Using Rolling Circles to Generate Caustic Envelopes Resulting from Reflected Light

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.