Rectifiability and finite variation

TL;DR

This paper establishes that the length of a path in two-dimensional space can be determined precisely if and only if its variation is known in every possible direction, linking geometric length to directional variations.

Contribution

It provides a novel equivalence between path length and directional variation, deepening understanding of rectifiability in geometric analysis.

Findings

Path length equals the supremum of directional variations.

Path length can be computed from directional variations in all directions.

The result characterizes rectifiability via directional variation.

Abstract

We show that the length of a path in $ \mathbf{R}^2 $ can be computed if and only if its variation in every direction can.