Using Rolling Circles to Generate Caustic Envelopes Resulting from Reflected Light

TL;DR

This paper introduces a geometric method using rolling circles to generate and analyze caustic envelopes created by reflected light from smooth curves, providing a unified approach for various light sources.

Contribution

It presents a novel geometric construction linking caustic envelopes to rolling circles on associated curves, applicable to different light sources including those at infinity.

Findings

Unified geometric framework for caustic envelopes

Application to classical examples of caustics

Method for all radiants at infinity

Abstract

Given any smooth plane curve {\alpha}(s)representing a mirror that reflects light the usual way and any radiant light source at a point in the plane, the reflected light will produce a caustic envelope. For such an envelope, we show that there is an associated curve \b{eta}(s) and a family of circles C(s) that roll on \b{eta}(s) without slipping such that there is a point on each circle that will trace the caustic envelope as the circles roll. For a given curve {\alpha}(s) and for all radiants at infinity there is a single curve \b{eta}(s) and family of circles C(s) that roll on \b{eta}(s) so that the different points on C(s) will simultaneously trace out, as the circles roll, all caustic envelopes from these radiants at infinity. We explore many classical examples using this method.