# An Algebraic Analysis of Conchoids to Algebraic Curves

Sendra Pons, J. Rafael and Sendra Pons, Juana (2008). An Algebraic Analysis of Conchoids to Algebraic Curves. "Applicable Algebra in Engineering Communication and Computing", v. 19 (n. 5); pp. 413-428. ISSN 0938-1279. https://doi.org/10.1007/s00200-008-0081-1.

## Description

Title: An Algebraic Analysis of Conchoids to Algebraic Curves Sendra Pons, J. Rafael Sendra Pons, Juana Article Applicable Algebra in Engineering Communication and Computing October 2008 0938-1279 19 Algebraic Curves, analysis, curve C. E.U.I.T. Telecomunicación (UPM) Matemática Aplicada a la Ingeniería Técnica de Telecomunicación [hasta 2014] Recognition - No derivative works - Non commercial

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## Abstract

We study the conchoid to an algebraic affine plane curve C from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside C, the notion of conchoid involves a point A in the affine plane (the focus) and a nonzero 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 atmost 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 C, and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve.

Item ID: 2948 https://oa.upm.es/2948/ oai:oa.upm.es:2948 10.1007/s00200-008-0081-1 http://www.springerlink.com/content/100499/ Memoria Investigacion 03 May 2010 12:35 20 Apr 2016 12:33

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