MAT 331, Spring 2000

**Project 0: Billiard in a circle**

*Due Monday, February 21*

Imagine a round room (everything is happening in two-dimensional world!) whose walls are perfect mirrors. Let us shoot out a *laser beam* from a point in this room that we picked at random, in a random direction.

- 1.
- Show how the room looks like after the beam has made 1000 reflections in the mirrors.
- 2.
- If the beam is deadly (as I suppose it goes with lasers), one would like to know if there is a ``safe island'' in the room that the beam won't ever touch. Find the island, and show it on your picture.
- 3.
- Is it possible at all for a beam trajectory to be
*periodic*? If it is, show a couple of those periodic trajectories. How would you describe them? Have you ever encounter a periodic trajectory in your experiments with random beams? What would be your guess about the*probability*of obtaining a periodic trajectory when shooting at random?