Caustics of plane curves, their birationality and matrix projections

TL;DR

This paper investigates the properties of caustics of plane curves, establishing their birationality under general conditions, and extends these results to curves in the space of symmetric matrices with implications for matrix projections.

Contribution

It proves the birationality of the caustic map for general source points and extends the birationality result to curves in the space of symmetric matrices, identifying special cases.

Findings

The caustic map is birational for a general source point S.

The projection B --- BS is birational on certain curves D in the symmetric matrix space.

Special cases where D lies in a specific plane are exceptions to birationality.

Abstract

After recalling the notion of caustics of plane curves and basic equations, we first show the birationality of the caustic map for a general source point S in the plane. Then we prove more generally a theorem for curves D in the projective space of 3x3 symmetric matrices B. For a general 3x1 vector S the projection to the plane given by B --- BS is birational on D, unless D is not a line and D is contained in a plane of the form Delta_v = {B | Bv = 0}.