New Associated Curves k-Principle Direction Curves and N_k Slant Helix

TL;DR

This paper introduces a new orthonormal frame for spatial curves, leading to the definition of N_k slant helix, which characterizes curves with a constant angle property, enhancing curve analysis methods.

Contribution

It proposes a novel orthonormal frame based on associated curves, enabling the characterization of N_k slant helix with specific constant angle properties.

Findings

Defined the N_k slant helix concept.

Derived conditions for the N_k slant helix.

Established properties of the new orthonormal frame.

Abstract

The Frenet frame is generally known an orthonormal vector frame for curves. But, it does not always meet the needs of curve characterizations. In this study, with the help of associated curves of any spatial curve we obtained a new orthonormal frame which has the property that the second vector makes a constant angle with a fixed direction for the spatial curve. Then, the curve is named as the N_k slant helix and the special conditions are obtained effectively.