Degree and class of caustics by reflection for a generic source

Abstract

Abstract. Given any irreducible algebraic (mirror) curve C ⊆ P 2 := P 2 (C) and any (light position) S ∈ P 2 , the caustic by reflection ΣS(C) of C from S is the Zariski closure of the envelope of the reflected lines got from the lines coming from S after reflection on C. In [7,8], we established formulas for the degree and class (with multiplicity) of ΣS(C) for any C and any S. In this paper, we prove the birationality of the caustic map for a generic S in P 2 . Moreover, we give simple formulas for the degree and class (without multiplicity) of ΣS(C) for any C and for a generic S in P 2 .

On the Halphen transform of algebraic space curves

“…Consider a rational curve parametrized by a morphism γ : P 1 → P 3 . Due to (8), C Q is the image of the rational map ψ : P 1 P 3 given by ψ C,Q :…”

“…ψ 1 , ψ 2 , ψ 3 , ψ 4 ) for the four coordinates of α given by (13) (resp. of Ψ given by (8)). Recall that Q(m) = t m · M · m for some symmetric matrix M = (m i,j ) i,j .…”

Caustics of Plane Curves, Their Birationality and Matrix Projections

“…At the moment of writing up the references for the present article, I became aware, by searching on the arXiv, that they have written an independent and different proof of birationality of the caustic map for general source, in [JP13].…”

On caustics by reflection of algebraic surfaces

“…Let us also mention the work of Bruce, Giblin and Gibson [3,1,2] in the real case. A precise computation of the degree and class of caustics by reflection of planar algebraic curves has been done in [9,10,11]. The idea was based on the fact that the caustic by reflection of an irreducible algebraic curve C of P 2 from source S 0 ∈ P 2 is the Zariski closure of the image of C by a rational map.…”