Alhazen's Billiard Problem
Alexander Zouev 000051 - 060
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
`
a
b
c
b
y A + y B and r is the c 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
-1-
Alexander Zouev 000051 - 060
Table of Contents
Heading Page
Introduction
3
Pre-examination of the problem
4
Initial approach
7
Analysis of specific scenarios
8
Forming a general
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
`
a
b
c
b
y A + y B and r is the c 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
-1-
Alexander Zouev 000051 - 060
Table of Contents
Heading Page
Introduction
3
Pre-examination of the problem
4
Initial approach
7
Analysis of specific scenarios
8
Forming a general
Bibliography: Dörrie 127 Highfield, Roger. “Don Solves the Last Puzzle Left by Ancient Greeks.” Daily Telegraph. April 1, 1997, Issue 676. Henderson, Tom. “Reflection and Its Importance”. The Physics Classroom. Dated 2004. Viewed 12 March 2005. Heinrich Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solutions. Dover Publications New York, 1965. 197-200 8 7