On the degree of caustics of reflection

Given a point S and any irreducible algebraic curve C in P^2 (with any type of singularities), we consider the caustic of reflection defined as the Zariski closure of the envelope of the reflected lines from the point S on the curve C. We identify this caustic with the Zariski closure of the image of C by a rational map. Thanks to a general fundamental lemma, we give a formula for the degree of the caustic of reflection in terms of multiplicity numbers of C (or of its branches). Our formula holds in every case. We also give some precisions about Pl\"ucker formulas.
Comment: 35 pages, 1 figure