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On the Shape of Curves that are Rational in Polar Coordinates

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Authors
Alcázar Arribas, Juan GerardoUniversity of Alcalá Author; Díaz Toca, Gema María
Identifiers
Permanent link (URI): http://hdl.handle.net/10017/20444
DOI: 10.1016/j.cagd.2012.09.001
ISSN: 0167-8396
Publisher
Elsevier
Date
2012
Affiliation
Universidad de Alcalá. Departamento de Física y Matemáticas. Unidad docente Matemáticas
Funders
Ministerio de Ciencia e Innovación
Bibliographic citation
Computer Aided Geometric Design, v. 29, n. 9, p. 665-675.
Keywords
Polar coordinates
Planar curve
Spiral
Rational curve
Description / Notes
The final version of this paper was published as Alcázar J.G., Díaz Toca G.M. (2012), On the Shape of Curves that are Rational in Polar Coordinates, Computer Aided Geometric Design, Vol. 29 Issue 9, pp. 665-675.
Project
info:eu-repo/grantAgreement/MICINN//MTM2011-25816-C02-01/ES/ALGORITMOS Y APLICACIONES EN GEOMETRIA DE CURVAS Y SUPERFICIES/
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/submittedVersion
Publisher's version
http://dx.doi.org/10.1016/j.cagd.2012.09.001
Rights
(c) Computer Aided Geometric Design, 2012
Access rights
info:eu-repo/semantics/openAccess
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Abstract
In this paper we provide a computational approach to the shape of curves which are rational in polar coordinates. Our study includes theoretical aspects on the shape of these curves, and algorithmic results which eventually lead to an algorithm for plotting the ``interesting parts" of the curve, i.e. the parts showing the main geometrical features of it. On the theoretical side, we prove that these curves, with the exceptions of lines and circles, cannot be algebraic (in cartesian coordinates), we characterize the existence of infinitely many self-intersections, and we connect this with certain phenomena which are not possible in the algebraic world, namely the existence of limit circles, limit points, or spiral branches. On the practical side, we provide an algorithm which has been implemented in the computer algebra system Maple to visualize this kind of curves. Our implementation makes use (and improves some aspects of) the command "polarplot" currently available in Maple for plotting curves in polar form.
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