Exercise 9: Reflection in different types of tables

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In this exercise, we will discuss a ball's motion in different types of table. Circular stadium, square table, square table with a circular interior wall located in the center and slightly off-center are included.

Intruduction and background

The famous Alhazen problem has to do with a circular billiard and there are two versions of the problem. The first case is to find at the edge of the circular billiard two points B, C such that a billiard ball moving from a given point A inside the circle of the billiard after reflection at B, C passes through the point A again (see Figure 1A). It is obvious that if O is the center of the circle and the points O, A, B, C are collinear then the problem is trivial.
The second case is, given two fixed points A and B inside the circle, to find a point P on the edge of the circular billiard such that the ball moving from A after one reflection at P will pass from B (see Figure 1B). It is obvious again that if the points A, B and O are on a diameter of the circle then the problem is trivial.

We can see billiard competition on TV. There are some incredible scene on the movie. Maybe they are good at geometry┑(￣Д ￣)┍

Content

• Problem 3.31
  Study the behavior for other types of tables. One interesting possibility is a square table with a circular table with a circular interior wall located either in the center, or slightly off-center. Another possibility is an elliptial table.

• Here I use Vpython simulate the motion of ball in square table and elliptical table.
  

  

• Figures
  Figure 1 Motion in square table. We can see the motion is not chaotic.
  
  Figure 2 Motion in circular stadium. There is an area in the center that the ball can't reach.
  
  And this is the phase space plot——$v_x$ versus x
  
  Figure 3 Square table with a elliptial interior wall located in the center
  
  Figure 4 Square table with a elliptial interior wall located slightly off the center.
  
  For code click the link.

Reference

• Debdyuti Banerjee and Nikolaos Dergiades, Debdyuti Banerjee and
  Nikolaos Dergiades,Forum Geometricorum Volume 12 (2012) 193–196.

• Chen Feng, Exercise_10: Chapter 3 problem 3.31: Billiard ball on diverse table