On the class of caustics by reflection

Alfrederic Josse (1) , Françoise Pene (2)

1 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique
2 LM - Laboratoire de mathématiques de Brest

Given any light position S in the complex projective plane P^2 and any algebraic curve C of P^2 (with any kind of singularities), we consider the incident lines coming from S (i.e. the lines containing S) and their reflected lines after reflection off the mirror curve C. The caustic by reflection is the Zariski clusure of the envelope of these reflected lines. We introduce the notion of reflected polar curve and express the class of the caustic by reflection in terms of intersection numbers of C with the reflected polar curve, thanks to a fundamental lemma established in [14]. This approach enables us to get an explicit formula for the class of the caustic by reflection in every case in terms of intersection numbers of the initial curve C.

Mots clés: caustic; class; polar; intersection number; pro-branch; Plucker formula

Domaines: Géométrie algébrique [math.AG]

Alfrederic Josse, Françoise Pene. On the class of caustics by reflection. 2013. ⟨hal-00744994v2⟩