## Phase Space:

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What do these dots represent? Each time the ball hits the wall, we measure the point on the wall at which the ball hits and the angle at which the ball hits. The horizontal axis in the phase space is a measure of the point on the wall that gets hit. The vertical axis records the angle with which the ball bounces off the wall. Thus the two coordinates of the dot give information about where the ball hit the wall and with which angle.

## PERIODIC ORBIT:

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A billiard path is periodic if it repeats itself after some number of bounces. The period of the orbit is the number of bounces it takes until the path repeats. For example, on a square table, a ball that moves vertically upward will be periodic. It will repeat its motion exactly after two bounces. We say its period is two. Also, a ball that is moving horizontally will be periodic with period two.

## CAUSTIC:

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A caustic for a billiard is a curve that is repeatedly grazed by a billiard trajectory. This curve is different from the wall of the table. For the circular billiard, circles inside the original circular table are caustics. Billiard motion inside an ellipse also has caustics but the caustics are no longer circles.