Curves That Must Be Retraced

Authors Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo

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Xiaoyang Gu
Jack H. Lutz
Elvira Mayordomo

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Xiaoyang Gu, Jack H. Lutz, and Elvira Mayordomo. Curves That Must Be Retraced. In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, pp. 149-160, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


We exhibit a polynomial time computable plane curve ${\bf \Gamma}$ that has finite length, does not intersect itself, and is smooth except at one endpoint, but has the following property. For every computable parametrization $f$ of ${\bf\Gamma}$ and every positive integer $m$, there is some positive-length subcurve of ${\bf\Gamma}$ that $f$ retraces at least $m$ times. In contrast, every computable curve of finite length that does not intersect itself has a constant-speed (hence non-retracing) parametrization that is computable relative to the halting problem.
  • Computable analysis
  • computable curve
  • computational complexity
  • Hausdorff measure
  • rectifiable curve


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