## 2 Theoretical Tooth Surface Modeling  
### Tooth Surface Modeling of the Gear  
The method described in this paper is the generating method, in which both sides of the gear teeth are processed simultaneously with a double-sided cutter disc containing an inner cutter and an outer cutter. When machining large wheels, the cutter head rotates around its own axis while revolving around the axis of the production wheel. The position relationship between the large wheel blank and the cutter head is shown in Figure 1.  
\(\beta_{2}\) is the coordinate system of the gear machine tool; \(P_{2}\) is the unit vector of the axis direction of the gear; \(\alpha_{02}\) is the pressure angle of the gear cutter head; \(\beta_{02}\) is the coordinate system of the gear cutter head; \(S_{2}\) is the radial cutter position; \(q_{2}\) is the angular cutter position; \(E_{02}\) is the vertical wheel position; \(X_{02}\) is the axial wheel position; \(X_{B2}\) is the bed position; \(\delta_{M2}\) is the root angle of machine tool; \(r_{0}\) is the nominal radius of cutter head; \(W_{2}\) is the angular cutter head radius; \(r_{t2}\) is cutter tip arc radius; \(O_{e}\) is the tool tip arc center; \(M_{0}\) is the apex of the tool tip; \(\theta_{2}\) is the angle between the section of cutter disk axis passing \(\overline{O_{0}M_{0}}\) and \(\textbf{{i}}_{2}\); \(r_{02}\) is the radius of the tool tip. Let \(\overline{\left|{MM_{0}}\right|}\)=\(s_{2}\); then, in \(\beta_{02}\), the radius vector \(\textbf{{r}}_{\text{C2}}\) and normal vector \(\textbf{{n}}_{\text{C2}}\) of \(M\) points on the cutting surface can be represented as follows:  
\[\textbf{{r}}_{\text{c2}}(s_{2},\theta_{2})=\left[ \begin{array}{c}(r_{02}-s_{ 2}sin\alpha_{02})cos\theta_{2}\\ (r_{02}-s_{2}sin\alpha_{02})sin\theta_{2}\\ -s_{2}cos\alpha_{02}\end{array} \right] \tag{1}\]  
\[\textbf{{n}}_{\text{c2}}(\theta_{2})=\left[ \begin{array}{c}cos\alpha_{02}cos \theta_{2}\\ cos\alpha_{02}sin\theta_{2}\\ -sin\alpha_{02}\end{array} \right] \tag{2}\]\(\theta_{02}\) is the angle between \(\overline{O_{e}M}\) and the axis of the cutter head, and the radius vectors \(\tau_{e2}\) and normal vectors \(\mathbf{n}_{e2}\) of \(\mathbf{M}\) on the cutter tip arc can be represented as  
\[\mathbf{r}_{e2}(\theta_{02},\theta_{2})=\left[ \begin{array}{c}(r_{O \theta 2}\mp r_{e2}sin\theta_{02})cos\theta_{2}\\ (r_{O\theta 2}\mp r_{e2}sin\theta_{02})sin\theta_{2}\\ r_{e2}(cos\theta_{02}-1)\end{array} \right] \tag{3}\]  
\[\mathbf{n}_{e2}(\theta_{02},\theta_{2})=\left[ \begin{array}{c}sin\theta_{02}cos \theta_{2}\\ sin\theta_{02}sin\theta_{2}\\ \mp cos\theta_{02}\end{array} \right] \tag{4}\]  
The cutter head rotates \(\Delta q_{2}\) around \(\mathbf{k}\), and the relationship between \(\beta_{02}\) and \(\beta_{2}\) can be obtained:  
\[M_{\beta_{02}}(\Delta q_{2})=\left[ \begin{array}{cccc}1&0&0&S_{2}\cos(q_{2 }+\Delta q_{2})\\ 0&1&0&S_{2}sin(q_{2}+\Delta q_{2})\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{5}\]  
Gear coordinate system \(\beta_{g}\) moves \(E_{02}\) along \(j\):  
\[M_{E_{02}}=\left[ \begin{array}{cccc}1&0&0&1\\ 0&1&0&E_{02}\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{6}\]  
Move \(X_{B2}\) along \(\mathbf{k}\):  
\[M_{X_{B2}}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&X_{B2}\\ 0&0&0&1\end{array} \right] \tag{7}\]  
Then, rotate (90\({}^{\circ}\)-\(\delta_{M2}\)) around \(j\),  
\[M_{S_{h\Omega}}=\left[ \begin{array}{cccc}sin\delta_{M2}&0&cos\delta_{M2}&0 \\ 0&1&0&0\\ -cos\delta_{M2}&0&sin\delta_{M2}&0\\ 0&0&0&1\end{array} \right] \tag{8}\]  
Move \(X_{2}\) along \(\mathbf{P}_{2}\):  
\[M_{X_{2}}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&X_{2}\\ 0&0&0&1\end{array} \right] \tag{9}\]  
The cutter head rotates \(\Delta q_{2}\) around \(\mathbf{k}\) and the gear rotates \(\psi_{2}=i_{02}\Delta q_{2}\) around \(\mathbf{P}_{2}\):  
\[M_{\psi_{2}}(\Delta q_{2})=\left[ \begin{array}{cccc}cos(\psi_{2})&-sin(\psi _{2})&0&0\\ sin(\psi_{2})&cos(\psi_{2})&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}