2012/10/24 by Alfrederic Josse, Josse, Alfrederic, Françoise Pène +2 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Meromorphic and Entire Functions #math.AG
paper · pdf · doi:10.48550/arxiv.1210.6551
21 pages, 1 figure
openalex publication_date 2012/10/24 · arxiv created 2013/04/14 · arxiv updated 2013/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given any light position S in the complex projective plane P2 and any algebraic curve C of P2 (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.