Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD.[1] 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 Jack M. Elkin [de].
Which point on the surface of the spherical mirror can reflect a ray of light from the candle to the observer's eye?
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 solve the problem, "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.[2][1] ( Alhazen himself never used this algebraic rewriting of the problem)
Alhazen's solutionedit
This section needs expansion. You can help by adding to it. (September 2021)
Ibn al-Haytham solved the problem using conic sections and a geometric proof.
An algebraic solution to the problem was finally found first in 1965 by Jack M. Elkin (an actuarian), by means of a quartic polynomial.[8] Other solutions were rediscovered later: in 1989, by Harald Riede;[9] in 1990 (submitted in 1988), by Miller and Vegh;[10] and in 1992, by John D. Smith[3] and also by Jörg Waldvogel.[11]
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.[14] 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.[15] 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.[15]
^ abcSmith, John D. (1992), "The Remarkable Ibn al-Haytham", The Mathematical Gazette, 76 (475): 189–198, doi:10.2307/3620392, JSTOR 3620392, S2CID 118597450
^Drexler, Michael; Gander, Martin J. (1998), "Circular Billiard", SIAM Review, 40 (2): 315–323, doi:10.1137/S0036144596310872, ISSN 0036-1445
^Fujimura, Masayo; Hariri, Parisa; Mocanu, Marcelina; Vuorinen, Matti (2018), "The Ptolemy–Alhazen Problem and Spherical Mirror Reflection", Computational Methods and Function Theory, 19 (1): 135–155, arXiv:1706.06924, doi:10.1007/s40315-018-0257-z, ISSN 1617-9447, S2CID 119303124
^Baker, Marcus (1881), "Alhazen's Problem", American Journal of Mathematics, 4 (1/4): 327–331, doi:10.2307/2369168, ISSN 0002-9327, JSTOR 2369168
^Alperin, Roger (2002-07-18), "Mathematical Origami: Another View of Alhazen's Optical Problem", in Hull, Thomas (ed.), Origami^{3}, A K Peters/CRC Press, doi:10.1201/b15735, ISBN978-0-429-06490-6
^Elkin, Jack M. (1965), "A deceptively easy problem", Mathematics Teacher, 58 (3): 194–199, doi:10.5951/MT.58.3.0194, JSTOR 27968003
^Riede, Harald (1989), "Reflexion am Kugelspiegel. Oder: das Problem des Alhazen", Praxis der Mathematik (in German), 31 (2): 65–70
^Miller, Allen R.; Vegh, Emanuel (1990), "Computing the grazing angle of specular reflection", International Journal of Mathematical Education in Science and Technology, 21 (2): 271–274, doi:10.1080/0020739900210213, ISSN 0020-739X
^Waldvogel, Jörg. "The Problem of the Circular Billiard.." Elemente der Mathematik 47.3 (1992): 108-113. [1]
^Highfield, Roger (1 April 1997), , Electronic Telegraph, 676, archived from the original on November 23, 2004, retrieved 2008-09-24
^Agrawal, Amit; Taguchi, Yuichi; Ramalingam, Srikumar (2011), , IEEE Conference on Computer Vision and Pattern Recognition, archived from the original on 2012-03-07
^ abAgrawal, Amit; Taguchi, Yuichi; Ramalingam, Srikumar (2010), , European Conference on Computer Vision, archived from the original on 2012-03-07
December 15, 2023
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Alhazen s problem also known as Alhazen s billiard problem is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD 1 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 Jack M Elkin de Which point on the surface of the spherical mirror can reflect a ray of light from the candle to the observer s eye Contents 1 Geometric formulation 2 Alhazen s solution 3 Algebraic solution 4 Generalization 5 ReferencesGeometric formulation editFurther information 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 solve the problem 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 2 1 Alhazen himself never used this algebraic rewriting of the problem Alhazen s solution editThis section needs expansion You can help by adding to it September 2021 Ibn al Haytham solved the problem using conic sections and a geometric proof Algebraic solution editLater mathematicians such as Christiaan Huygens James Gregory Guillaume de l Hopital 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 3 4 5 6 7 An algebraic solution to the problem was finally found first in 1965 by Jack M Elkin an actuarian by means of a quartic polynomial 8 Other solutions were rediscovered later in 1989 by Harald Riede 9 in 1990 submitted in 1988 by Miller and Vegh 10 and in 1992 by John D Smith 3 and also by Jorg Waldvogel 11 In 1997 the Oxford mathematician Peter M Neumann proved there is no ruler and compass construction for the general solution of Alhazen s problem 12 13 although in 1965 Elkin had already provided a counterexample to Euclidean construction 3 Generalization editRecently Mitsubishi Electric Research Labs researchers solved the extension of Alhazen s problem to general rotationally symmetric quadric mirrors including hyperbolic parabolic and elliptical mirrors 14 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 15 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 15 References edit a b Weisstein Eric W Alhazen s Billiard Problem MathWorld O Connor John J Robertson Edmund F Abu Ali al Hasan ibn al Haytham MacTutor History of Mathematics Archive University of St Andrews a b c Smith John D 1992 The Remarkable Ibn al Haytham The Mathematical Gazette 76 475 189 198 doi 10 2307 3620392 JSTOR 3620392 S2CID 118597450 Drexler Michael Gander Martin J 1998 Circular Billiard SIAM Review 40 2 315 323 doi 10 1137 S0036144596310872 ISSN 0036 1445 Fujimura Masayo Hariri Parisa Mocanu Marcelina Vuorinen Matti 2018 The Ptolemy Alhazen Problem and Spherical Mirror Reflection Computational Methods and Function Theory 19 1 135 155 arXiv 1706 06924 doi 10 1007 s40315 018 0257 z ISSN 1617 9447 S2CID 119303124 Baker Marcus 1881 Alhazen s Problem American Journal of Mathematics 4 1 4 327 331 doi 10 2307 2369168 ISSN 0002 9327 JSTOR 2369168 Alperin Roger 2002 07 18 Mathematical Origami Another View of Alhazen s Optical Problem in Hull Thomas ed Origami 3 A K Peters CRC Press doi 10 1201 b15735 ISBN 978 0 429 06490 6 Elkin Jack M 1965 A deceptively easy problem Mathematics Teacher 58 3 194 199 doi 10 5951 MT 58 3 0194 JSTOR 27968003 Riede Harald 1989 Reflexion am Kugelspiegel Oder das Problem des Alhazen Praxis der Mathematik in German 31 2 65 70 Miller Allen R Vegh Emanuel 1990 Computing the grazing angle of specular reflection International Journal of Mathematical Education in Science and Technology 21 2 271 274 doi 10 1080 0020739900210213 ISSN 0020 739X Waldvogel Jorg The Problem of the Circular Billiard Elemente der Mathematik 47 3 1992 108 113 1 Neumann Peter M 1998 Reflections on Reflection in a Spherical Mirror American Mathematical Monthly 105 6 523 528 doi 10 1080 00029890 1998 12004920 JSTOR 2589403 MR 1626185 Highfield Roger 1 April 1997 Don solves the last puzzle left by ancient Greeks Electronic Telegraph 676 archived from the original on November 23 2004 retrieved 2008 09 24 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 archived from the original on 2012 03 07 a b Agrawal Amit Taguchi Yuichi Ramalingam Srikumar 2010 Analytical Forward Projection for Axial Non Central Dioptric and Catadioptric Cameras European Conference on Computer Vision archived from the original on 2012 03 07 Retrieved from https en wikipedia org w index php title Alhazen 27s problem amp oldid 1163207102, wikipedia, wiki, book, books, library,
