AN EXAMINATION ON THE MANNHEIM FRENET RULED SURFACE BASED ON NORMAL VECTOR FIELDS IN E$^{3}$

Abstract

In this paper we consider six special Frenet ruled surfaces along to the Mannheim pairs $\left\{ \alpha ^{\ast },\alpha \right\} $. First we define and find the parametric equations of Frenet ruled surfaces which are called $% Mannheim$ $Frenet$ $ruled$ $surface$, along Mannheim curve $\alpha ,$ in terms of the Frenet apparatus of Mannheim curve $\alpha .$ Later, \ we find only one matrix gives us all nine positions of normal vector fields of these six Frenet ruled surfaces and $Mannheim$ $Frenet$ $ruled$ $surface$ in terms of Frenet apparatus of Mannheim curve $\alpha $ too. Further using that matrix we have some results such as; normal ruled surface and $Mannheim$ $% normal$ $ruled$ $surface$ of Mannheim curve $\alpha $ have perpendicular normal vector fields along the curve $\varphi _{2}\left( s\right) =\alpha +% \frac{\tan \theta }{k_{1}\tan \theta -k_{2}}V_{2},$ \ under the condition $% \tan \theta \neq \frac{k_{2}}{k_{1}}$.

Keywords

• Mannheim curve,Frenet ruled surface

References

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