Associated Curves in $E^3$ from a Different Point of View

TL;DR

This paper introduces a novel perspective on associated curves in three-dimensional space, defining tangent, principal normal, and binormal associations, and explores their properties, solutions, and applications with specific examples.

Contribution

It defines new types of associated curves based on vector positions relative to a curve's osculating, normal, and rectifying planes, and derives their frames and curvatures.

Findings

Involutes are tangent associated curves.

Bertrand and Mannheim curves are principal normal associated curves.

Examples demonstrate the application of the theory.

Abstract

In this paper, tangent-, principal normal-, and binormal-wise associated curves are defined such that each of these vectors of any given curve lies on the osculating, normal, and rectifying plane of its mate, respectively. For each associated curve, a new moving frame and the corresponding curvatures are found, and in addition to this the possible solutions for distance functions between the curve and its associated mate are discussed. In particular, it is seen that the involute curves belong to the family of tangent associated curves, the Bertrand and the Mannheim curves belong to the principal normal associated curves. Finally, as an application, we present some examples and map a given curve together with its mate and their frames.