Curves That Must Be Retraced
Computational Complexity
2009-05-19 v2
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.
Cite
@article{arxiv.0802.4312,
title = {Curves That Must Be Retraced},
author = {Xiaoyang Gu and Jack H. Lutz and Elvira Mayordomo},
journal= {arXiv preprint arXiv:0802.4312},
year = {2009}
}