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Phase Space: |
| Polygon |
Ellipse |
Circle |
Stadium |
| 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.
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PERIODIC ORBIT: |
| Polygon |
Ellipse |
Circle |
Stadium |
| 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. |
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CAUSTIC: |
| Polygon |
Ellipse |
Circle |
Stadium |
| 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. |
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