## 3 Ruled Surface for Face Gears  
### Tooth Surface Equation of Ruled Surfaces  
A ruled surface is a special surface composed of a family of straight lines, that is, a line passing through each point on the curve falls on that surface, as shown in Figure 2a, and the general expression can be represented by the following equation:  
\[g(u,v)=a(u)+v\cdot b(u) \tag{9}\]  
where \(a(u)\) represents the leading line, \(b(u)\) is the unit vector on the straight line passing through a point \(a(u)\) on the leading line, and the distance from a point on the ruled surface to a point \(a(u)\) on the leading line is \(v\). Figure 1: Generation of the face gear. In the meshing process of conventional face gear pairs, a special contact line \(L_{\rm r}\) is shown in Figure 2b, in which \(L_{\rm r}\) extends from the top of the toe to the root of the heel, and the line \(L_{\rm r}\) is an involute extending in the longitudinal direction of the face gear tooth that is tangent to the surface \(\Sigma_{\rm c}\) and \(\Sigma_{\rm f}\). If this special line \(L_{\rm r}\) is considered as a leading line, a family of lines that are tangent to \(L_{\rm r}\) at different cross sections or to the end face involute at different points that are arranged in the longitudinal direction of the face gear tooth and change with respect to the pressure angle are generatrices, so the tooth surface of the face gear can be seen as a ruled surface, as shown in Figure 3b, which greatly facilitates its manufacturing process. Compared with the implicit equation given in the reference [29; 30], the explicit expression of the ruled surface is derived in this paper. The ruled surface of a orthogonal-straight tooth face gear can be obtained by the following process. At any point on this line \(L_{\rm r}\), the linear velocities of the pinion and face gear are equal, which are represented as follows:  
\[r_{\rm q}w_{\rm c}=L_{\rm q}w_{\rm f}\sin\gamma=\frac{L_{\rm q}w_{\rm f}N_{\rm f }\sin\gamma}{N_{\rm c}} \tag{10}\]  
where \(r_{\rm q}\) and \(L_{\rm q}\) are the distance from point \(q\) on the line to axes \(z_{\rm c}\) and \(z_{\rm f}\), respectively, and \(w_{\rm c}\) and \(w_{\rm f}\) are the angular velocities of the pinion and face gear, respectively. As shown in Figure 3a, any point on the involute, \(r_{\rm q}\) can be expressed as follows:  
\[r_{\rm q}=\frac{r_{\rm c}}{\cos(\theta_{\rm s}+\theta_{0})} \tag{11}\]  
Figure 3: Schematic diagram of the development of ruled surface: (**a**) generation coordinate system of ruled surface, and (**b**) generating result of ruled surface for face gear. Figure 2: Schematic diagrams of ruled surface and the special contact line of conventional face gear pair: (**a**) definition of ruled surface, and (**b**) a special contact line in surfaces \(\Sigma_{\rm c}\) and \(\Sigma_{\rm f}\). Hence, \(L_{\rm q}\) can be expressed as shown in Equation (12). \[L_{\rm q}=\frac{r_{\rm c}N_{\rm f}}{N_{\rm c}\cos(\theta_{\rm s}+\theta_{0}){\rm sin }\gamma} \tag{12}\]  
By taking Equation (12) into consideration, the position vector \(R_{2}\) of the ruled surface can be represented by the following formulas:  
\[A=|O_{\rm a}O_{\rm b}|=r_{\rm c}\tan(\theta_{0}+\theta_{s})-r_{\rm c}\theta_{s} \tag{13}\]  
\[L_{0}=|O_{2}O_{\rm c}|=\frac{r_{\rm q}}{\tan\gamma}+L_{\rm q} \tag{14}\]  
\[R_{2}=M_{2\rm c}M_{\rm cb}M_{\rm ba}R_{\rm a}=M_{\rm 2a}R_{\rm a} \tag{15}\]  
\[M_{\rm 2s}= \begin{bmatrix}1&0&0&0\\ 0&-\cos\gamma&-\sin\gamma&-L_{0}\sin\gamma\\ 0&\sin\gamma&-\cos\gamma&-L_{0}\cos\gamma\\ 0&0&0&1\end{bmatrix} M_{\rm cb}= \begin{bmatrix}1&0&0&0\\ 0&1&0&-r_{\rm q}\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} \]  
\[M_{\rm ba}= \begin{bmatrix}\cos(\theta_{0}+\theta_{s})&\sin(\theta_{0}+\theta_{ s})&0&-A\cos(\theta_{0}+\theta_{s})\\ -\sin(\theta_{0}+\theta_{s})&\cos(\theta_{0}+\theta_{s})&0&A\sin(\theta_{0}+ \theta_{s})\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix} R_{\rm a}= \begin{bmatrix}0\\ u_{\rm a}\\ 0\\ 1\end{bmatrix}