An Algebraic Analysis of Conchoids to Algebraic Curves

TL;DR

This paper provides an algebraic geometric analysis of conchoids to algebraic curves, defining their properties, components, and relationships to initial curves, with implications for curve characterization.

Contribution

It introduces a formal algebraic definition of conchoids, analyzes their components, and establishes criteria for identifying conchoids of other curves.

Findings

Conchoids are algebraic curves with at most two irreducible components, except for specific circles.

Most conchoids have at least one simple component, except for lines through the focus.

Simple components are birationally equivalent to the original curve in reducible cases.

Abstract

We study conchoids to algebraic curve from the perspective of algebraic geometry, analyzing their main algebraic properties. We introduce the formal definition of conchoid of an algebraic curve by means of incidence diagrams. We prove that, with the exception of a circle centered at the focus and taking $d$ as its radius, the conchoid is an algebraic curve having at most two irreducible components. In addition, we introduce the notions of special and simple components of a conchoid. Moreover we state that, with the exception of lines passing through the focus, 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 the initial curve, and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve.