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On caustics by reflection of algebraic surfaces

Alfrederic Josse and Françoise Pène
From the journal Advances in Geometry

Abstract

Given a point S (the light position) in ℙ3 and an algebraic surface 𝓩 (the mirror) of ℙ3, the caustic by reflection ΣS(𝓩) of 𝓩 from S is the Zariski closure of the envelope of the reflected lines 𝓡m obtained by reflection of (Sm) on 𝓩 at m 𝓩. We use the ramification method to identify ΣS(𝓩) with the Zariski closure of the image, by a rational map, of an algebraic 2-covering space of 𝓩. We also give a general formula for the degree (with multiplicity) of caustics (by reflection) of algebraic surfaces of ℙ3.

MSC 2010: 14H50; 14E05; 14N05; 14N10

Communicated by: A. Sommese


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Received: 2014-9-8
Revised: 2015-3-3
Published Online: 2016-10-15
Published in Print: 2016-10-1

© 2016 by Walter de Gruyter Berlin/Boston

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