## 2 Geometric Concepts  
Consider a rectangular coordinate system in three-dimensional Euclidean space denoted by \(u=(u_{1},u_{2},u_{3})\) with metric defined as \(\langle u,u\rangle=du_{1}^{2}+du_{2}^{2}+du_{3}^{2}\). For any curve \(\gamma=\gamma(s):I\subset\ R\rightarrow\mathbb{E}^{3}\), where \(s\) represents the arc-length parameter, we define the moving Frenet frame along \(\gamma\) as \(\mathcal{F}=\{T(s),N(s),B(s)\}\). The Frenet equations for the curve \(\gamma\) can be expressed as  
\[\left[ \begin{array}{c}T^{\prime}(s)\\ N^{\prime}(s)\\ B^{\prime}(s)\end{array} \right]=\left[ \begin{array}{cc}0&\kappa(s)&0\\ -\kappa(s)&0&\tau(s)\\ 0&-\tau(s)&\tau(s)\end{array} \right]\left[ \begin{array}{c}T(s)\\ N(s)\\ B(s)\end{array} \right]. \tag{1}\]  
Here, \(\kappa(s)\) and \(\tau(s)\) represent the curvature and torsion of the curve \(\gamma\), respectively. The vectors \(T,N\), and \(B\) are mutually orthonormal vectors that satisfy the following conditions:  
\[\left\{ \begin{array}{c}\langle T,T\rangle=\langle N,N\rangle=\langle B,B \rangle=1,\\ \langle T,N\rangle=\langle N,B\rangle=\langle B,T\rangle=0,\\ \det(T,N,B)=1.\end{array} \right. \tag{2}\]  
**Definition 1** ([23; 24]). : _A ruled surface (R-S) is a surface constructed from straight lines parametrized by \(\gamma(s)\) and \(X(s)\). It can be represented parametrically as_  
\[Q(s,v)=\gamma(s)+v\,X(s), \tag{3}\]  
_where \(\gamma=\gamma(s):I\subset\ R\rightarrow\mathbb{E}^{3}\) is the directrix or base curve and \(X(s)\) represents a unit vector in the direction of the ruling of the (R-S). If there exists a common perpendicular line for two constructive rulings on the (R-S), the point where this perpendicular intersects the main rulings is called a central point. The locus of these central points is known as the striction curve, and its parametrization on the (R-S) (3) is given by [4]_  
\[\gamma^{*}(s)=\gamma(s)-\frac{\langle\gamma^{\prime}(s),X^{\prime}(s)\rangle}{ \|X^{\prime}(s)\|^{2}}X(s). \tag{4}\]  
_If \(\|X^{\prime}(s)\|=0\), then the (R-S) does not have any striction curve, and it is identified as cylindrical. In such a case, the base curve can serve as a striction curve._  
**Definition 2** ([25]). : _The unit normal vector field \(U_{n}\) on the surface \(Q\) is defined by_  
\[U_{n}=\frac{Q_{s}\wedge Q_{v}}{\|Q_{s}\wedge Q_{v}\|}, \tag{5}\]  
_where \(Q_{s}=\frac{\partial Q(s,v)}{\partial s}\) and \(Q_{v}=\frac{\partial Q(s,v)}{\partial v}\)._  
**Definition 3** ([25]). : _Let \(\kappa_{g}\), \(\kappa_{n}\), \(\tau_{g}\) be the geodesic curvature, normal curvature, and geodesic torsion, respectively, associated with the curve \(\gamma(s)\) on the surface \(Q\). They can be defined by the following formula:_  
\[\kappa_{g}=\langle U_{n}\wedge T,T^{\prime}\rangle,\qquad\kappa_{n}=\langle \gamma^{\prime\prime},U_{n}\rangle,\qquad\tau_{g}=\langle U_{n}\wedge U_{n}^{ \prime},T^{\prime}\rangle.\]  
**Definition 4** ([25]). : _The curve \(\gamma(s)\) lying on a surface \(Q\) is a geodesic curve, an asymptotic line, and a principal line if and only if \(\kappa_{g}=0\), \(\kappa_{n}=0\), and \(\tau_{g}=0\), respectively._  
**Definition 5** ([25]). : _Let \(K\), \(H\), and \(\lambda\) denote the Gaussian curvature (GC), mean curvature (MC), and distribution parameter, respectively. They can be defined by the following formulas:_  
\[K =\frac{h_{11}h_{22}-h_{12}^{2}}{g_{11}g_{22}-g_{12}^{2}}, \tag{6}\] \[H =\frac{g_{11}h_{22}+g_{22}h_{11}-2g_{12}h_{12}}{2\left(g_{11}g_{ 22}-g_{12}^{2}\right)},\] (7) \[\lambda =\frac{\det(\gamma^{\prime},X,X^{\prime})}{\|X^{\prime}\|^{2}}. \tag{8}\]  
_where \(g_{ij}\) and \(h_{ij}\), \(i,j=1,2\) represent the first fundamental quantities and second fundamental quantities, respectively and they can be expressed as_  
\[g_{11}=\langle Q_{s},Q_{s}\rangle,\;\;g_{12}=\langle Q_{s},Q_{v}\rangle,\;\;g_ {22}=\langle Q_{v},Q_{v}\rangle. \tag{9}\]  
\[h_{11}=\langle Q_{ss},U_{n}\rangle,\;\;h_{12}=\langle Q_{sv},U_{n}\rangle,\;\; h_{22}=\langle Q_{vv},U_{n}\rangle. \tag{10}\]  
**Definition 6** ([26]).