Regarding the problem of circular billiards raised last week, our regular commentator Manuel Amorós provides the following solution:
“Let us consider two balls A and B placed on the diameter of a circular table of radius R. The balls are placed on both sides of the center C (if they are on the same side, the only trivial solution is to throw in the direction of the diameter). Ball A is at a distance a from the center and ball B at a distance b from the center.
Let P be the point where ball A must hit the table. Let α be the angle formed by CP with the diameter:
cos α = R(ba)/(2ab)
The problem only has a non-trivial solution when said cosine is between -1 and 1″.
In the case of the balls placed on a diameter and on both sides of the centre of the table, it is not difficult to find the solution graphically with a ruler and compass (I invite my astute readers to look for it). But if the billiard balls (or the light source and the object to be illuminated in the circular mirror) are not on the same diameter, that is, in the non-simplified Alhazen problem, the construction with a ruler and compass is not possible, and when trying to solve it algebraically we find a fourth-degree equation. And it is not only difficult to solve this equation, but even to formulate it, in contrast to the apparent graphic simplicity of the problem. So much so that, at the end of the 15th century, and after searching in vain for a mathematical solution to the Alhazen problem, Leonardo da Vinci proposed a mechanical solution using an ingenious pantograph designed ad hoc. An algebraic solution was not found until 1997, by the British mathematician Peter M. Neumann.
Bunimovich Stadium and Sinai Billiards
Since we have been talking about conventional rectangular billiards and circular billiards (less conventional but which also exists physically) in the last few weeks, it is inexcusable to talk about the sum of both: Bunimovich billiards, designed by the Russian-American mathematician and physicist Leonid Bunimovich, known for his important contributions to the study of dynamical systems, which is a rectangle limited by semicircles on two opposite sides (due to its shape, it is also known as Bunimovich stadium). It is not a physical billiards, but an ideal one, and the “balls” are point particles with rectilinear and uniform movement that, when colliding with the contour, are reflected specularly without loss of energy (elastic collision), that is, always maintaining the same speed. It is the best known of the “dynamic billiards”, which are idealizations of the game of billiards with different contours and possible internal obstacles. Its purpose is to model different types of particle movements and study Hamiltonian systems (but that is another article).
Another famous dynamic billiards is the one devised by Yakov Sinai, one of the greatest living mathematicians (also Russian-American, like Bunimovich): it is a square with a circle inside, in which particles also bounce in a mirror-like manner, and is used, among other things, to model the behaviour of an ideal gas.
Can you determine where on the square perimeter the particle in the figure will hit after colliding for the fourth time with the central circle of the Sinai billiards? Even more difficult: Where on the circle will it hit for the fifth time? The side of the square measures 29 cm, the diameter of the central circle measures 14 cm, and the numbers in blue indicate the measurements of the corresponding segments.
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