## 2 Numerical Model of the Face Gear Surface  
The tooth surface of the face gear was generated using an involute spur shaper cutter through a certain movement; the coordinate system is shown in Figure 2. The coordinate systems \(S_{50}\) and \(S_{20}\) were fixed to describe the initial positions of the spur gear cutter \(S\) and the machined face gear. The coordinate systems \(S_{S}\) and \(S_{2}\) rotated with the spur gear cutter \(S\) and the machined face gear, respectively. The coordinate origin of \(S_{S}\), \(S_{2}\), \(S_{50}\), and \(S_{20}\) was located at the intersection of the axis of cutting tool \(S\) and the rotation axis of the machined face gear. The tooth surface \(\Sigma_{s}\) of the spur gear cutter was established according to the involute equation as follows:  
\[r_{s}(u_{s},\theta_{s})=\left\{ \begin{array}{c}\pm r_{bs}(\sin(\theta_{os}+ \theta_{s})-\theta_{s}\cos(\theta_{os}+\theta_{s}))\\ -r_{bs}(\cos(\theta_{os}+\theta_{s})+\theta_{s}\sin(\theta_{os}+\theta_{s})) \\ u_{s}\end{array} \right. \tag{1}\]  
where \(u_{s}\) and \(\theta_{s}\) are the tooth surface parameters of the cutter gear; \(r_{bs}\) is the radius of the base circle; '\(\pm\)' represents the involutes \(\gamma-\gamma\) and \(\beta-\beta\), respectively; and \(\theta_{os}\) is expressed as:  
\[\theta_{os}=\frac{\pi}{2N_{s}}-\tan(\alpha_{0})+\alpha_{0} \tag{2}\]  
where \(N_{s}\) is the number of teeth of the gear cutter and \(\alpha_{0}\) is the pressure angle. The unit normal vector \(n_{s}\) of a point \(P\) on \(\Sigma_{s}\) is expressed as follows:  
\[n_{s}=\frac{\frac{\partial r_{s}}{\partial\theta_{s}}\times\frac{\partial r_{ s}}{\partial\theta_{s}}}{\left|\frac{\partial r_{s}}{\partial\theta_{s}} \times\frac{\partial r_{s}}{\partial\theta_{s}}\right|} \tag{3}\]  
The cutter gear was used as a generating gear, and the face gear tooth surface \(\Sigma_{2}\), which was conjugated to the cutter tooth surface, was developed in accordance with the relative  
Figure 2: Diagrams showing the processing of the face gear: (a) parameters of the involute surface of the gear cutter and (b) generating coordinate system of face gear. motion relationship between the two tooth surfaces. Based on the meshing principle, the tooth surface \(\Sigma_{2}\) could be obtained according to the following equation:  
\[\left\{ \begin{array}{l}r_{2}(u_{s},\theta_{s},\varphi_{s})=M_{2,20}M_{20,50}M_{ 50,5}r_{s}(u_{s},\theta_{s})\\ n_{s}\cdot v_{s}^{S2}=f(u_{s},\theta_{s},\varphi_{s})=0\end{array} \right. \tag{4}\]  
where \(M_{2,20}\), \(M_{20,50}\), and \(M_{50,5}\) are the coordinate conversion matrices, \(v_{s}^{S2}\) is the relative motion speed of the cutting tool and the tooth surface of face gear at the meshing point, and the position vector \(r_{2}\) indicates the tooth surface equation of \(\Sigma_{2}\). The transition surface \(\Sigma_{2}^{*}\) of the face gear was formed using the tooth top of the cutting tool. This meant that the top line of the cutting tool was transformed into the surface \(\Sigma_{2}^{*}\) formed in the fixed coordinate system of the face gear using a homogeneous coordinate transformation. The angle parameter \(\theta_{s}\) in the tooth profile equation of the cutting tool was replaced by the parameter at the addendum circle \(\theta_{s}^{*}\), and the equation of the tooth top could be derived. Similar to the principle of generating the tooth surface, the transition surface equation \(r_{2}^{*}(u_{s},\varphi_{s})\) expressed in the coordinate system \(S_{2}\) was as follows:  
\[r_{2}^{*}(u_{s},\varphi_{s})=M_{2,20}M_{20,50}M_{50,5}r_{s}^{*}(u_{s},\theta_{ s}^{*}) \tag{5}\]  
If undercutting did not occur, there was a common tangent \(L\) between the tooth fillet surface \(\Sigma_{2}^{*}\) and the working surface \(\Sigma_{2}\) and its equation was as follows:  
\[\left\{ \begin{array}{l}r_{L}(u_{s}^{*},\varphi_{s})=M_{2,20}M_{20,50}M_{50,5 }r_{s}(\theta_{s}^{*},u_{s})\\ f(u_{s}^{*},\theta_{s}^{*},\varphi_{s})=0\end{array} \right. \tag{6}\]  
The tooth surface of the face gear is a complex spatial surface.