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<p>We define the notion of <i>approximate affine equivalence</i> for polynomially parametrized curves and trigonometric curves, i.e., curves parametrized by truncated Fourier series, whose coefficients are floating point numbers, therefore known up to finite precision, and provide symbolic-numeric algorithms in any dimension to compute the approximate affine equivalences between such curves. In order to do this, we first develop the notion of approximate equivalence <i>in norm</i>, which leads to concrete algorithms, and then we analyze the relationship between closeness in norm, and closeness in terms of the Hausdorff distance. The algorithms have been implemented in the computer algebra system <Emphasis FontCategory="NonProportional">Maple</Emphasis>; we report on experiments carried out with these algorithms to provide evidence of their efficiency.</p>

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Approximate Affine Equivalences of Polynomial and Trigonometric Curves

  • 期刊论文

  • 发表日期：

    2026年6月15日

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  • Juan Gerardo Alcázar,
  • Carlos Hermoso,
  • Hüsnü Anıl Çoban,
  • Uğur Gözütok

摘要

We define the notion of approximate affine equivalence for polynomially parametrized curves and trigonometric curves, i.e., curves parametrized by truncated Fourier series, whose coefficients are floating point numbers, therefore known up to finite precision, and provide symbolic-numeric algorithms in any dimension to compute the approximate affine equivalences between such curves. In order to do this, we first develop the notion of approximate equivalence in norm, which leads to concrete algorithms, and then we analyze the relationship between closeness in norm, and closeness in terms of the Hausdorff distance. The algorithms have been implemented in the computer algebra system Maple; we report on experiments carried out with these algorithms to provide evidence of their efficiency.