Alhazen's problem

The problem comprises drawing lines from two points, meeting at a third point on the circumference of a circle and making equal angles with the normal at that point. Thus, its main application in optics is to solve the problem, "Find the point on a spherical concave mirror at which a ray of light coming from a given point must strike in order to be reflected to another point." This leads to an equation of the fourth degree.[2][1]

Ibn al-Haytham solved the problem using conic sections and a geometric proof. He derived a formula for the sum of fourth powers, where previously only the formulas for the sums of squares and cubes had been stated.

His method can be readily generalized to find the formula for the sum of any integral powers, although he did not himself do this (perhaps because he only needed the fourth power to calculate the volume of the paraboloid he was interested in). He used his result on sums of integral powers to perform what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.[3]

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.[4][5][6][7][8]

An algebraic solution to the problem was finally found first in 1965 by Jack M. Elkin, an actuarian.[9] Other solutions were rediscovered later: in 1989, by Harald Riede;[10] in 1990 (submitted in 1988), by Miller and Vegh;[11] and in 1992, by John D. Smith[4] and also by Jörg Waldvogel[12]

In 1997, the Oxford mathematician Peter M. Neumann proved the theorem that there is no ruler-and-compass construction for the general solution of Alhazen's problem[13][14] (although in 1965 Elkin had already provided a counterexample to Euclidean construction).[4]

Recently, Mitsubishi Electric Research Labs researchers solved the extension of Alhazen's problem to general rotationally symmetric quadric mirrors including hyperbolic, parabolic and elliptical mirrors.[15] 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.[16] 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.[16]