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Alhazen's problem
Alhazen's problem is a mathematical problem in optics concerning reflection in a spherical mirror. It asks for the point in the mirror where one given point reflects to another. The special case of a concave spherical mirror is also known as Alhazen's billiard problem, as it can be formulated equivalently as constructing a reflected path from one billiard ball to another on a circular billiard table. Other equivalent formulations ask for the shortest path from one point to the other that touches the circle, or for an ellipse that is tangent to the circle and has the given points as its foci.
Although special cases of this problem were studied by Ptolemy in the 2nd century CE, it is named for the 11th-century Arab mathematician Alhazen (Hasan Ibn al-Haytham), who formulated it more generally and presented a solution in his Book of Optics. It has no straightedge and compass construction; instead, al-Haytham and others including Christiaan Huygens found solutions involving the intersection of conic sections. According to Roberto Marcolongo, Leonardo da Vinci invented a mechanical device to solve the problem. Later mathematicians, starting with Jack M. Elkin in 1965, solved the problem algebraically as the solution to a quartic equation, and used this equation to prove the impossibility of solving the problem with straightedge and compass.
21st-century researchers have extended this problem and the methods used to solve it to mirrors of other shapes and to non-Euclidean geometry, and have applied fast computational methods for its solution to modeling light reflection off the lakes of Titan.
The problem comprises drawing lines from two points, meeting at a third point on the circumference (boundary) of a circle and making equal angles with the normal at that point (specular reflection). It belongs to geometrical optics (in which light is modeled using rays rather than waves or particles), and catoptrics, the use of mirrors to control light: it can be used to find the path of a ray of light that starts at one point of space, is reflected from a spherical mirror, and passes through a second point. Although this is a three-dimensional problem, it can immediately be reduced to the two-dimensional problem of reflection in a circular mirror in the plane, because its solution lies entirely within the plane formed by the two points and the center of the sphere.
The same problem can be formulated with the two given points inside the circle instead of outside. For two points near each other within the circle, in general position, there will be two solutions, but points that are farther apart have four solutions. Any solution describes the path of a billiards ball reflected within a circular billiards table, as Lewis Carroll once suggested for billiards play. If the two segments of the reflection path are extended to chords of the circle, the two chords make equal angles to the circle and therefore have equal length. Thus, these chords form the two equal sides of an isosceles triangle inscribed within the circle, with the two given points on two sides of this triangle. Another equivalent form of Alhazen's problem asks to construct a triangle with these properties.
Another way of describing the problem, for points inside or outside the circle, is that it seeks an ellipse having the two given points as its foci, tangent to the given circle. The point of tangency is the solution point to Alhazen's problem. A ray from one focus of the ellipse to this point of tangency will be reflected by the ellipse to the other focus (). Furthermore, because the given circle has the same angle at the point of tangency, it will reflect the same ray in the same way. In an ellipse, all single-reflection paths from one focus to another have equal lengths, so the smallest ellipse tangent to the circle produces the shortest path from one given point to the circle and then to the other point. The idea that light rays follow shortest paths is Hero's principle, later reformulated in Fermat's principle that light rays follow quickest paths. More generally, as James Gregory observed, for any analogous three-dimensional reflection problem, the point of reflection is also a point of tangency of an ellipsoid having the source and destination of the reflected ray as its foci.
Ptolemy included the problem of reflection in a circular mirror in his Optics (written in the second century CE), but was only able to solve certain special cases; al-Haytham formulated and solved the problem more generally. Al-Haytham was inspired by Ptolemy's work, and modeled his own book on Ptolemy's, but differed from it in important ways; for instance, Ptolemy used a model of visual perception in which visual rays travel outward from the eye to the objects it sees, while al-Haytham reversed this to the still-used model in which light rays travel inward from objects to the eye.
By the time of Pappus of Alexandria, in the 4th century AD, Greek mathematicians had categorized geometric solutions into three types: straightedge and compass constructions, constructions using conic sections, and neusis constructions involving a marked ruler, preferring the earlier categories of solution over the later ones. Ibn al-Haytham's solution is of the second type, using hyperbola, through which he develops a neusis construction. In his 1881 survey of the problem, Marcus Baker calls al-Haytham's solution "excessively prolix and intricate", and quotes Isaac Barrow as expressing a similar opinion. Later in the 11th century, Yusuf al-Mu'taman ibn Hud, a king of the Taifa of Zaragoza in Spain, simplified al-Haytham's lemmas somewhat, but did not make a significant advance on the problem. The work of al-Haytham became known in the rest of Europe through manuscript Latin translations in the 12th or 13th century, and a translation was published in Basel in 1572. Later geometric solutions by Christiaan Huygens, René-François de Sluse, and Guillaume de l'Hôpital used the same idea of an auxiliary conic section: a hyperbola for Huygens, a parabola for Sluse, and both methods for l'Hôpital. Baker cites Huygens's solution as "the most elegant the problem has ever received".
