The Ptolemy–Alhazen Problem and Spherical Mirror Reflection

Abstract

An ancient optics problem of Ptolemy, studied later by Alhazen, is discussed. This problem deals with reflection of light in spherical mirrors. Mathematically this reduces to the solution of a quartic equation, which we solve and analyze using a symbolic computation software. Similar problems have been recently studied in connection with ray-tracing, catadioptric optics, scattering of electromagnetic waves, and mathematical billiards, but we were led to this problem in our study of the so-called triangular ratio metric.2010 Mathematics Subject Classification. 30C20, 30C15, 51M99.

Triangular ratio metric in the unit disk

“…This metric was studied recently in Refs. [10][11][12], and our goal here is to continue this investigation. We introduce new methods for estimating the triangular ratio metric in terms of several other metrics and establish several results with sharp constants.…”

“…This is a very simple task if the domain is, for instance, a half-plane or a polygon, but solving the triangular ratio distance in the unit disk is a complicated problem with a very long history (see Ref. [11]). However, there are two special cases where this problem becomes trivial: if the points x and y in the unit disk are collinear with the origin or at the same distance from the origin, there are explicit formulas for the triangular ratio metric.…”

Barrlund's distance function and quasiconformal maps

“…The equation (1.6) appears also in [18, (1), p. 525] and [23, p.194 line 1]. Note that in [7] we study this topic from a different point of view.…”

“…Now for the case of the unit disk G = D and z 1 , z 2 ∈ D and the extremal point z 0 ∈ ∂D , for the definition (1.2), the connection between the triangular ratio metric Remark 1.8. After the publication of [7], we have learned more about the history of the Ptolemy-Alhazen problem: e.g. the book of A.M. Smith [22] provides a historical account of Alhazen's work on optics.…”

Intrinsic Quasi-Metrics

“…In this section, we will focus on the inequalities between the hyperbolic-type metrics and quasi-metrics in the case of the unit disk. Calculating the exact value of the triangular ratio metric in the unit disk is not a trivial task, but instead quite a difficult problem with a very long history, see [3] for more details. However, we already know from Corollary 4.9 that the quasi-metric w G serves as a lower bound for the triangular ratio metric in convex domains G and this helps us considerably.…”