On caustics by reflection of algebraic surfaces

TL;DR

This paper studies the geometric properties of caustics formed by reflecting light off algebraic surfaces in projective space, providing a formula for their degree and a method to identify their structure.

Contribution

It introduces a ramification method to characterize caustics by reflection and derives a general degree formula for these caustics of algebraic surfaces.

Findings

Caustics are identified as the Zariski closure of the image of a rational map.

A general formula for the degree of caustics is established.

The method applies to algebraic surfaces in projective three-space.

Abstract

Given a point S (the light position) in P^3 and an algebraic surface Z (the mirror) of P^3, the caustic by reflection of Z from S is the Zariski closure of the envelope of the reflected lines got by reflection of the incident lines (Sm) on Z at m in Z. We use the ramification method to identify the caustic by reflection with the Zariski closure of the image, by a rational map, of an algebraic 2-covering space of Z. We also give a general formula for the degree (with multiplicity) of caustics (by reflection) of algebraic surfaces.