Characterizations of Special Curves

TL;DR

This paper introduces new ways to characterize special curves such as helices and Bertrand curves without directly using their curvatures, instead relying on derivatives of Frenet vectors.

Contribution

It provides novel characterizations of special curves based on derivatives of Frenet vectors, avoiding the traditional curvature-based methods.

Findings

New characterizations of general helices, slant helices, Bertrand, and Mannheim curves.

Characterizations use norms of derivatives of Frenet vectors instead of curvatures.

Simplifies analysis of special curves by avoiding curvature calculations.

Abstract

In this study, the new characterizations of special curves are investigated without using the curvatures of these special curves: general helices, slant helices, Bertrand curves, Mannheim curves. The curvatures are given by the help of the norms of the derivatives of Frenet vectors.