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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.
- Show how the room looks like after the beam has made 1000 reflections in the mirrors.
- 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.
- 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?