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Alhazen's problem
Alhazen's problem is a mathematical problem in optics concerning reflection in a spherical mirror. It asks for the point in the mirror where one given point reflects to another. The special case of a concave spherical mirror is also known as Alhazen's billiard problem, as it can be formulated equivalently as constructing a reflected path from one billiard ball to another on a circular billiard table. Other equivalent formulations ask for the shortest path from one point to the other that touches the circle, or for an ellipse that is tangent to the circle and has the given points as its foci.
Although special cases of this problem were studied by Ptolemy in the 2nd century CE, it is named for the 11th-century Arab mathematician Alhazen (Hasan Ibn al-Haytham), who formulated it more generally and presented a solution in his Book of Optics. It has no straightedge and compass construction; instead, al-Haytham and others including Christiaan Huygens found solutions involving the intersection of conic sections. According to Roberto Marcolongo, Leonardo da Vinci invented a mechanical device to solve the problem. Later mathematicians, starting with Jack M. Elkin in 1965, solved the problem algebraically as the solution to a quartic equation, and used this equation to prove the impossibility of solving the problem with straightedge and compass.
21st-century researchers have extended this problem and the methods used to solve it to mirrors of other shapes and to non-Euclidean geometry, and have applied fast computational methods for its solution to modeling light reflection off the lakes of Titan.
The problem comprises drawing lines from two points, meeting at a third point on the circumference (boundary) of a circle and making equal angles with the normal at that point (specular reflection). It belongs to geometrical optics (in which light is modeled using rays rather than waves or particles), and catoptrics, the use of mirrors to control light: it can be used to find the path of a ray of light that starts at one point of space, is reflected from a spherical mirror, and passes through a second point. Although this is a three-dimensional problem, it can immediately be reduced to the two-dimensional problem of reflection in a circular mirror in the plane, because its solution lies entirely within the plane formed by the two points and the center of the sphere.
The same problem can be formulated with the two given points inside the circle instead of outside. For two points near each other within the circle, in general position, there will be two solutions, but points that are farther apart have four solutions. Any solution describes the path of a billiards ball reflected within a circular billiards table, as Lewis Carroll once suggested for billiards play. If the two segments of the reflection path are extended to chords of the circle, the two chords make equal angles to the circle and therefore have equal length. Thus, these chords form the two equal sides of an isosceles triangle inscribed within the circle, with the two given points on two sides of this triangle. Another equivalent form of Alhazen's problem asks to construct a triangle with these properties.
Another way of describing the problem, for points inside or outside the circle, is that it seeks an ellipse having the two given points as its foci, tangent to the given circle. The point of tangency is the solution point to Alhazen's problem. A ray from one focus of the ellipse to this point of tangency will be reflected by the ellipse to the other focus (). Furthermore, because the given circle has the same angle at the point of tangency, it will reflect the same ray in the same way. In an ellipse, all single-reflection paths from one focus to another have equal lengths, so the smallest ellipse tangent to the circle produces the shortest path from one given point to the circle and then to the other point. The idea that light rays follow shortest paths is Hero's principle, later reformulated in Fermat's principle that light rays follow quickest paths. More generally, as James Gregory observed, for any analogous three-dimensional reflection problem, the point of reflection is also a point of tangency of an ellipsoid having the source and destination of the reflected ray as its foci.
Ptolemy included the problem of reflection in a circular mirror in his Optics (written in the second century CE), but was only able to solve certain special cases; al-Haytham formulated and solved the problem more generally. Al-Haytham was inspired by Ptolemy's work, and modeled his own book on Ptolemy's, but differed from it in important ways; for instance, Ptolemy used a model of visual perception in which visual rays travel outward from the eye to the objects it sees, while al-Haytham reversed this to the still-used model in which light rays travel inward from objects to the eye.
By the time of Pappus of Alexandria, in the 4th century AD, Greek mathematicians had categorized geometric solutions into three types: straightedge and compass constructions, constructions using conic sections, and neusis constructions involving a marked ruler, preferring the earlier categories of solution over the later ones. Ibn al-Haytham's solution is of the second type, using hyperbola, through which he develops a neusis construction. In his 1881 survey of the problem, Marcus Baker calls al-Haytham's solution "excessively prolix and intricate", and quotes Isaac Barrow as expressing a similar opinion. Later in the 11th century, Yusuf al-Mu'taman ibn Hud, a king of the Taifa of Zaragoza in Spain, simplified al-Haytham's lemmas somewhat, but did not make a significant advance on the problem. The work of al-Haytham became known in the rest of Europe through manuscript Latin translations in the 12th or 13th century, and a translation was published in Basel in 1572. Later geometric solutions by Christiaan Huygens, René-François de Sluse, and Guillaume de l'Hôpital used the same idea of an auxiliary conic section: a hyperbola for Huygens, a parabola for Sluse, and both methods for l'Hôpital. Baker cites Huygens's solution as "the most elegant the problem has ever received".
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