A simple formula for the computation of branches and asymptotes of curves and some applications
Identifiers
Permanent link (URI): http://hdl.handle.net/10017/55106DOI: 10.1016/j.cagd.2022.102084
ISSN: 0167-8396
Publisher
Elsevier
Date
2022-03-02Affiliation
Universidad de Alcalá. Departamento de Automática; Universidad de Alcalá. Departamento de Ciencias de la Computación; Universidad de Alcalá. Departamento de Física y Matemáticas. Unidad docente MatemáticasFunders
Agencia Estatal de Investigación
Bibliographic citation
Campo Montalvo, E., Fernández de Sevilla, M. & Pérez Díaz, S. 2022, "A simple formula for the computation of branches and asymptotes of curves and some applications", Computer Aided Geometric Design, vol. 94, art. no. 102084.
Keywords
Parametrization
Curves
Infinity branches
Asymptotes
Perfect curves
Project
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-113192GB-I00/ES/VISUALIZACION MATEMATICA: FUNDAMENTOS, ALGORITMOS Y APLICACIONES/
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/publishedVersion
Publisher's version
https://doi.org/10.1016/j.cagd.2022.102084Rights
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
Access rights
info:eu-repo/semantics/openAccess
Abstract
In this paper, we obtain a simple formula based on the computation of some derivatives for determining the branches and the asymptotes of curves that are defined by a parametrization. For this purpose, we use some previous results and notions presented in Blasco and Pérez-Díaz, 2014a, Blasco and Pérez-Díaz, 2014b, Blasco and Pérez-Díaz, 2015, Blasco and Pérez-Díaz, 2020. From these results, we show how the generalized asymptotes of the input curve can be easily computed and we present some applications related to the ramification index and degree of the asymptote, the infinity form and the multiplicity of the infinity points. Furthermore, we show how to construct all the families of parametric curves having some given asymptotes. We develop this method for the plane case but it can be trivially adapted for dealing with rational curves in n-dimensional space. In addition, the formulaes presented can be similarly obtained for curves defined by a parametrization not necessarily rational.
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