Length of parallel curves

TL;DR

This paper establishes a proportional relationship between the length difference of a closed curve and its parallel at small distances, and its rotation index, with applications in estimating rotation index via the Cauchy-Crofton formula.

Contribution

It introduces a new theoretical result linking length difference and rotation index, and applies it to estimate the rotation index using integral geometry.

Findings

Length difference is proportional to rotation index.

The rotation index can be estimated using the Cauchy-Crofton formula.

Provides a new geometric insight into curve analysis.

Abstract

We prove that the length difference between a closed periodic curve and its parallel curve at a sufficiently small distance is proportional to the rotation index. As an application, the rotation index of a curve could be estimated by means of Cauchy-Crofton formula.