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Alhazen's problem

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Ibn al-Haytham's work on catoptrics in Book V of the Book of Optics contains the important Islamic mathematical problem known as Alhazen's problem.

Geometric formulationEdit

The problem comprises drawing lines from two points in the plane of a circle meeting at a point on the circumference and making equal angles with the normal at that point. This is equivalent 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 canon off the edge of the table and hit another ball at a second given point. 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." This leads to an equation of the fourth degree.[1][2][3]

Sums of powers and integral calculusEdit

This eventually led Ibn al-Haytham to derive the earliest formula for the sum of the fourth powers, and using an early proof by mathematical induction, he developed a method for determining the general formula for the sum of any integral powers. He used his result on sums of integral powers to perform an integration, in order to find the volume of a paraboloid. He was thus able to find the integrals for polynomials up to the fourth degree, and came close to finding a general formula for the integrals of any polynomials. This was fundamental to the development of infinitesimal and integral calculus.[4]

He also contributed to calculus with the following geometric infinite series:[5]

 \frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots  for |x|<1 [6]


Ibn al-Haytham solved the problem using conic sections and a geometric proof, but Alhazen's problem remained influential in Europe, when later mathematicians such as Christiaan Huygens, James Gregory, Guillaume de l'Hôpital, Isaac Barrow, and many others, attempted to find an algebraic solution to the problem, using various methods, including analytic methods of geometry and derivation by complex numbers.[7]

Over the centuries, Ibn al-Haytham's contributions to integral calculus showed up in the work of various mathematicians. According to mathematics historian Victor J. Katz: "Ibn al-Haytham's formula for the sum of fourth powers shows up in other places in the Islamic world over the next few centuries. It appears in the work of Abu-l-Hasan ibn Haydur (d. 1413), who lived in what is now Morocco, and in the work of Abu Abdallah ibn Ghazi (1437-1514), who also lived in Morocco. Furthermore, one also finds the formula in The Calculator's Key of Ghiyath al-Din Jamshid al-Kashi (d. 1429), a mathematician and astronomer whose most productive years were spent in Samarkand, now in Uzbekistan, in the court of Ulugh Beg." Ibn al-Haytham's equation also later showed up in Indian mathematics, specifically the Kerala school of astronomy and mathematics in the 16th century.[4]

An algebraic solution to Alhazen's problem was finally found in 1997 by the Oxford mathematician Peter M. Neumann.[8] Recently, Mitsubishi Electric Research Labs (MERL) researchers Amit Agrawal, Yuichi Taguchi and Srikumar Ramalingam solved the extension of Alhazen's problem to general rotationally symmetric quadric mirrors including hyperbolic, parabolic and elliptical mirrors .[9] They showed that the mirror reflection point can be computed by solving an eighth degree equation in the most general case. If the camera (eye) is placed on the axis of the mirror, the degree of the equation reduces to six .[10] Alhazen's problem can also be extended to multiple refractions from a spherical ball. 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.[10]

References Edit

  1. O'Connor, John J.; Robertson, Edmund F., "Abu Ali al-Hasan ibn al-Haytham", MacTutor History of Mathematics archive, University of St Andrews,
  2. MacKay, R. J.; Oldford, R. W. (August 2000), "Scientific Method, Statistical Method and the Speed of Light", Statistical Science 15 (3): 254–78
  3. Weisstein, Eric. "Alhazen's Billiard Problem". Mathworld. Retrieved on 2008-09-24.
  4. 4.0 4.1 Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine68 (3): 163–174 [165-9 & 173-4]
  5. Edwards, C. H., Jr. 1979. The Historical Development of the Calculus. New York: Springer-Verlag.
  6. "On the Use of Series in Hindu Mathematics" (1936). Osiris 1: 606–628. doi:10.1086/368443. 
  7. John D. Smith (1992), "The Remarkable Ibn al-Haytham", The Mathematical Gazette 76 (475): 189–198
  8. Highfield, Roger (1 April 1997), "Don solves the last puzzle left by ancient Greeks", Electronic Telegraph 676, archived from the original on 2004-11-23,, retrieved 2008-09-24
  9. Agrawal, Amit; Taguchi, Yuichi; Ramalingam, Srikumar (2011), Beyond Alhazen's Problem: Analytical Projection Model for Non-Central Catadioptric Cameras with Quadric Mirrors, IEEE Conference on Computer Vision and Pattern Recognition,
  10. 10.0 10.1 Agrawal, Amit; Taguchi, Yuichi; Ramalingam, Srikumar (2010), Analytical Forward Projection for Axial Non-Central Dioptric and Catadioptric Cameras, European Conference on Computer Vision,

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