# Mathematics (EE): Alhazen's Problem

Extracts from this document...

Introduction

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

Heading | Page |

Introduction | 3 |

Pre-examination of the problem | 4 |

Initial approach | 7 |

Analysis of specific scenarios | 8 |

Forming a general solution | 10 |

Solution | 15 |

Verification of solution | 16 |

Other possible solutions | 18 |

Further investigation | 20 |

Bibliography | 21 |

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.

Middle

11

12

chord a (cm)

9.9

10.1

11

11.8

12.3

12.1

11.6

10.6

10

9.8

11.8

10.7

chord b (cm)

12.2

12.4

12.2

11.8

11.4

11.3

11.8

12.2

11.5

11.2

11.4

11.8

chord a - chord b (cm)

-2.3

-2.3

-2.2

0

0.9

0.8

-0.2

-1.6

-1.5

-1.4

0.4

-1.1

Solution

* * * *

Chords are drawn going through ball A to each of the 12 points, and the same for ball B. The lengths of the chords are measured and recorded in a table (below). The solutions to where ball A must hit to bounce off and hit ball B can possibly be found by looking at where the corresponding chords are equal to one another, in other words where chord a – chord b = 0. By making a table showing chord a – chord b we could perhaps find possible solutions.

From looking at the changing “chord a – chord b” we can see that solutions should be at point 4, next to point 7, between points 10 and 11, and between points 11 and 12. However there appears to be an apparent paradox as although our results suggest that there is a solution between the points 11 and 12 and also between the points 6 and 7 on the circumference, by looking at the graph one can see that these chords leading to the points are in fact the same chords and the points would therefore definitely not work as solutions (unless these chords are in fact the diameter, as we will see in the following example).

Analysis of specific scenarios:

Let us analyze another more specific scenario. In figure 6, ball A and ball B both lie on the diameter, and are equidistant from the centre of the circle at point C. Possible solutions can be found at points 1, 2, 3 and 4 as shown on the diagram. In this case we have four places where we can strike one of the balls so that it rebounds and hits the other.

Conclusion

Further Investigation:

Although Alhazen’s Problem might have already been solved and analyzed in a numerous amount of creative ways, one cannot help but wonder how many extended study questions can be proposed relating to Alhazen’s Problem. For instance, what would happen if the table were triangular, or hexagonal? Perhaps one could try to interpret Alhazen’s Problem not as two points on a two-dimensional circle, but as two points in a three-dimensional sphere.

Bibliography

Alperin, Roger C. Trisections and Totally Real Origami. MAA Monthly 2005.

Viewed August 05 2006.

< http://www.math.sjsu.edu/%7Ealperin/TRFin.pdf>

Drexler, M Gander, M. Circular Billiard. SIREV, 1998 volume 40 issue 2. 315 – 323. SIREV Journal 1998.

Dörrie, Heinrich. 100 Great Problems of Elementary Mathematics: Their History and Solutions. Dover Publications New York, 1965.

Guimarães, L. C. ; Bellemain, F. . Reflections on the Problema Alhazeni. The 10th

International Congress on Mathematical Education, 2004. Aug 02 2006.

<www.descartes.ajusco.upn.mx/varios/tsg10/articulos/Bellemain_49_revised_paper.doc>

Henderson, Tom. “Reflection and Its Importance”. The Physics Classroom. Dated 2004. Viewed 12 March 2005. <http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1c.html>

Highfield, Roger. “Don Solves the Last Puzzle Left by Ancient Greeks.” Daily _ Telegraph. April 1, 1997, Issue 676.

Klaff, Jack. “The World May be Divided into Two Types of People – Alhazen’s Billiard Problem.” Viewed 19 February 2005. <http://www.jackflaff.com/hos.htm>

Klingens, Dick. “Reflectie Binnen een Cirkelvorminge Spiegel”. Wisfaq!. Dated 1 October 2003. Viewed February 25 2005. http://www.wisfaq.nl/showrecord3.asp?id=14816

Sabra, Abdelhamid I. 4 Points of Reflection Applet, “Alhazen’s Applet”. Viewed August 20 2006. Harvard University.

<http://www.people.fas.harvard.edu/~sabra/applets/>

Weisstein, Eric W. “Alhazen’s Billiard Problem”. Mathworld. Dated 1999. Viewed February 25 2006. <http://mathworld.wolfram.com/AlhazensBilliardProblem.html>

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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