Determining the asymptotic family of an implicit curve
type: Item
instance of: scholarly article
title: Determining the asymptotic family of an implicit curve
zbMATH Open document ID: 1506.14116
author: E. Campo-Montalvo
author: M. Fernández de Sevilla
author: R. Magdalena Benedicto
author: Sonia Pérez Díaz
published in: Computer Aided Geometric Design
publication date: 2022-10-04T00:00:00Z
review text: This paper considers the problem of determining the asymptotic family of a plane algebraic curve \(\mathcal C\) which is assumed to have only regular points at infinity. The asymptotic family of \(\mathcal C\) is the set of algebraic plane curves that have the same asymptotic behaviour as \(\mathcal C\). The main result states that these curves form the vector space of all algebraic plane curves of the degree \(d = \deg \mathcal C\) whose defining equations share the homogeneous forms of degree \(d\) and of degree \(d-1\). This generalizes results of [\textit{A. Blasco} and \textit{S. Pérez-Díaz}, Comput. Aided Geom. Des. 31, No. 2, 81--96 (2014; Zbl 1301.14033)] for curves with only one regular infinity branch. The characterization of asymptotic families simplifies the computation of generalized asymptotes of \(\mathcal C\) (plane algebraic curves of minimal degree, one for each infinity branch, whose distance to \(\mathcal C\) converges to zero as the curves tend to infinity) as it is possible to ignore the irrelevant coefficients of the low-degree homogeneous forms. It also allows to compute in a straightforward way an asymptotic family from its set of regular \(g\)-asymptotes by simply multiplying the \(g\)-asymptotes' equations and extracting the two homogeneous forms of highest degree. This paper is self-contained with precise definitions of central concepts, collections of previous results, and examples illustrating results and theories. Proofs are collected in a concluding section. An obvious question for future research is the generalization of the results to the case of curves with non-regular points at infinity.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 7596221
zbMATH Keywords: implicit algebraic plane curve
zbMATH Keywords: parametric plane curve
zbMATH Keywords: infinity branches
zbMATH Keywords: asymptotes
zbMATH Keywords: perfect curves
zbMATH Keywords: approaching curves
Wikidata QID: Q114202250
MaRDI profile type: Publication
full work available at URL: j.cagd.2022.102146
OpenAlex ID: W4293833214
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DOI: 10.1016/J.CAGD.2022.102146
description: scientific article; zbMATH DE number 7596221