Abstract
In this paper we present two algorithms for computing the g-asymptotes or generalized asymptotes, of a plane algebraic curve, \(\mathscr {C}\), implicitly or parametrically defined. The asymptotes of a curve \(\mathscr {C}\) reflect the status of \(\mathscr {C}\) at points with sufficiently large coordinates. It is well known that an asymptote of a curve \(\mathscr {C}\) is a line such that the distance between \(\mathscr {C}\) and the line approaches zero as they tend to infinity. However, a curve \(\mathscr {C}\) may have more general curves than lines describing the status of \(\mathscr {C}\) at infinity. These curves are known as g-asymptotes or generalized asymptotes. The pseudocodes of these algorithms are presented, as well as the corresponding implementations. For this purpose, we use the algebra software Maple. A comparative analysis of the algorithms is carried out, based on some properties of the input curves and their results to analyze the efficiency of the algorithms and to establish comparative criteria. The results presented in this paper are a starting point to generalize this study to surfaces or to curves defined by a non-rational parametrization, as well as to improve the efficiency of the algorithms. Additionally, the methods developed can provide a new and different approach in prediction (regression) or classification algorithms in the machine learning field.
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References
Blasco, A., Pérez-Díaz, S.: Asymptotes and perfect curves. Computer Aided Geometric Design 31(2), 81–96 (2014). https://doi.org/10.1016/j.cagd.2013.12.004
Blasco, A., Pérez-Díaz, S.: Asymptotic behavior of an implicit algebraic plane curve. Computer Aided Geometric Design 31(7), 345–357 (2014). https://doi.org/10.1016/j.cagd.2014.04.002
Blasco, A., Pérez-Díaz, S.: Asymptotes of space curves. Journal of Computational and Applied Mathematics 278, 231–247 (2015). https://doi.org/10.1016/j.cam.2014.10.013
Blasco, A., Pérez-Díaz, S.: A new approach for computing the asymptotes of a parametric curve. Journal of Computational and Applied Mathematics 364, 112350–118 (2020). https://doi.org/10.1016/j.cam.2019.112350
Bradley, E., Stolle, R.: Automatic construction of accurate models of physical systems. Annals of Mathematics and Artificial Intelligence 17, 1–28 (1996). https://doi.org/10.1007/BF02284622
Fernández de Sevilla, M., Magdalena Benedicto, R., Pérez-Díaz, S.: Computing branches and asymptotes of meromorphic functions.Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A Matemáiticas. 117(109) (2023). https://doi.org/10.1007/s13398-023-01441-7
Campo-Montalvo, E., Fernández de Sevilla, M., Pérez-Díaz, S.: A simple formula for the computation of branches and asymptotes of curves and some applications. Computer Aided Geometric Design 94, 102084 (2022). https://doi.org/10.1016/j.cagd.2022.102084
Cheever, E.A., Li, Y.: A tool for construction of bode diagrams from piecewise linear asymptotic approximations. International Journal of Engineering Education 21(2), 335–340 (2005)
Jain, A.K., Mao, J., Mohiuddin, K.M.: Artificial neural networks: A tutorial. Computer 29(3), 31–44 (1996). https://doi.org/10.1109/2.485891
Kečkić, J.D.: A method for obtaining asymptotes of some curves. The Theaching of Mathematics III(1), 53–59 (2000)
LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521, 436–444 (2015). https://doi.org/10.1038/nature14539
Maxwell, E.A.: An Analytical Calculus, vol. 3. Cambridge University Press, Cambridge, United Kingdom (1962)
Pérez-Díaz, S.: On the problem of proper reparametrization for rational curves and surfaces. Computer Aided Geometric Design 23(4), 307–323 (2006). https://doi.org/10.1016/j.cagd.2006.01.001
Salas, S.L., Etgen, G.J., Hilles, E.: Calculus. One and Several Variables. John Wiley and Sons Inc, New Jersey, NJ, USA (1995)
Sendra, J.R., Winkler, F., Pérez-Díaz, S.: Rational Algebraic Curves: A Computer Algebra Approach. Algorithms and Computation in Mathematics, Springer, Berlin, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73725-4
Zeng, G.: Computing the asymptotes for a real plane algebraic curve. Journal of Algebra 316(2), 680–705 (2007). https://doi.org/10.1016/j.jalgebra.2007.03.030
Acknowledgements
The author S. Pérez-Díaz is partially supported by Ministerio de Ciencia, Innovación y Universidades - Agencia Estatal de Investigación/PID2020-113192GB-I00 (Mathematical Visualization: Foundations, Algorithms and Applications). The author R. Magdalena Benedicto is partially supported by the State Plan for Scientific and Technical Research and Innovation of the Spanish MCI (PID2021-127946OB-I00)and by the European Union, Digital Europe Program 21-22 Call Cloud Data and TEF (CitCom.ai 101100728). The author S. Pérez-Díaz belongs to the Research Group ASYNACS (Ref. CCEE2011/R34).
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Ministerio de Ciencia, Innovación y Universidades - Agencia Estatal de Investigación/PID2020-113192GB-I00 (Mathematical Visualization: Foundations, Algorithms and Applications).
State Plan for Scientific and Technical Research and Innovation of the Spanish MCI (PID2021-127946OB-I00) and by the European Union, Digital Europe Program 21-22 Call Cloud Data and TEF (CitCom.ai 101100728).
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Fernández de Sevilla, M., Magdalena Benedicto, R. & Pérez-Díaz, S. Design and implementation of symbolic algorithms for the computation of generalized asymptotes. Ann Math Artif Intell 91, 537–561 (2023). https://doi.org/10.1007/s10472-023-09856-z
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DOI: https://doi.org/10.1007/s10472-023-09856-z
Keywords
- Algebraic curves
- Infinity branches
- Convergent branches
- Approaching curves
- Generalized asymptotes
- Algorithm performance
- Hardware usage