Conchoidal transform of two plane curves

TL;DR

This paper generalizes the classical conchoid construction of plane curves using algebraic and geometric methods, analyzing properties like irreducibility, singularities, degree, and genus for the conchoid of generic and specific curves.

Contribution

It introduces a new general framework for conchoids of plane curves, providing algebraic and geometric definitions, and characterizes their properties including irreducibility and singularities.

Findings

Conchoids of generic curves are irreducible.

The degree and genus of conchoids are explicitly determined.

Criteria for irreducibility and reconstructing original curves from conchoids are established.

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 $\PP^2 \times \PP^2$. 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.