The Ptolemy-Alhazen problem and quadric surface mirror reflection

Abstract

We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as the Alhazen problem. For the spherical mirror problem, we focus on the reflection property of an ellipse, and show that the catacaustic curve of the unit circle follows naturally from the equation obtained from the reflection property of an ellipse. Moreover, we provide an algebraic equation that solves Alhazen's problem for quadric surface mirrors.

Figures (5)

• Figure 1: The catacaustics of the unit circle with radiant points $c=0.5$ (left) and $c=0.8$ (right). The thick (red) curves indicate the catacaustics. The thick (blue) and thin (cyan) dotted circles represent $E_1(c,z)=0$ and $E_2(c,z)=0$, respectively.
• Figure 2: Solution using an ellipse (left). Solution using the circle of Apollonius (right).
• Figure 3: The leaning ellipse indicates $C$. The points $u_3$ and $u_6$ are tangent points of the thick and dotted ellipses with $C$, respectively. The foci of these thick and dotted ellipses are both $-1,1$. Here, the points $u_1$ and $u_5$ are tangent points of $C$ and the hyperbolas with foci $-1,1$.
• Figure 4: The leaning hyperbola indicates $C$. The two points $u_1$ and $u_5$ are tangent points of the dotted and thick ellipses with $C$, respectively. The foci of these dotted and thick ellipses are both $-1, 1$. Moreover, the two points $u_3$ and $u_4$ are tangent points of $C$ and the hyperbolas with foci $-1,1$.
• Figure 5: The level sets of $G=\{\vert \vert z-(-1/2-1/2i)\vert -\vert z -(1-i)\vert \vert <4/5 \}$ (left) and $G=\{\vert z-3/2\vert +\vert z-(-1/3-1/2i)\vert <11/5 \}$ (right). Note that each red curve passes through all edge points of the contour curves.

Theorems & Definitions (20)

• Lemma 2.2: See, for example fhmv
• Proposition 2.5
• proof
• Corollary 2.7
• proof
• Remark 2.9
• Remark 2.12
• Lemma 3.1
• proof
• Proposition 3.2