Extended Essay – Mathematics Alhazen’s Billiard Problem
Antwerp International School May 2007
Word Count: 3017 -0-
Alexander Zouev 000051 - 060
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, x 2 @ y 2 P + r 2 ` yp @ xm a + xy2M ”, where P b c b c b c
= y A A xB + yB A x A , M = y A A yB @ x A A xB , p = x A + xB , m = b c
y A + y B 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
Alexander Zouev 000051 - 060
Table of Contents
Pre-examination of the problem
Analysis of specific scenarios
Forming a general solution
Verification of solution
Other possible solutions
Alexander Zouev 000051 - 060 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:
On a circular billiards 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?
Heinrich Dörrie also described the problem as “find in a given circle an isosceles Jack Klaff, “The World May be Divided into Two Types of People – Alhazen’s Billiard Problem.” Viewed 19 February 2005. Heinrich Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solutions. Dover Publications New York, 1965. 197-200 3 4 2 1
Eric W Weisstein, “Alhazen’s Billiard Problem”. Mathworld. Dated 1999. Viewed February 25 2006.
Alexander Zouev 000051 - 060 triangle whose legs pass through two given points inside the circle”. 5 My primary