Dynamical Billiards
- Author:
- Anson Ho
- Topic:
- Conic Sections, Ellipse, Geometry, Hyperbola, Reflection
Dynamical Billiards in an Ellipse
This demonstration shows what would happen if you were to play billiards with one ball, on an elliptical table, and without energy loss. The initial trajectory can be adjusted to get different kinds of paths, some of which are periodic. The option is also given to change the eccentricity of the ellipse, down to the special case of the circle.
After many internal reflections (here 50), we can sometimes see a conic section that is either an ellipse or a hyperbola inside the ellipse. This is both confocal (sharing the same foci) and conformal (angle-preserving) to the elliptical "table". This conic section that we see is tangential to all of the paths of the billiard ball after each reflection and is called a caustic - an option is provided to hide or show this.
If the billiard ball passes through either one of the foci of the elliptical table, it necessarily passes through the other ellipse after reflection (this can be understood in terms of Fermat's principle, and the definition of the ellipse in terms of the distance from the two foci). This process happens forever, and the path of the ball converges towards the horizontal axis. This is difficult to see in this simulation because it is difficult to get the billiard ball to pass exactly through either focus point, although the effect can be seen to some extent.