## 3 Construction of Ruled Surfaces in \(\mathbb{E}^{3}\)  
We consider the (R-S) with circular helix curve \(\gamma(s)\) (a family of curves with constant curvature \(\kappa(s)=\kappa\) and constant torsion \(\tau(s)=\tau\)) as a base curve. Therefore, the (R-S) can be constructed by  
\[Q(s,v)=\gamma(s)+vX(s),\;\;v\in  
For simplicity, we choose  
\[\hat{b}=\tau-\hat{a}u_{1}\,,\quad\hat{d}=\kappa-\hat{a}u_{3}. \tag{23}\]  
Then  
\[Q_{s}\wedge Q_{v}=-\hat{b}vT+(\hat{a}u_{2}v-u_{3})N+(u_{2}-\hat{d}v)B. \tag{24}\]  
Also, we have  
\[\|Q_{s}\wedge Q_{v}\|^{2}=(\hat{b}v)^{2}+(\hat{a}u_{2}v-u_{3})^{2}+(u_{2}-\hat{ d}v)^{2}. \tag{25}\]  
By straightforward computation, we obtain  
\[ \begin{array}{l}\|Q_{s}\wedge Q_{v}\|^{2}=\delta^{2},\\ \delta^{2}=1-u_{1}^{2}-2u_{2}kv+\hat{c}v^{2}.\end{array}  \tag{26}\]  
Since the unit normal vector field \(U_{n}\) to the (R-S) is defined by (5), then by using (24) and (26), we obtain  
\[U_{n}=\frac{1}{3}\Big{(}-\hat{b}vT+(\hat{a}u_{2}v-u_{3})N+(u_{2}-\hat{d}v)B \Big{)}.\]  
By using (20) and (23), we obtain the unit normal vector as the following explicit formula:  
\[ \begin{array}{l}U_{n}=\frac{1}{\delta}\Big{(}(u_{1}u_{3}\kappa-(u_{2}^{2}+u_ {3}^{2})\tau)vT+(u_{1}u_{2}\tau v-(1-u_{2}\kappa v)u_{3})N\\ \qquad\qquad+(u_{2}+u_{1}u_{3}\tau v-\kappa(u_{1}^{2}+u_{2}^{2})v)B\Big{)},\\ \delta=\Big{(}1-u_{1}^{2}-2u_{2}\kappa v+(\kappa^{2}+\tau^{2}-(\kappa u_{3}+ \tau u_{1})^{2}\,)v^{2}\Big{)}^{\frac{1}{2}}\end{array}  \tag{27}\]  
\(\square\)  
**Theorem 2**. : _Consider the (R-S) given by (14). Then,_  
\[h_{11}=\frac{1}{\delta}(-\hat{a}\delta^{2}+\hat{b}u_{1})\,\quad h_{12}=\frac{ \hat{b}}{\delta}\,\quad h_{22}=0, \tag{28}\]  
_where_  
\[ \begin{array}{l}\delta^{2}=1-u_{1}^{2}-2u_{2}\kappa v+\hat{c}v^{2},\\ \hat{a}=\kappa u_{3}+\tau u_{1}\,\quad\hat{b}=\tau-\hat{a}u_{1}\,\quad\hat{c}= \kappa^{2}+\tau^{2}-\hat{a}^{2}.\end{array} \]  
Proof. : Taking the second partial derivatives of (19) with respect to \(s\) and \(v\), then we obtain  
\[ \begin{array}{l}Q_{ss}=-\kappa(\kappa u_{1}-\tau u_{3})vT+(\kappa-u_{2}( \kappa^{2}+\tau^{2})v)N+\tau(\kappa u_{1}-\tau u_{3})vB,\\ Q_{sv}=(-\kappa u_{2})T+(\kappa u_{1}-\tau u_{3})N+(\tau u_{2})B,\\ Q_{vv}=0.\end{array}  \tag{29}\]  
Since the curvature tensors \(h_{ij}\) are defined by (10), then by using (18) and the first equation of (29), we have  
\[ \begin{array}{l}h_{11}=\frac{1}{\delta}\Big{(}(\kappa\hat{b}(\kappa u_{1}- \tau u_{3})-\hat{a}u_{2}^{2}(\kappa^{2}+\tau^{2})-\tau\hat{d}(\kappa u_{1}- \tau u_{3}))v^{2}\\ \quad\quad+((\kappa^{2}+\tau^{2})u_{2}u_{3}+\kappa\hat{a}u_{2}+\tau u_{2}( \kappa u_{1}-\tau u_{3}))v-\kappa u_{3}\Big{)}.\end{array} \]  
After some complicated computations, we obtain:  
\[h_{11}=\frac{1}{\delta}\Big{(}-\kappa u_{3}+(\kappa u_{3}+\tau u_{1})\Big{(}2 u_{2}\kappa v-(\kappa^{2}+\tau^{2}-(\kappa u_{3}+\tau u_{1})^{2})v^{2}\Big{)} \Big{)}.\]We can rewrite \(h_{11}\) in the following simple form:  
\[h_{11}=\frac{1}{\delta}(-\hat{a}\delta^{2}+\hat{b}u_{1}). \tag{30}\]  
Also, by using (18) and the second equation of (29), we have  
\[h_{12}=\frac{1}{\delta}\Big{(}\kappa\hat{b}u_{2}v+(\kappa u_{1}-\tau u_{3})(-u_ {3}+\hat{a}u_{2}v)+\tau u_{2}(u_{2}-\hat{d}v)\Big{)}.\]  
Explicitly, we obtain  
\[h_{12}=\frac{1}{\delta}\Big{(}-u_{1}u_{3}\kappa+(u_{2}^{2}+u_{3}^{2})\tau\Big{)}.\]  
Or we can obtain the following simple form for \(h_{12}\) as  
\[h_{12}=\frac{\hat{b}}{\delta}\, \tag{31}\]  
In addition, the metric tensor \(h_{22}\) can be given by using (18) and the third equation of (29) as  
\[h_{22}=0. \tag{32}\]  
\(\square\)  
**Lemma 1**. : _Consider the (R-S) given by (14); then, \(h^{ij}\) are given by_  
\[h^{11}=0\,\quad h^{12}=\frac{\delta}{\hat{b}}\,\quad h^{22}=-\frac{\delta}{ \hat{b}^{2}}(-\hat{a}\delta^{2}+\hat{b}u_{1}), \tag{33}\]  
_where_  
\[ \begin{array}{l}\delta^{2}=1-u_{1}^{2}-2u_{2}\kappa v+\hat{c}v^{2},\\ \hat{a}=\kappa u_{3}+\tau u_{1}\,\quad\hat{b}=\tau-\hat{a}u_{1}\,\quad\hat{c}= \kappa^{2}+\tau^{2}-\hat{a}^{2}.\end{array} \]  
**Lemma 2**.