When quoting this document, please refer to the following
DOI: 10.4230/OASIcs.CCA.2009.2267
URN: urn:nbn:de:0030-drops-22674
URL: http://drops.dagstuhl.de/opus/volltexte/2009/2267/
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Gu, Xiaoyang ; Lutz, Jack H. ; Mayordomo, Elvira
Contributed Papers

### Curves That Must Be Retraced

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### Abstract

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.

### BibTeX - Entry

@InProceedings{gu_et_al:OASIcs:2009:2267,
author =	{Xiaoyang Gu and Jack H. Lutz and Elvira Mayordomo},
title =	{{Curves That Must Be Retraced}},
booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
series =	{OpenAccess Series in Informatics (OASIcs)},
ISBN =	{978-3-939897-12-5},
ISSN =	{2190-6807},
year =	{2009},
volume =	{11},
editor =	{Andrej Bauer and Peter Hertling and Ker-I Ko},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},