Rational parametrization of conchoids to algebraic curves

Sendra Pons, Juana (2010). Rational parametrization of conchoids to algebraic curves. "Applicable Algebra in Engineering, Communication and Computing", v. 21 (n. 4); pp. 285-308. ISSN 0938-1279. https://doi.org/10.1007/s00200-010-0126-0.

Description

Title: Rational parametrization of conchoids to algebraic curves
Author/s:
  • Sendra Pons, Juana
Item Type: Article
Título de Revista/Publicación: Applicable Algebra in Engineering, Communication and Computing
Date: May 2010
ISSN: 0938-1279
Volume: 21
Subjects:
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

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Abstract

We study the rationality of each of the components of the conchoid to an irreducible algebraic affine plane curve, excluding the trivial cases of the lines through the focus and the circle centered at the focus and radius the distance involved in the conchoid. We prove that conchoids having all their components rational can only be generated by rational curves. Moreover, we show that reducible conchoids to rational curves have always their two components rational. In addition, we prove that the rationality of the conchoid component, to a rational curve, does depend on the base curve and on the focus but not on the distance. As a consequence, we provide an algorithm that analyzes the rationality of all the components of the conchoid and, in the affirmative case, parametrizes them. The algorithm only uses a proper parametrization of the base curve and the focus and, hence, does not require the previous computation of the conchoid. As a corollary, we show that the conchoid to the irreducible conics, with conchoid-focus on the conic, are rational and we give parametrizations. In particular we parametrize the Limaçons of Pascal. We also parametrize the conchoids of Nicomedes. Finally, we show how to find the foci from where the conchoid is rational or with two rational components.

More information

Item ID: 9457
DC Identifier: http://oa.upm.es/9457/
OAI Identifier: oai:oa.upm.es:9457
DOI: 10.1007/s00200-010-0126-0
Official URL: http://www.springerlink.com/content/4u41x42g74345881/
Deposited by: Memoria Investigacion
Deposited on: 14 Nov 2011 12:22
Last Modified: 20 Apr 2016 17:52
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