1. On the degree of caustics of reflection
- Author
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Josse, Alfrederic, Pene, Françoise, Laboratoire de Mathématiques de Brest, Laboratoire de mathématiques de Brest (LM), Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)-Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), and Pene, Francoise
- Subjects
Mathematics - Algebraic Geometry ,MSC : 14N05, 14N10, 14H50, 14E05 ,[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG] ,FOS: Mathematics ,14N05, 14N10, 14H50, 14E05 ,intersection number ,polar ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,pro-branch ,Plücker formula ,Algebraic Geometry (math.AG) ,caustic ,degree - Abstract
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
- Published
- 2011