Conchoidal transform of two plane curves

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 P ² × P ². 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.