A new class of curves generalizing helix and rectifying curves

TL;DR

This paper introduces a new class of curves called f-rectifying curves, which generalize helix and rectifying curves by defining a specific vector lying in the rectifying plane, characterized through curvature and torsion.

Contribution

The paper defines f-rectifying curves, explores their properties, and provides classification and characterization in terms of curvature and torsion, extending known curve classes.

Findings

f-rectifying curves include helix and rectifying curves as special cases

Characterization of f-rectifying curves via curvature and torsion functions

Examples illustrating the properties of f-rectifying curves

Abstract

In this paper, we introduce a new class of curves \alpha called a f-rectifying curves, which its f-position vector defined by {\alpha}_{f}(s)=\int f(s)T(s)ds always lie in the rectifying plane of \alpha, where f is an integrable function and T is the speed curve of {\alpha}. In particular case, when the function f=0 or constant, the class of f-rectifying curves are helix or rectifying curves, respectively. The classification and the characterization of such curves in terms of their curvature and the torsion functions are given with a physical interpretation. We close this study with some examples.