On the degree of caustics of reflection

Alfrederic Josse (1) , Françoise Pene (2)

1 Laboratoire de Mathématiques de Brest
2 LM - Laboratoire de mathématiques de Brest

Given a point S and an irreducible algebraic curve C in P^2, we consider the caustic of reflection defined as 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 of the degree of the caustic of reflection in terms of multiplicity numbers of pro-branches of C. Our formula holds in the most general case. We also give some precisions about Plücker formulas.

Mots clés: caustic; degree; polar; intersection number; pro-branch; Plücker formula

Domaines: Géométrie algébrique [math.AG]

Alfrederic Josse, Françoise Pene. On the degree of caustics of reflection. 2011. ⟨hal-00652633v2⟩