Curves That Must Be Retraced

Xiaoyang Gu; Jack H. Lutz; Elvira Mayordomo

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arXiv · 0802.4312

Curves That Must Be Retraced

Xiaoyang Gu·Jack H. Lutz·Elvira Mayordomo

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Abstract

We exhibit a polynomial time computable plane curve 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 GAMMA and every positive integer n, there is some positive-length subcurve of GAMMA that f retraces at least n 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.

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Xiaoyang Gu, Jack H. Lutz, Elvira Mayordomo. 2009-05-19. Curves That Must Be Retraced. https://arxiv.org/abs/0802.4312

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