflected shots is a new circle with the same centre. Perhaps the existence of periodical paths is more easy to be seen.

** General situation**

The definition of a hyperbola resembles the definition of an ellipse: given two fixed point F1 and F2 in a plane, a hyperbola is the locus of all points P in that plane for which the

In the following animated picture 50 reflections are shown.

Also in this case one can imagine that exceptions exist. An example: if we push the ball along the minor axis of the ellipse it will continuously bounce along the same axis.

*
Herman Serras, July 2007
*

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