On the class of caustics by reflection

TL;DR

This paper provides a new algebraic approach to compute the caustic by reflection for any algebraic curve in the projective plane, expressing its class explicitly via intersection numbers.

Contribution

It introduces the reflected polar curve concept and derives an explicit formula for the caustic's class based on intersection theory.

Findings

Explicit formula for caustic class in terms of intersection numbers

General method applicable to curves with singularities

Fundamental lemma linking reflected polar and caustic class

Abstract

Given any light position S in the complex projective plane P^2 and any algebraic curve C of P^2 (with any kind of singularities), we consider the incident lines coming from S (i.e. the lines containing S) and their reflected lines after reflection off the mirror curve C. The caustic by reflection is the Zariski clusure of the envelope of these reflected lines. We introduce the notion of reflected polar curve and express the class of the caustic by reflection in terms of intersection numbers of C with the reflected polar curve, thanks to a fundamental lemma established in [14]. This approach enables us to get an explicit formula for the class of the caustic by reflection in every case in terms of intersection numbers of the initial curve C.