## 2 Numerical Model of the Face Gear Surface
In this study, the coordinates of discrete tooth surface points of the face gear were obtained using the rotating projection mapping method. First, the minimum inner radius \(R_{1}\) and the maximum outer radius \(R_{2}\) of the face gear were obtained by solving the conditions to avoid undercutting and tooth tip sharpening. The tooth top position \(r_{sm}\) and the tooth root position \(r_{\text{as}}\) were obtained through the parameters of the cutting tool, and then the range of the tooth surface of the face gear on the projection plane was determined, as shown in Figure 3. \(O_{L}\) is the intersection of the cutting tool axis with the face gear axis. Finally, the grid planning of the projection plane was carried out, the mapping relationship between the grid points and the tooth surface coordinates was established and the grid parameters were obtained by solving the nonlinear equations. Figure 3: The mapping between the tooth surface and the rotating projection plane. In Figure 3, the point \(M\) on the working surface of the face gear is three-dimensional, and its coordinates are \((x_{M}(\theta_{s},\,\varphi_{s}),\,y_{M}(\theta_{s},\,\varphi_{s}),\,z_{M}( \theta_{s},\,\varphi_{s}))\). The corresponding point of \(M\) in the rotating projection plane is \(M^{\prime}\), and its coordinates are \((RL,\,ZL)\). The equation could be obtained from the rotational projection relation:  
\[\left\{ \begin{array}{l}z_{M}(\theta_{s},\,\varphi_{s})=ZL\\ x_{M}^{2}(\theta_{s},\,\varphi_{s})+y_{M}^{2}(\theta_{s},\,\varphi_{s})=RL^{2 }\end{array} \right. \tag{7}\]  
Similarly, the point \(T\) on the filled surface of the face gear was three-dimensional, and its coordinates are \((x_{T}(u_{s},\,\varphi_{s}),\,y_{T}\,(u_{s},\,\varphi_{s}),\,z_{T}(u_{s},\, \varphi_{s}))\). The corresponding point of \(T\) in the rotating projection plane is \(T^{\prime}\), and its coordinates are \((RL,\,ZL)\). The equation can be obtained from the rotational projection relation as follows:  
\[\left\{ \begin{array}{l}z_{T}(u_{s},\,\varphi_{s})=ZL\\ x_{T}^{2}(u_{s},\,\varphi_{s})+y_{T}^{2}(u_{s},\,\varphi_{s})=RL^{2}\end{array} \right. \tag{8}\]  
Equations (7) and (8) are nonlinear equations, which need to be provided with initial values near the zero point when solved using [PERSON]'s method. However, [PERSON]'s method is sensitive to the initial value, and if the initial value is not appropriate, a solution that does not match the expectation may be obtained. In this study, a parametric partitioning method was used to provide the initial values [31]. As shown in Figure 3, \(ABC\) is the tooth profile of the face gear at a certain radius, where \(AB\) is the working tooth profile, \(BC\) is the transition tooth profile, and point \(B\) is on the transition line. The equation parameters of the working tooth surface are \((\theta_{s},\,\varphi_{s})\) and those of the transition tooth surface are \((\varphi_{s},\,u_{s})\). \(\theta_{As}\) and \(\theta_{Bs}\) denote the values of parameter \(\theta_{s}\) when the cutting tool meshes with the face gear at points \(A\) and \(B\), respectively. \(\varphi_{Bs}\) and \(\varphi_{Cs}\) denote the values of parameters \(\varphi_{s}\) when the tool and face gear are engaged at points \(B\) and \(C\), respectively. The initial parameter values of the equations of \(AB\) and \(BC\) were calculated separately, and the steps were as follows. 1. On the working tooth profile, the parameters \(\theta_{As}\), \(\theta_{Bs}\), and \(\varphi_{As}\) were obtained according to Equations (9) and (10). Divide \(\theta_{As}\) to \(\theta_{Bs}\) equally into \(\theta_{ns}\) subsections, substitute them into Equation (11) as known parameters, and then solve Equation (11) to obtain \(\varphi_{ns}\). Take \((\theta_{ns},\,\varphi_{ns})\) as the initial value for solving Equation (7). 2.