Abstract
This paper is framed within the problem of analyzing the rationality of the components of two classical geometric constructions, namely the offset and the conchoid to an algebraic plane curve and, in the affirmative case, the actual computation of parametrizations. We recall some of the basic definitions and main properties on offsets (see [13]), and conchoids (see [15]) as well as the algorithms for parametrizing their rational components (see [1] and [16], respectively). Moreover, we implement the basic ideas creating two packages in the computer algebra system Maple to analyze the rationality of conchoids and offset curves, as well as the corresponding help pages. In addition, we present a brief atlas where the offset and conchoids of several algebraic plane curves are obtained, their rationality analyzed, and parametrizations are provided using the created packages.
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Arrondo, E., Sendra, J., Sendra, J.R.: Parametric Generalized Offsets to Hypersurfaces. Journal of Symbolic Computation 23(2–3), 267–285 (1997)
Arrondo, E., Sendra, J., Sendra, J.R.: Genus Formula for Generalized Offset Curves. Journal of Pure and Applied Algebra 136(3), 199–209 (1999)
Farouki, R.T., Neff, C.A.: Analytic Properties of Plane Offset Curves. Comput. Aided Geom. Des. 7, 83–99 (1990)
Farouki, R.T., Neff, C.A.: Algebraic Properties of Plane Offset Curves. Comput. Aided Geom. Des. 7, 100–127 (1990)
Hoffmann, C.M.: Geometric and Solid Modeling. Morgan Kaufmann Publis. (1993)
Kerrick, A.H.: The limaçon of Pascal as a basis for computed and graphic methods of determining astronomic positions. J. of the Instit. of Navigation 6, 5 (1959)
Menschik, F.: The hip joint as a conchoid shape. J. of Biomechanics 30(9) (September 1997) 971-3 9302622
Peternell, M., Gruber, D.: Conchoid surfaces of quadrics. Journal of Symbolic Computation (2013), doi:10.1016/j.jsc, 07.003
Peternell, M., Gruber, D., Sendra, J.: Conchoid surfaces of rational ruled surfaces. Comp. Aided Geom. Design 28, 427–435 (2011)
Peternell, M., Gruber, D., Sendra, J.: Conchoid surfaces of spheres. Comp. Aided Geom. Design 30(1), 35–44 (2013)
Peternell, M., Gotthart, L., Sendra, J., Sendra, J.: The Relation Between Offset and Conchoid Constructions. arXiv:1302.1859v2 [math.AG] (June 10, 2013)
Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Computer Aided Geometric Design 15, 223–249 (1998)
Sendra, J., Sendra, J.R.: Algebraic Analysis of Offsets to Hypersurfaces. Mathematische Zeitschrif 234, 697–719 (2000)
Sendra, J., Sendra, J.R.: Rationality Analysis and Direct Parametrization of Generalized Offsets to Quadrics. Applicable algebra in Engineering, Communication and Computing 11, 111–139 (2000)
Sendra, J., Sendra, J.R.: An Algebraic Analysis of Conchoids to Algebraic Curves. Applicable Algebra in Engineering, Communication and Computing 19, 413–428 (2008)
Sendra, J., Sendra, J.R.: Rational parametrization of conchoids to algebraic curves, Applicable Algebra in Engineering. Communication and Computing 21(4), 413–428 (2010)
Sendra, J.R., Sevilla, D.: Radical Parametrizations of Algebraic Curves by Adjoint Curves. Journal of Symbolic Computation 46, 1030–1038 (2011)
Sendra, J.R., Sevilla, D.: First Steps Towards Radical Parametrization of Algebraic Surfaces. Computer Aided Geometric Design 30(4), 374–388 (2013)
Sultan, A.: The Limaçon of Pascal: Mechanical Generating Fluid Processing. J. of Mechanical Engineering Science 219(8), 813–822 (2005) ISSN.0954-4062
Weigan, L., Yuang, E., Luk, K.M.: Conchoid of Nicomedes and Limaçon of Pascal as Electrode of Static Field and a Wavwguide of High Frecuency Wave. Progres. In: Electromagnetics Research Symposium, PIER, vol. 30, pp. 273–284 (2001)
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Sendra, J., Gómez, D., Morán, V. (2014). Rational Conchoid and Offset Constructions: Algorithms and Implementation. In: Aranda-Corral, G.A., Calmet, J., Martín-Mateos, F.J. (eds) Artificial Intelligence and Symbolic Computation. AISC 2014. Lecture Notes in Computer Science(), vol 8884. Springer, Cham. https://doi.org/10.1007/978-3-319-13770-4_15
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DOI: https://doi.org/10.1007/978-3-319-13770-4_15
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