# Mathematics (EE): Alhazen's Problem

Alexander Zouev

000051 - 060

Extended Essay – Mathematics

Alhazen’s Billiard Problem

Antwerp International School

May 2007

Word Count: 3017

Abstract

The research question of this Mathematics Extended Essay is, “on a circular table there are two balls; at what point along the circumference must one be aimed at in order for it to strike the other after rebounding off the edge”.  In investigating this question, I first used my own initial approach (which involved measuring various chord lengths), followed by looking at a number of special cases scenarios (for example when both balls are on the diameter, or equidistant from the center) and finally forming a general solution based on coordinate geometry and trigonometric principles.  The investigation included using an idea provided by Heinrich Dorrie and with the use of diagrams and a lengthy mathematical analysis with a large emphasis on trigonometric identities, a solution was found. The conclusion reached is, “if we are given the coordinate plane positions of billiard ball A with coordinates (xA, yA) and billiard ball B with coordinates (xB, yB), and also the radius of the circle, the solution points are at any of the points of intersection of the circular table with the hyperbola,”, where P =, M = , p = ,  m =  and r is the radius.  The solution was verified by considering specific examples through technology such as Autograph software and a TI-84 graphing calculator.  Finally I briefly looked at various other solutions to the problem and also considered further research questions.

Word Count : 234

Table of Contents

Extended Essay – Mathematics

Alhazen’s Billiard Problem

Introduction:

Regarded as one of the classic problems from two dimensional geometry, Alhazen’s Billiard Problem has a truly rich history.  The problem is believed to have been first introduced by Greek astronomer Ptolemy back in 150 AD1 and then eventually noticed by 17th century Arabic mathematician Abu Ali al Hassan ibn Alhaitham (whose name was later Latinized into Alhazen)2.

Alhazen made reference to this problem in one of his published works entitled Optics and presented it in the form, “Find the given point on a spherical mirror at which a ray of light coming from a given point must strike in order to be reflected” 3.  Nowadays, this problem is often referred to as the “Billiard Problem” because it involves locating the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in order to carom (bounce) once off the edge and strike another ball at a second given point.4

The focus question of this extended essay will be:

Heinrich Dörrie also described the problem as “find in a given circle an isosceles triangle whose legs pass through two given points inside the circle”. 5    My primary reason for choosing to investigate this focus question is that the I.B Higher Level Mathematics Programme at our school is at times limited with regards to the study of geometry and trigonometry.  Investigating this problem gave me an opportunity to fill this void.  That being said, the problem was in itself also very appealing to me as I personally enjoy playing billiards or pool and was eager to find out about the mathematics of the game.

The problem appeared in the Daily Telegraph news in 1997 when Dr Peter Nueman, an Oxford don of Queen’s Collage, managed to provide a new solution to the problem.  Inspired by early mathematician Descartes, Nueman cleverly translated the billiards table geometry simply into x and y coordinates on two axes. 6 This is a method I intend to use further into my extended essay.  Please note that this essay (and the solution to the focus question) is narrowed down to emphasize the algebraic solution to Alhazen’s Problem - however in the conclusion, other methods are briefly discussed.

Pre-examination of the problem:

The great difficulty with this investigation lies within two concepts.  First of all, the balls in question are randomly scattered on the table with no specific locations – in other words our solution would need to be generalized for any set of billiard balls.  Second of all, the balls need to be treated as fixed points. To begin this investigation one should first consider where and how many possibilities there can be on a circular pool table that would allow for a ball to strike once off the edge and then hit another ball.  Moreover, what exactly characterizes the direction of a ball bouncing off a circular table border?  The law of reflection states that that the  and  are equal, with each angle being measured from the normal to the boundary (line indicating the border)7.  In figure 1, the incident path θi must have an angle equal to the reflected path θr.

The boundary in our case would be a tangent line drawn to the point on the border of the circle where the ball A bounces off the circular side to ball B (Figure 2).

Another way to express this problem is, “to describe in a given circle an isosceles triangle whose legs pass through two given points made inside the circle” 8.  This is useful because it allows us to relate the billiard balls to chords within the circle.  Observe Figure 3: ball A and ball B are located within the circular billiard table with the table’s center ...