Alhazen's problem
{{Short description|Mathematical problem in geometrical optics}}
File:Alhazen-pb.svg can reflect a ray of light from the candle to the observer's eye?]]
Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD.{{Mathworld |title=Alhazen's Billiard Problem |urlname=AlhazensBilliardProblem |mode=cs2 }}
It is named for the 11th-century Arab mathematician Alhazen (Ibn al-Haytham), who presented a geometric solution in his Book of Optics. The algebraic solution involves quartic equations and was found in 1965 by {{ill|Jack M. Elkin|de}}.
Geometric formulation
{{see|Geometrical optics}}
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 (specular reflection). Thus, its main application in optics is to "Find the point on a spherical convex 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.{{MacTutor|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}
Alhazen himself never used this algebraic rewriting of the problem.
Alhazen's solution
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Ibn al-Haytham solved the problem using conic sections and a geometric proof.
Algebraic solution
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.{{citation | last1 = Smith | first1 = John D. | year = 1992 | title = The Remarkable Ibn al-Haytham | journal = The Mathematical Gazette | volume = 76 | issue = 475 | pages = 189–198 | doi=10.2307/3620392 | jstor = 3620392 | s2cid = 118597450 }}{{citation |last1=Drexler |first1=Michael |last2=Gander |first2=Martin J. |title=Circular Billiard |journal=SIAM Review |volume=40 |issue=2 |year=1998 |pages=315–323 |issn=0036-1445 |doi=10.1137/S0036144596310872 |bibcode=1998SIAMR..40..315D |url=https://archive-ouverte.unige.ch/unige:6305}}{{citation |last1=Fujimura |first1=Masayo |last2=Hariri |first2=Parisa |last3=Mocanu |first3=Marcelina |last4=Vuorinen |first4=Matti |title=The Ptolemy–Alhazen Problem and Spherical Mirror Reflection |journal=Computational Methods and Function Theory |volume=19 |issue=1 |year=2018 |pages=135–155 |issn=1617-9447 |doi=10.1007/s40315-018-0257-z |arxiv=1706.06924 |s2cid=119303124}}{{citation |last1=Baker |first1=Marcus |title=Alhazen's Problem |journal=American Journal of Mathematics |volume=4 |issue=1/4 |year=1881 |pages=327–331 |issn=0002-9327 |doi=10.2307/2369168 |jstor=2369168}}{{citation | chapter=Mathematical Origami: Another View of Alhazen's Optical Problem |author-last=Alperin |author-first=Roger | editor-last=Hull | editor-first=Thomas | title=Origami3 | publisher=A K Peters/CRC Press | date=2002-07-18 | isbn=978-0-429-06490-6 | doi=10.1201/b15735 | ref={{sfnref | A K Peters/CRC Press}}}}
An algebraic solution to the problem was finally found first in 1965 by Jack M. Elkin (an actuary), by means of a quartic polynomial.{{Citation |last=Elkin |first=Jack M. |title=A deceptively easy problem |journal=Mathematics Teacher |volume=58 |issue=3 |pages=194–199 |year=1965 |doi=10.5951/MT.58.3.0194 |jstor=27968003}}
Other solutions were rediscovered later:
in 1989, by Harald Riede;{{Citation |last=Riede |first=Harald |title=Reflexion am Kugelspiegel. Oder: das Problem des Alhazen |journal=Praxis der Mathematik |volume=31 |issue=2 |pages=65–70 |year=1989 |language=de}}
in 1990 (submitted in 1988), by Miller and Vegh;{{citation |last1=Miller |first1=Allen R. |last2=Vegh |first2=Emanuel |title=Computing the grazing angle of specular reflection |journal=International Journal of Mathematical Education in Science and Technology |volume=21 |issue=2 |year=1990 |pages=271–274 |issn=0020-739X |doi=10.1080/0020739900210213}}
and also by Jörg Waldvogel.Waldvogel, Jörg. [http://eudml.org/doc/141532 "The Problem of the Circular Billiard"]. Elemente der Mathematik 47.3 (1992): 108–113.
In 1997, the Oxford mathematician Peter M. Neumann proved that there is no ruler-and-compass construction for the general solution of Alhazen's problem{{Citation |last=Neumann |first=Peter M. |author-link=Peter M. Neumann |title=Reflections on Reflection in a Spherical Mirror |journal=American Mathematical Monthly |volume=105 |issue=6 |pages=523–528 |year=1998 |jstor=2589403 |mr=1626185 |doi=10.1080/00029890.1998.12004920}}{{Citation |last=Highfield |first=Roger |date=1 April 1997 |title=Don solves the last puzzle left by ancient Greeks |journal=Electronic Telegraph |volume=676 |url=https://www.telegraph.co.uk/htmlContent.jhtml?html=/archive/1997/04/01/ngre01.html |access-date=2008-09-24 |url-status=dead |archive-url=https://web.archive.org/web/20041123051228/http://www.telegraph.co.uk/htmlContent.jhtml?html=%2Farchive%2F1997%2F04%2F01%2Fngre01.html |archive-date=November 23, 2004 }}
(although in 1965 Elkin had already provided a counterexample to Euclidean construction).
Generalization
Researchers have extended Alhazen's problem to general rotationally symmetric quadric mirrors, including hyperbolic, parabolic and elliptical mirrors.{{Citation |last1=Agrawal |first1=Amit |last2=Taguchi |first2=Yuichi |last3=Ramalingam |first3=Srikumar |year=2011 |title=Beyond Alhazen's Problem: Analytical Projection Model for Non-Central Catadioptric Cameras with Quadric Mirrors |publisher=IEEE Conference on Computer Vision and Pattern Recognition |url=http://www.umiacs.umd.edu/~aagrawal/cvpr11/fp/fp.html |url-status=dead |archive-url=https://web.archive.org/web/20120307040949/http://www.umiacs.umd.edu/~aagrawal/cvpr11/fp/fp.html |archive-date=2012-03-07 }} 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.{{Citation |last1=Agrawal |first1=Amit |last2=Taguchi |first2=Yuichi |last3=Ramalingam |first3=Srikumar |year=2010 |title=Analytical Forward Projection for Axial Non-Central Dioptric and Catadioptric Cameras |publisher=European Conference on Computer Vision |url=http://www.umiacs.umd.edu/~aagrawal/eccv10/fp/fp.html |url-status=dead |archive-url=https://web.archive.org/web/20120307042704/http://www.umiacs.umd.edu/~aagrawal/eccv10/fp/fp.html |archive-date=2012-03-07 }} 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.