# Pattern Classification and Gear Design of Spatial Noncircular
## 4 Tooth Profile Classification Design of Spatial Noncircular Gear
### Tooth Surface Equation  
The accurate tooth surface model of the curve face gear pair was obtained by the method shown in Figure 11. Firstly, the tooth surface equation of the generating rack was derived, then the tooth surface equation of the generating gear was obtained, and finally the tooth surface equations of the noncircular gear and curve face gear were obtained. Figure 11: The tooth surface generating process of curve face gear pair. (1) Tooth surface equation of the generating rack  
The tooth surface \(\Sigma_{r}\) of the generating rack includes straight tooth, helical tooth, and curvilinear tooth. Figure 12 is the tooth surface coordinate system when the tooth profile is a helical tooth or curvilinear tooth. When the helix angle \(\beta_{r}\) is 0, it is a straight tooth generating rack. The global coordinate systems of both the helical and curvilinear tooth generating rack are expressed as \(S_{r}\) (\(O_{r}\)-\(X_{r}Y_{r}Z_{r}\)), abbreviated as \(S_{r}\), as shown in Figure 12a,d. In Figure 12b, \(S_{t}\) (\(O_{t}\)-\(X_{t}Y_{t}Z_{t}\)), abbreviated as \(S_{t}\), is the cross-section coordinate system of helical rack in the tooth width direction. \(O_{t}X_{t}\) coincides with the rack pitch line and \(O_{t}Y_{t}\) coincides with the cogging symmetry line. \(H_{a}\) and \(H_{b}\) are the addendum and dedendum, respectively. \(M\) is the module, \(\alpha_{r}\) is the profile angle and \(u_{r}\) is the tooth height direction parameter. In Figure 12e, \(S_{n}\) (\(O_{n}\)-\(X_{n}Y_{n}Z_{n}\)), abbreviated as \(S_{n}\), is the normal section coordinate system of curvilinear rack. The normal section tooth profile of the curvilinear rack is the same as the cross-section tooth profile of the helical rack in the tooth width direction. Figure 12c,f show the transformation process from coordinate systems \(S_{t}\) and \(S_{n}\) to \(S_{r}\), respectively. \(\beta_{r}\) and \(b_{r}\) are the helical angle and tooth width parameter of the helical rack respectively. \(\theta_{r}\) and \(\rho_{r}\) are the arc angle and arc radius of the curvilinear rack, respectively. \(m\) is the module of the rack. The tooth profile equations in \(S_{t}\) and \(S_{n}\) can be expressed as  
\[\mathbf{r}_{t}(u_{r})=\mathbf{r}_{n}(u_{r})=\left[ \begin{array}{c}\pm u_{r}\sin \alpha_{r}\pm\frac{\pi m}{4}\\ u_{r}\cos\alpha_{r}\\ 0\\ 1\end{array} \right], \tag{12}\]  
where, the positive and negative signs, respectively, represent the left and right tooth surfaces of the rack. Figure 12: The tooth surface coordinate system of generating rack. The transformation matrix from \(S_{g}\) to \(S_{c}\) is  
\[\mathbf{M}_{cg}=\left[ \begin{array}{cccc}\cos\varphi&-sin\varphi&0&0\\ sin\varphi&\cos\varphi&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]. \tag{16}\]  
Then, the transformation matrix from \(S_{r}\) to \(S_{c}\) can be expressed as  
\[\mathbf{M}_{cr}=\mathbf{M}_{cg}\mathbf{M}_{gr}=\left[ \begin{array}{cccc}\cos\varphi&-sin \varphi&0&-r_{c}(\varphi\cos\varphi-sin\varphi)\\ sin\varphi&\cos\varphi&0&-r_{c}(\varphi\sin\varphi+\cos\varphi)\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]. \tag{17}\]  
The tooth surface equation of the generating gear in \(S_{c}\) can be expressed by \(r_{c}\left(u_{r},b_{r},\theta_{r}\right)\). \[r_{c}(u_{r},b_{r},\theta_{r})=\mathbf{M}_{cr}\mathbf{r}_{r}(u_{r},b_{r},\theta_{r}). \tag{18}\]  
Figure 14 is the generating process from the generating gear to the curve face gear pair. \(\sigma_{c}\), \(\sigma_{1}\), and \(\sigma_{2}\) are the pitch curve of the generating gear, noncircular gear, and curve face gear, respectively. When \(\sigma_{1}\) is purely rolling around \(\sigma_{2}\), \(\sigma_{c}\) is purely rolling inside \(\sigma_{1}\). According to the relative motion relationship between \(\sigma_{1}\) and \(\sigma_{2}\), the relative motion relationship between \(\sigma_{c}\) and \(\sigma_{2}\) can be obtained. The transformation matrix from the coordinate system \(S_{c}\) to \(S_{1}\) is  
\[\mathbf{M}_{1c}=\left[