Alhazen's hyperbolic billiard problem (Q1950950)

Given two points \(A\) and \(B\) inside a circle \(\Omega\), al-Hazen's problem asks about the location of a point \(C\) on \(\Omega\) so that the reflection of a light ray from \(A\) to \(C\) passes through \(B\). Thinking of \(\Omega\) as a circular billiard table, and of \(A\) and \(B\) as balls, finding \(C\) is equivalent to finding the way one must strike \(A\) so that it strikes \(B\) after rebounding from the cushion. The problem is also equivalent to finding \(C\) so that the chords drawn from \(C\) through \(A\) and \(B\) are equal, and also to finding \(C\) whose distances from \(A\) and \(B\) have a minimal (or maximal) sum. This problem appears as Problem 41 in \textit{H. Dörrie}'s book [100 great problems of elementary mathematics. Their history and solution. New York: Dover Publications, Inc. (1965; Zbl 0496.00001)]. It is attributed to the Arab mathematician and physicist al-Hassan ibn al-Haytham (ca. 965 -- ca. 1039), whose first name was latinized into al-Hazen. The problem leads to a fourth degree equation, and its solution point \(C\) can be realized as the intersection of a circle and a hyperbola. Interestingly, the problem led ibn al-Haytham to discover a closed formula for the sum \(1^4 + 2^4 + \cdots + n^4\) (in the same way the quadrature of the parabola led Archimedes to find a closed formula for \(1^2 + 2^2 + \cdots + n^2\)). For a general location of \(A\) and \(B\), the point \(C\) is not constructible by a straightedge and compass. However, it is shown by \textit{R. C. Alperin} in [Am. Math. Mon. 112, No. 3, 200--211 (2005; Zbl 1085.51022)] that \(C\) is always origami constructible. The authors of the paper under review consider the hyperbolic exact analogue of this problem. They prove that there is a homeomorphism between \(\Omega\) and the hyperbolic disc \(\Omega'\) that takes a pair \((A,B)\) for which \(C\) is constructible by straightedge and compasses to a pair \((A',B')\) for which \(C'\) is constructible by hyperbolic straightedge and compasses. Such pairs form a small subset of \(\Omega' \times \Omega'\) -- precisely, a set of measure zero.