Alhazen's Billiard Problem

In a given Circle, find an Isosceles Triangle whose Legs pass through two given Points inside the Circle. This can be restated as: from two Points in the Plane of a Circle, draw Lines meeting at the Point of the Circumference and making equal Angles with the Normal at that Point.

The problem is called the billiard problem because it corresponds to finding the Point on the edge of a circular ``Billiard'' table at which a cue ball at a given Point must be aimed in order to carom once off the edge of the table and strike another ball at a second given Point. The solution leads to a Biquadratic Equation of the form

$$H(x^2-y^2)-2Kxy+(x^2+y^2)(hy-kx)=0.$$

The problem is equivalent to the determination of the point on a spherical mirror where a ray of light will reflect in order to pass from a given source to an observer. It is also equivalent to the problem of finding, given two points and a Circle such that the points are both inside or outside the Circle, the Ellipse whose Foci are the two points and which is tangent to the given Circle.

The problem was first formulated by Ptolemy in 150 AD, and was named after the Arab scholar Alhazen, who discussed it in his work on optics. It was not until 1997 that Neumann proved the problem to be insoluble using a Compass and Ruler construction because the solution requires extraction of a Cube Root. This is the same reason that the Cube Duplication problem is insoluble.

See also Billiards, Billiard Table Problem, Cube Duplication

References

Dörrie, H. ``Alhazen's Billiard Problem.'' §41 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 197-200, 1965.

Hogendijk, J. P. ``Al-Mutaman's Simplified Lemmas for Solving `Alhazen's Problem'.'' From Baghdad to Barcelona/De Bagdad à Barcelona, Vol. I, II (Zaragoza, 1993), pp. 59-101, Anu. Filol. Univ. Barc., XIX B-2, Univ. Barcelona, Barcelona, 1996.

Lohne, J. A. ``Alhazens Spiegelproblem.'' Nordisk Mat. Tidskr. 18, 5-35, 1970.

Neumann, P. Submitted to Amer. Math. Monthly.

Riede, H. ``Reflexion am Kugelspiegel. Oder: das Problem des Alhazen.'' Praxis Math. 31, 65-70, 1989.

Sabra, A. I. ``ibn al-Haytham's Lemmas for Solving `Alhazen's Problem'.'' Arch. Hist. Exact Sci. 26, 299-324, 1982.