Bertrand Legendre curves in the unit tangent bundle over Euclidean plane

TL;DR

This paper studies Bertrand Legendre curves in the unit tangent bundle over the Euclidean plane, exploring their properties, existence conditions, and a bijective mapping between sets of such curves.

Contribution

It introduces a framework for analyzing Bertrand Legendre curves, providing existence conditions, inverse operations, and a bijective mapping between curve sets.

Findings

Established existence conditions for Bertrand regular plane curves.

Defined an inverse operation for Bertrand Legendre curves.

Proved a bijective mapping between sets of Legendre curves.

Abstract

We investigate not only the associated curves of regular plane curves, but also those of Legendre curves. As associated curves, we consider Bertrand regular plane curves and Bertrand Legendre curves. These curves contain parallel, evolute and involute curves, as well as evolutoid and involutoid curves. Since associated curves may have singular points even if the original curve is regular, Legendre curves provide a suitable framework for investigating the properties of such curves. We give existence conditions of Bertrand regular plane curves and Bertrand Legendre curves. Moreover, we give an inverse operation for Bertrand Legendre curves. Furthermore, we define a mapping between sets of Legendre curves using Bertrand Legendre curves and prove that this mapping is bijective up to equivalence relations.