An algebraic analysis of conchoids to algebraic curves

Abstract

We study the conchoid to an algebraic affine plane curve ${\mathcal C}$ from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside ${\mathcal C}$ , the notion of conchoid involves a point A in the affine plane (the focus) and a non-zero field element d (the distance). We introduce the formal definition of conchoid by means of incidence diagrams. We prove that the conchoid is a 1-dimensional algebraic set having at most two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to ${\mathcal C}$ , and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve.