  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.
