The Ptolemy-Alhazen problem with source at infinity

Abstract

We study the well-known Ptolemy-Alhazen problem on reflection of light at the surface of a spherical mirror in the case when the source of light is very far from the mirror.

Figures (8)

• Figure 1: If the unit circle is a mirror, then a light signal from the source $z_1$ to the observer at $z_2$ is reflected at the point $w.$ Moreover, $s_{\mathbb{B}^2}(z_1,z_2)$ is equal to the eccentricity of the maximal ellipse.
• Figure 2: Light rays reach the unit circle from the right side.
• Figure 3: The light ray reflects at the point $w$ on the unit circle and reaches the point $f$.
• Figure 4: Tangential parabola and the four roots of a quartic equation on the unit circle. See Theorem \ref{['claim2']}.
• Figure 5: A light signal is reflected at the point $w$ of the unit circle and reaches the point $a$. It comes from an infinitely distant location in a specific direction.

Theorems & Definitions (10)

• Theorem 2.3
• Corollary 2.6
• Theorem 3.1
• proof
• Theorem 3.4
• proof
• Lemma 4.1
• proof
• Proposition 4.5
• proof