Abstract
In this paper we present two packages, implemented in the computer algebra system Maple, for dealing with offsets and conchoids to algebraic curves, respectively. Help pages and procedures are described. Also in an annex, we provide a brief atlas, created with these packages, and where the offset and the conchoid of several algebraic plane curves are obtained, their rationality is analyzed, and parametrizations are computed. Practical performance of the implemented algorithms shows that the packages execute in reasonable time; we include time cost tables of the computation of the offset and conchoid curves of two rational families of curves using the implemented packages.
Similar content being viewed by others
References
Arrondo, E., Sendra, J., Sendra, J.R.: Parametric generalized offsets to hypersurfaces. J. Symb. Comput. 23(2–3), 267–285 (1997)
Arrondo, E., Sendra, J., Sendra, J.R.: Genus formula for generalized offset curves. J. Pure Appl. Algebra 136(3), 199–209 (1999)
Farouki, R.T., Ne, C.A.: Analytic properties of plane offset curves. Comput. Aided Geom. Des. 7, 83–99 (1990)
Farouki, R.T., Ne, 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 Publisher (1993)
Kerrick, A.H: The limaçon of Pascal as a basis for computed and graphic methods of determining astronomic positions. J. Instit. Navig. 6, 5 (1959)
Menschik, F: The hip joint as a conchoid shape. J. Biomech. 30(9), 971–3 9302622 (1997)
Peternell, M., Gruber, D.: Conchoid surfaces of quadrics. J. Symb. Comput. (2013). doi:http://dx.doi.org/10.1016/j.jsc.2013.07.003
Peternell, M., Gruber, D., Sendra, J.: Conchoid surfaces of rational ruled surfaces. Comp. Aided Geom. Des. 28, 427–435 (2011)
Peternell, M., Gruber, D., Sendra, J.: Conchoid surfaces of spheres. Comp. Aided Geom. Des. 30(1), 35–44 (2013)
Peternell, M., Gotthart, L., Sendra, J., Sendra, J.R.: Offsets, conchoids and pedal surfaces. J. Geom. 106(2), 321–339 (2015)
Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Comput. Aided Geom. Des. 15, 223–249 (1998)
Sendra, J, Sendra, J.R.: Algebraic Analysis of Offsets to Hypersurfaces. Math. Z. 234, 697–719 (2000)
Sendra, J, Sendra, J.R.: Rationality analysis and direct parametrization of generalized offsets to quadrics. AAECC 11, 111–139 (2000)
Sendra, J, Sendra, J.R.: An algebraic analysis of conchoids to algebraic curves. AAECC 19, 413–428 (2008)
Sendra, J, Sendra, J.R.: Rational parametrization of conchoids to algebraic curves. AAECC 21(4), 413–428 (2010)
Sendra, J.R., Sevilla, D.: Radical parametrizations of algebraic curves by adjoint curves. J. Symb. Comput. 46, 1030–1038 (2011)
Sendra, J.R., Sevilla, D.: First steps towards radical parametrization of algebraic surfaces. Comput. Aided Geom. Des. 30(4), 374–388 (2013)
Sultan, A: The Limaçon of Pascal: Mechanical generating fluid processing. J. Mech. Eng. Sci. 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. In: Progress in Electromagnetics Research Symposium, PIER, vol. 30, pp 273–284 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sendra, J., Sánchez-Pascuala, D.G. & Morán, V. Design and implementation of maple packages for processing offsets and conchoids. Ann Math Artif Intell 80, 47–64 (2017). https://doi.org/10.1007/s10472-016-9504-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-016-9504-z
Keywords
- Symbolic mathematical software
- Maple
- Offset variety
- Conchoid variety
- Pedal construction
- Rational parametrization