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
Author/s:
  • Sendra Pons, J. Rafael
  • Sendra Pons, Juana
Item Type: Article
Título de Revista/Publicación: Applicable Algebra in Engineering Communication and Computing
Date: October 2008
ISSN: 0938-1279
Volume: 19
Subjects:
Freetext Keywords: Algebraic Curves, analysis, curve C.
Faculty: E.U.I.T. Telecomunicación (UPM)
Department: Matemática Aplicada a la Ingeniería Técnica de Telecomunicación [hasta 2014]
Creative Commons Licenses: Recognition - No derivative works - Non commercial

Full text

[img]
Preview
PDF - Requires a PDF viewer, such as GSview, Xpdf or Adobe Acrobat Reader
Download (1MB) | Preview

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.

More information

Item ID: 2948
DC Identifier: http://oa.upm.es/2948/
OAI Identifier: oai:oa.upm.es:2948
DOI: 10.1007/s00200-008-0081-1
Official URL: http://www.springerlink.com/content/100499/
Deposited by: Memoria Investigacion
Deposited on: 03 May 2010 12:35
Last Modified: 20 Apr 2016 12:33
  • Logo InvestigaM (UPM)
  • Logo GEOUP4
  • Logo Open Access
  • Open Access
  • Logo Sherpa/Romeo
    Check whether the anglo-saxon journal in which you have published an article allows you to also publish it under open access.
  • Logo Dulcinea
    Check whether the spanish journal in which you have published an article allows you to also publish it under open access.
  • Logo de Recolecta
  • Logo del Observatorio I+D+i UPM
  • Logo de OpenCourseWare UPM