Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms

Jinhua Qian,Young Ho Kim,Xueqian Tian

doi:10.3390/math8060919

Jinhua Qian, Young Ho Kim + Show 1 more

Open Access

https://doi.org/10.3390/math8060919

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Abstract

In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector fields, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme.

Highlights

The space associate curves for which there exist some relations between their Frenet frames or curvatures compose a large class of fascinating subjects in the curve theory, such as Bertrand curve, Mannheim curve, central trace of osculating sphere, involute-evolute curves etc. [1,2,3]
In Euclidean 3-space, the classical Bertrand curves are characterized by constant distance between the corresponding points of the partner curves and by constant angle between tangent vector fields of the partner curves
Due to the causal character of vectors, some simple problems become a little complicated and strange, such as the arc-length of null curves can not be defined similar to the definition in Euclidean space; the angles between different type of vectors need to be classified according to different conditions [5,6]

Summary

Introduction

The space associate curves for which there exist some relations between their Frenet frames or curvatures compose a large class of fascinating subjects in the curve theory, such as Bertrand curve, Mannheim curve, central trace of osculating sphere, involute-evolute curves etc. [1,2,3]. In Minkowski space, the curves are divided into spacelike, timelike and lightlike (null) curves according to the causal character of their tangent vectors. Some particular curves such as the helix, the Bertrand curve, the Mannheim curve and the normal curve, the osculating curve and the rectifying curve have been surveyed by some researchers [5,7,8,9,10,11,12].

Preliminaries

Associate Normal Curves of a Pseudo Null Curve on de-Sitter Space

Associate Normal Curves of a Pseudo Null Curve on Hyperbolic Space

The Relationships of the Normal Partner Curves

Examples