On the Degree of Caustics by Reflection

Given a point S ∈ ?²: = ?²(?) and an irreducible algebraic curve 𝒞 of ?² (with any type of singularities), we consider the lines ? _m obtained by reflection of the lines (S m) on 𝒞 (for m ∈ 𝒞). The caustic by reflection Σ _S (𝒞) is classically defined as the Zariski closure of the envelope of the reflected lines ? _m . We identify this caustic with the Zariski closure of Φ(𝒞), where Φ is some rational map. We use this approach to give general and explicit formulas for the degree (with multiplicity) of caustics by reflection. Our formulas are expressed in terms of intersection numbers of the initial curve 𝒞 (or of its branches). Our method is based on a fundamental lemma for rational map thanks to the notion of Φ-polar and on the computation of intersection numbers. In particular, we use precise estimates related to the intersection numbers of 𝒞 with its polar at any point and to the intersection numbers of 𝒞 with its Hessian curve. These computations are linked with generalized Plücker formulas for the class and for the number of inflection points of 𝒞.