Direction curves of tangent indicatrix of a curve

TL;DR

This paper introduces new associated curves derived from the tangent indicatrix of a curve in Euclidean 3-space, providing methods to construct helices and slant helices from spherical curves, with theoretical relationships and examples.

Contribution

It defines new associated curves as integral curves of a vector field from Frenet vectors and explores their relationships, enabling construction of special helices from spherical curves.

Findings

Relationships between curvatures of the new associated curves

Methods to construct helices and slant helices from spherical curves

Examples illustrating the construction methods

Abstract

In this paper, we define some new associated curves as integral curves of a vector field generated by Frenet vectors of tangent indicatrix of a curve in Euclidean 3-space. We give some relationships between curvatures of these curves. By using these associated curves, we give some methods to construct helices and slant helices from some special spherical curves such as circles on unit sphere, spherical helices, and spherical slant helices. Finally, we give some related examples.