Abstract

Abstract In this paper, some new integral curves are defined in three-dimensional Euclidean space by using a new frame of a polynomial spatial curve. The Frenet vectors, curvature and torsion of these curves are obtained by means of new frame and curvatures. We give the characterizations and properties of these integral curves under which conditions they are general helix. Also, the relationships between these curves in terms of being some kinds of associated curves are introduced. Finally, an example is illustrated.

Highlights

  • Differential geometry uses the technique of calculus to understand shapes and their properties

  • Curve theory is a significant subsection of differential geometry, in the sense of that shapes

  • For a space curve α in three-dimensional Euclidean space E3, the adapted frames are the collection of triples {v1, v2, v3}, where vi form an orthonormal basis of E3 such that v1 is tangent to α and v2, v3 are picked permissively on the plane which is normal to v1

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Summary

Introduction

Differential geometry uses the technique of calculus to understand shapes and their properties. Dede [4] has defined a new frame which is called Frenet-like curve frame. This new frame is computed in terms of using the cross product of the first and highest order derivatives of the curve to calculate the binormal vector of the curve. The advantage of this frame is that it decreases the number of singular points where they cannot be defined in the Frenet frame. Some integral curves with a new frame 1333 rotation around the tangent vector of the curve. We characterize them in terms of Bertrand pairs, Mannheim pairs, involute-evolute pairs and Salkowski curves

Preliminaries
New integral curves
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