A canonical parameterization of paths in $\mathbb{R}^n$

TL;DR

This paper introduces a new, well-defined path length measure called $ extsf{len}$ for all continuous paths in $R^n$, enabling a canonical parametrization and analysis of path families.

Contribution

It defines a new path length $ extsf{len}$ that is finite, continuous, and invariant, extending canonical parametrization to all continuous paths in $R^n$.

Findings

$ extsf{len}$} is finite and continuous for all continuous paths.

Provides characterizations of reparameterizable path families.

Establishes a homeomorphism between certain arc families.

Abstract

For sufficiently tame paths in $\mathbb{R}^n$, Euclidean length provides a canonical parametrization of a path by length. In this paper we provide such a parametrization for all continuous paths. This parametrization is based on an alternative notion of path length, which we call $\mathsf{len}$. Like Euclidean path length, $\mathsf{len}$ is invariant under isometries of $\mathbb{R}^n$, is monotone with respect to sub-paths, and for any two points in $\mathbb{R}^n$ the straight line segment between them has minimal $\mathsf{len}$ length. Unlike Euclidean path length, the $\mathsf{len}$ length of any path is defined (i.e., finite) and $\mathsf{len}$ is continuous relative to the uniform distance between paths. We use this notion to obtain characterizations of those families of paths which can be reparameterized to be equicontinuous or compact. Finally, we use this parametrization to obtain a canonical homeomorphism between certain families of arcs.