The conchoid of a plane curve C is constructed using a fixed circle B in the affine plane. We generalize the classical definition so that we obtain a conchoid from any pair of curves B and C in the projective plane. We present two definitions, one purely algebraic through resultants and a more geometric one using an incidence correspondence in P2 X P2. We prove, among other things, that the conchoid of a generic curve of fixed degree is irreducible, we determine its singularities and give a formula for its degree and genus. In the final section we return to the classical case: for any given curve C we give a criterion for its conchoid to be irreducible and we give a procedure to determine when a curve is the conchoid of another.
Conchoidal transform of two plane curves
ALBANO, Alberto;ROGGERO, Margherita
2010-01-01
Abstract
The conchoid of a plane curve C is constructed using a fixed circle B in the affine plane. We generalize the classical definition so that we obtain a conchoid from any pair of curves B and C in the projective plane. We present two definitions, one purely algebraic through resultants and a more geometric one using an incidence correspondence in P2 X P2. We prove, among other things, that the conchoid of a generic curve of fixed degree is irreducible, we determine its singularities and give a formula for its degree and genus. In the final section we return to the classical case: for any given curve C we give a criterion for its conchoid to be irreducible and we give a procedure to determine when a curve is the conchoid of another.File | Dimensione | Formato | |
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