Alhazens billiard problem extended essay

If we extend the path that ball A must take to the opposite side of the circle, we have a chord — the same can be done for ball B. This analytical solution just described is only one of many methods known. 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.

Viewed Aug 02 The zeros of the functions should give us the x-values of the solution. Custom essays uk review Texas State of Arizona. T corresponding parts of congruent triangles are congruent Alexander Zouev — This explains why instead of looking at how one ball must be struck in order for it to strike the other after rebounding off the edge, we can look for an inscribed isosceles triangle whose legs pass through ball A and ball B.

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The lengths of the chords are measured and recorded in a table below. According to mathematics historian Victor J. Essay my hobby in english uniforms Relationship in my family essay video Privacy rights essay celebrities Led research paper Alhazens billiard problem extended essay my shyness essay dollar, tasks for essay writing format.

Dover Publications New York, Everyone in my school hand to hand in their Extended Essays in October. Research papers in english pdf management. Regarded as one of the classic problems from two dimensional geometry.

Figure 12 Indeed both methods give the same result, and using Autograph 3 we can clearly see how the hyperbola intersects the circle in only two places. There is another scenario where we arguably only have 2 solutions.

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Figure 4 12 1 2 11 A 3 10 C 4 B 9 5 8 7 6 Alexander Zouev — Chords are drawn going through ball A to each of the 12 points, and the same for ball B. However that being said, if ball A was aimed at point 1, then point 1 would also be a solution. It answers our focus question and works for any randomly located and infinitely small billiard balls.

They emphasize the fact that for any given ellipse, if a ball is placed in each focus then any point on the rim would be a solution such is the nature of the foci points.

Alhazen’s Billiard Problem Paper

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On-time Delivery Our time management skills are superb. Thus, its main application in optics is to solve the problem, "Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.

The great difficulty with this investigation lies within two concepts. I know that the solution points must be on the circumference of the circle they satisfy the equation of the circle. Before you download the paper, you can review the file and send it for revision, if there are any mistakes or inconsistencies you would like the writer to correct.

But, please, do not entrust your academic reputation to a random agency. Figure 2 B C A? Consider Figure 4, here I have randomly chosen two points to be my locations for ball A and ball B. Do you study Law or Medicine?

Tell us what you need to have done now! However there appears to be an apparent paradox as although our results suggest that there is a solution between the points 11 and 12 1 Figure 5 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.

In other words for ball A, mathematically points 1, 2, 3 and 4 are all solutions, but realistically only points 1, 2, and 4 are solutions because ball B would block the path of ball A before it can reach point 3.

Given a light source and a spherical ball of certain refractive index, the closest point on the spherical ball where the light is refracted to the eye of the observer can be obtained by solving a tenth degree equation.

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Alhazen’s Billiard Problem

Alhazen’s Billiard Problems Describe an isosceles triangle whose legs pass through two given points inside a circle. c O P 1 P 2 This problem comes from the Arabic mathematician Abu Ali al Hassan ibn al Hassan ibn.

ALHAZEN’S OPTICAL PROBLEM Roger C. Alperin 1. Fields and Constructions cular billiard problem considered also in [8]. One can use inversion in the given circle to convert one formulation to the other.

Yet another We restate Alhazens’s problem in terms of. The problem is called the billiard problem because it corresponds to finding 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 once off the edge of the table.

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Alhazens billiard problem solving any kind of outline which the greatest solution to stretch the neo colonial. Main parts of all, theories and or speech and style, are not successfully meet during their abortion.

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Alhazens billiard problem extended essay
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