Caustics of light rays and Euler's angle of inclination

TL;DR

This paper explores the optical interpretation of Euler's equations for plane curves, analyzing caustics and evolutes, and introduces new types of curves and a novel pantograph equation related to mirror reflections.

Contribution

It generalizes Euler's classical problem to optical caustics, introduces new curve types, and derives a new pantograph equation with analytic solutions.

Findings

Characterization of curves with constant-angle caustics

Identification of new curve types beyond classical evolutes

Development of a novel pantograph equation for mirror caustics

Abstract

Euler used intrinsic equations expressing the radius of curvature as a function of the angle of inclination to find curves similar to their evolutes. We interpret the evolute of a plane curve optically, as the caustic (envelope) of light rays normal to it, and study the Euler's problem for general caustics. The resulting curves are characterized when the rays are at a constant angle to the curve, generalizing the case of evolutes. Aside from analogs of classical solutions we encounter some new types of curves. We also consider caustics of parallel rays reflected by a curved mirror, where Euler's problem leads to a novel pantograph equation, and describe its analytic solutions.