Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in

geometrical optics Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of '' rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...

first formulated by

Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...

in 150 AD. 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 equation In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is :ax^4+bx^3+cx^2+dx+e=0 \, where ''a'' ≠ 0. The quartic is the highest order polynomi ...

s and was found in 1965 by .

Geometric formulation

The problem comprises drawing lines from two points, meeting at a third point on the

circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...

of a circle and making equal angles with the

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

at that point (

specular reflection Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface. The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...

). 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 ''Ray of Light'' is the seventh studio album by American singer-songwriter Madonna, released in early 1998 by Maverick Records. A stylistic and aesthetical departure from her previous work, ''Ray of Light'' is an electronica and techno-pop reco ...

coming from a given point must strike in order to be reflected to another point." This leads to an equation of the fourth degree. ( Alhazen himself never used this algebraic rewriting of the problem)

Alhazen's solution

Ibn al-Haytham solved the problem using

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...

s and a geometric proof.

Algebraic solution

Later mathematicians such as Christiaan Huygens, James Gregory,

Guillaume de l'Hôpital Guillaume François Antoine, Marquis de l'Hôpital (; sometimes spelled L'Hospital; 1661 – 2 February 1704), also known as Guillaume-François-Antoine Marquis de l'Hôpital, Marquis de Sainte-Mesme, Comte d'Entremont, and Seigneur d'Ouques-la- ...

, 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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s. An algebraic solution to the problem was finally found first in 1965 by Jack M. Elkin (an actuarian), by means of a

quartic polynomial In algebra, a quartic function is a function (mathematics), function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of Degree of a polynomial, degree four, called a quartic polynomial. A ''qua ...

. Other solutions were rediscovered later: in 1989, by Harald Riede; in 1990 (submitted in 1988), by Miller and Vegh; and in 1992, by John D. Smith and also by Jörg Waldvogel. In 1997, the

Oxford Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to the ...

mathematician Peter M. Neumann proved there is no

ruler-and-compass construction In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

for the general solution of Alhazen's problem (although in 1965 Elkin had already provided a counterexample to Euclidean construction).

Generalization

Recently, Mitsubishi Electric Research Labs researchers solved the extension of Alhazen's problem to general rotationally symmetric

quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...

mirrors including hyperbolic, parabolic and elliptical mirrors. 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 6 is a number, numeral, and glyph. 6 or six may also refer to: * AD 6, the sixth year of the AD era * 6 BC, the sixth year before the AD era * The month of June Science * Carbon, the element with atomic number 6 * 6 Hebe, an asteroid People ...

. Alhazen's problem can also be extended to multiple

refractions In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...

from a spherical ball. Given a light source and a spherical ball of certain

refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

, 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.

References

{{reflist, 2 Mathematical problems Mathematics in the medieval Islamic world Geometrical optics Ibn al-Haytham