Caustics of Light Rays and Euler's Angle of Inclination

  • Sergiy Koshkin University of Houston-Downtown
  • Ivan Rocha University of Houston-Downtown
Keywords: plane curve; evolute; intrinsic equations; angle of inclination; caustic; delay differential equation; pantograph equation

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.

Published
2020-10-22
How to Cite
Koshkin, S., & Rocha, I. (2020). Caustics of Light Rays and Euler’s Angle of Inclination. The PUMP Journal of Undergraduate Research, 3, 205-225. Retrieved from https://journals.calstate.edu/pump/article/view/2417