Curves That Must Be Retraced

TL;DR

The paper constructs a specific computable plane curve that, despite being smooth and non-self-intersecting, must be retraced infinitely often by any computable parametrization, highlighting limitations in computable curve parametrizations.

Contribution

It introduces a computable curve with finite length that cannot be parametrized without retracing, revealing fundamental constraints in computable curve representations.

Findings

Any computable parametrization retraces some subcurve infinitely often.

Every non-self-intersecting finite-length computable curve admits a non-retracing computable parametrization.

The constructed curve demonstrates inherent limitations in computable curve parametrizations.

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.