Curves That Must Be Retraced

We exhibit a polynomial time computable plane curve Γ 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 Γ and every positive integer n, there is some positive-length subcurve of Γ 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.