## 4 Special Ruled Surfaces and Their Characterizations
### The Ruled Surfaces with \(u_{2}=0\)  
Consider the (R-S) constructed by (14). For \(u_{2}=0\), \(u_{1}^{2}+u_{3}^{2}=1\), then, the equation for the (R-S) (14) takes the following form:  
\[Q(s,v)=\gamma(s)+v(u_{1}T+u_{3}B). \tag{61}\]  
In this case, we have,  
\[\delta^{2} =u_{3}^{2}+(\kappa u_{1}-\tau u_{3})^{2}v^{2},\] \[\hat{a} =\kappa u_{3}+\tau u_{1},\quad\hat{b}=-u_{3}(\kappa u_{1}-\tau u _{3}),\quad\hat{c}=(\kappa u_{1}-\tau u_{3})^{2}.\]  
**Lemma 18**.: _The (GC), (MC), and the distribution parameter \(\lambda\) for the (R-S) at \(u_{2}=0\) are given by_  
\[K =-\frac{u_{3}^{2}(\kappa u_{1}-\tau u_{3})^{2}}{(u_{3}^{2}+(\kappa u _{1}-\tau u_{3})^{2}v^{2})^{2}},\] \[H =-\frac{1}{2(u_{3}^{2}+(\kappa u_{1}-\tau u_{3})^{2}v^{2})^{3/2} }\Big{(}(\kappa u_{3}+\tau u_{1})(\kappa u_{1}-\tau u_{3})^{2}v^{2}\] \[+u_{3}(\kappa(u_{3}^{2}-u_{1}^{2})+2u_{1}u_{3}\tau)\Big{)},\] \[\lambda =\frac{-u_{3}}{(\kappa u_{1}-\tau u_{3})}.\]  
_Also, the (S-MC) and (S-GC) for the (R-S) at \(u_{2}=0\) are given by_  
\[H_{II} =-\frac{\sum_{i=0}^{4}A_{i}v^{i}}{2(u_{3}^{2}(\kappa u_{1}-\tau u _{3})^{2})\left(u_{3}^{2}+(\kappa u_{1}-\tau u_{3})^{2}v^{2}\right)^{\frac{3} {2}}}\,,\] \[A_{0} =u_{3}^{3}(\kappa u_{1}-\tau u_{3})^{2}\Big{(}\kappa(1+2u_{3}^{2 })+2u_{1}u_{3}\tau\Big{)},\] \[A_{1} =0,\] \[A_{2} =-u_{3}(\kappa u_{1}-\tau u_{3})^{4}\Big{(}\kappa(7u_{1}^{2}-5)- 7\tau u_{1}u_{3}\,\Big{)},\] \[A_{3} =0,\] \[A_{4} =2(\kappa u_{3}+\tau u_{1})\,(\kappa u_{1}-\tau u_{3})^{6}.\]  
_And_  
\[K_{II} =-\frac{\sum_{i=0}^{4}B_{i}v^{i}}{2(u_{3}^{2}(\kappa u_{1}-\tau u _{3})^{2})\left(u_{3}^{2}+(\kappa u_{1}-\tau u_{3})^{2}v^{2}\right)^{\frac{3} {2}}}\,,\] \[B_{0} =u_{3}^{3}(\kappa u_{1}-\tau u_{3})^{2}\Big{(}-\kappa(u_{1}^{2}- u_{3}^{2})+2u_{1}u_{3}\tau)\,\Big{)},\] \[B_{1} =0,\]\[B_{2} =u_{3}(\kappa u_{1}-\tau u_{3})^{4}\Big{(}\kappa(1+u_{3}^{2})+u_{1}u_{ 3}\tau\,\Big{)},\] \[B_{3} =0,\] \[B_{4} =(\kappa u_{3}+\tau u_{1})\,(\kappa u_{1}-\tau u_{3})^{6}.\]  
**Lemma 19**.: _The geodesic curvature \(\kappa_{g}\), the normal curvature \(\kappa_{n}\), and the geodesic torsion \(\tau_{g}\) associated with the base curve \(\gamma(s)\) at \(u_{2}=0\), take the following formula:_  
\[\kappa_{g}=\frac{-\kappa u_{1}(\kappa u_{1}-\tau u_{3})v}{\sqrt{u_{3}^{2}+( \kappa u_{1}-\tau u_{3})^{2}v^{2}}}\,\quad\kappa_{n}=\frac{-u_{3}\kappa}{\sqrt{u_{3}^{2}+( \kappa u_{1}-\tau u_{3})^{2}v^{2}}}\quad\tau_{g}=\frac{-u_{3}\kappa(\kappa u_{ 1}-\tau u_{3})^{2}v}{u_{3}^{2}+(\kappa u_{1}-\tau u_{3})^{2}v^{2}}.\]  
**Lemma 20**.: _The (R-S) given by (61) that constructed with \(u_{2}=0\) in \(\mathbb{E}^{3}\) is a flat, II-minimal, and II-flat surface at any point \((s,v)\) (also at the point \((s,0)\)) if and only if the following condition holds:_  
\[\frac{\tau}{\kappa}=\frac{u_{1}}{u_{3}}.\]  
**Lemma 21**.: _There are no minimal (R-S) with \(u_{2}=0\) at a point \((s,v)\) in \(\mathbb{E}^{3}\)._  
**Lemma 22**.: _The base curve for the (R-S) given by (61) is a geodesic curve and a principal line if and only if_  
\[\frac{\tau}{\kappa}=\frac{u_{1}}{u_{3}}.\]  
**Lemma 23**.: _The (R-S) given by (61) that constructed with \(u_{2}=0\) in \(\mathbb{E}^{3}\) is characterized by the following conditions at the point \((s,0)\):_  
* _It is minimal and II-flat if and only if the ratio of the torsion to curvature is equal to_ \[\frac{\tau}{\kappa}=\frac{u_{1}^{2}-u_{3}^{2}}{2u_{1}u_{3}}.\]
* _It is II-minimal if and only if the ratio of the torsion to curvature is equal to_ \[\frac{\tau}{\kappa}=-\frac{1+2u_{3}^{2}}{2u_{1}u_{3}},\quad\text{or}\quad \frac{\tau}{\kappa}=\frac{u_{1}}{u_{3}}.\]
* _The base curve for the (R-S) is a geodesic curve and a principal line._