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 vector 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 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. The problem is insoluble
using a compass and ruler construction
because the solution requires extraction of a cube root
(Elkin 1965, Reide 1989, Neumann 1998). This is the same reason that the cube
duplication problem is insoluble.
SEE ALSO: Billiards,
Cube
Duplication
REFERENCES:
Alperin, R. C. "Mathematical Origami: Another View of Alhazen's Optical
Problem." https://www.math.sjsu.edu/~alperin/Alhazen.pdf.
Alperin, R. C. "A Mathematical Theory of Origami Constructions and Numbers."
New York J. Math. 6, 119-133, 2000.
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.
Elkin, J. M. "A Deceptively Easy Problem." Math. Teacher 58,
194-199, 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.
Hungerbühler, N. "Geometrical Aspects of the Circular Billiard." Elem.
Math. 47, 114-117, 1992.
Huygens, C. Oeuvres Complètes, Vol. 6. The Hague, Netherlands:
M. Nijhoff, p. 462 and unnumbered facing page, 1895. https://gallica.bnf.fr/ark:/12148/bpt6k778547.
Huygens, C. Oeuvres Complètes, Vol. 7. The Hague, Netherlands:
M. Nijhoff, pp. 164-165 and 187-189, 1897. https://gallica.bnf.fr/ark:/12148/bpt6k77855k.
Lohne, J. A. "Alhazens Spiegelproblem." Nordisk Mat. Tidskr. 18,
5-35, 1970.
Neumann, P. M. "Reflections on Reflection in a Spherical Mirror."
Amer. Math. Monthly 105, 523-528, 1998.
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.
Waldvogel, J. "The Problem of the Circular Billiard." Elem. Math. 47,
108-113, 1992.
Referenced on Wolfram|Alpha:
Alhazen's Billiard Problem
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