On $\Lambda$-positioning of an arc between two parallel support lines

TL;DR

This paper proves that any rectifiable plane arc has a specific configuration of two parallel support lines with three consecutive points, and this configuration is unique for simple arcs, contributing to geometric support line theory.

Contribution

The paper establishes the existence and uniqueness of a particular support line configuration for rectifiable plane arcs, advancing geometric understanding.

Findings

Existence of two parallel support lines with three specific points

Uniqueness of the support line configuration for simple arcs

Characterization of the support line arrangement in rectifiable arcs

Abstract

We show that a rectifiable plane arc g has two parallel support lines and a triple of consecutive points g(r), g(s), g(t), r<s<t, so that g(s) lies on one line, while g(r) and g(t) lie on the other. If the arc is simple, such a pair of lines is unique.