The first mathemetician who thought to play "billiards in the round", or non-rectangular billiards, was Charles L. Dodgson, a.k.a. Lewis Carroll. Carroll had apparently designed his own game with a set of rules for a 2-person game. He even had a suitable table to play on. His rules specified that the game must have a cushion all around, no pockets, and three spots positioned in an equilateral triangle, where three different colored balls are initially placed. The point values are as follows:
Hitting two balls- 2 Points
Hitting a ball, cushion, then another ball- 3 Points
Hitting a cushion, cushion, ball- 4 Points
Hitting a cushion, ball, cushion, ball-5 Points
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Trajectory after 5 shots On an elliptacle table, the path of the ball shows more surpises. If the ball is shot from one of the foci, it then hits the cushion and then passes over the other focus. Theorectically, if there were no friction, the ball would continue to bounce then pass over a focus. The path of a ball which has bounced 100 times and shot from a focus is shown in the picture below.
If the ball is not hit from either foci, there are two possibilities. One possibility is if the ball were driven so that it does not pass between the two foci. The result after 100 bounces is a new elipse outlined with the same foci as the original table. That situation is shown below.
If the ball is not started at either foci and is driven between the two foci, the ball continues to pass between them after rebounding off the cushion. The resultafter 100 bounces is a hyperbola with the same foci as the original ellptical table. The scenario explained above is shown below.
Playing billiards on an even different type of table, a stadium shaped table, supplies even more surprises. A stadium table is made by replacing the two shorter ends of a rectangle table with half circles. If there is no friction, the trajectory of the ball seems almost random. The more times the ball hits the curved edges, the more chaos. Researches have also studied the effects of the trajectory of billiard balls on peanut, violin, moon, flipper, and droplet-shaped tables. The results are quite interesting.
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