Detecting affine equivalences between certain types of parametric curves, in any dimension


Alcázar J. G., ÇOBAN H. A., Gözütok U.

AIMS Mathematics, vol.9, no.6, pp.13750-13769, 2024 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 9 Issue: 6
  • Publication Date: 2024
  • Doi Number: 10.3934/math.2024670
  • Journal Name: AIMS Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Directory of Open Access Journals
  • Page Numbers: pp.13750-13769
  • Keywords: affine equivalence, algorithm, rational parametrization, symbolic computation, symmetry
  • Karadeniz Technical University Affiliated: Yes

Abstract

Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In this paper we generalized previous results by the authors to provide an algorithm for computing the affine equivalences between two parametric curves of certain types, in any dimension. In more detail, the algorithm is valid for rational curves, and for parametric curves with nonrational but meromorphic components, it admits an also meromorphic, and in fact rational, inverse. Unlike other algorithms already known for rational curves, the algorithm completely avoids polynomial system solving, and instead uses bivariate factoring as a fundamental tool. The algorithm has been implemented in the computer algebra system Maple and can be freely downloaded and used.