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 ${\mathtt{Maple}}$ and can be freely downloaded and used.
Citation: Juan Gerardo Alcázar, Hüsnü Anıl Çoban, Uğur Gözütok. Detecting affine equivalences between certain types of parametric curves, in any dimension[J]. AIMS Mathematics, 2024, 9(6): 13750-13769. doi: 10.3934/math.2024670
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 ${\mathtt{Maple}}$ and can be freely downloaded and used.
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Juan Gerardo Alcázar, Hüsnü Anıl Çoban, Uğur Gözütok. Detecting affine equivalences between certain types of parametric curves, in any dimension[J]. AIMS Mathematics, 2024, 9(6): 13750-13769. doi: 10.3934/math.2024670
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