## 2 Geometric Modeling of Conjugate Curve-Face Gear Tooth Surface
### Equations of Curve-Face Gear Surface  
The family of the shaper tooth surface \(\Sigma_{s}\) in coordinate system \(S_{2}\) can be expressed  
\[\overset{\rightarrow}{r}_{2}^{(2)}(\theta_{s},\phi_{s},u_{s})=M_{2s}(\theta_{s} )\overset{\rightarrow}{r}_{s}^{(s)}(\phi_{s},u_{s}) \tag{6}\]  
Moreover, the coordinate transformation in transition from \(S_{s}\) to \(S_{2}\) is represented by the matrix:  
\[M_{2s}(\theta_{s})=M_{2f}M_{fs}=\left[\begin{array}{cccc}\cos\theta_{2}\cos \theta_{s}&-\cos\theta_{2}\sin\theta_{s}&\sin\theta_{2}&0\\ -\sin\theta_{2}\cos\theta_{s}&\sin\theta_{2}\sin\theta_{s}&\cos\theta_{2}&0\\ -\sin\theta_{s}&-\cos\theta_{s}&0&-l(\theta_{2})\\ 0&0&0&1\end{array}\right] \tag{7}\]  
where, \(\theta_{s}\) and \(\theta_{2}\) are the angular displacements of shaper and curve-face gear, respectively, \(l(\theta_{2})\) is the axial displacements of shaper. Known in advance that the rotation angle of shaper can be calculated as \(\theta_{s}(\theta_{2})=\int_{0}^{\theta_{2}}\sqrt{R^{2}+f^{\prime 2}( \theta)}d\theta/r_{s}\), in terms of the fundamental transmission geometry relation through subsequent calculation in Section 3.1.  
To simplify the derivations of the equation of meshing, the frame coordinate system \(S\) is preferable to be used for the relative velocity vector \(\overset{\rightarrow}{v}_{s2}^{(f)}\) and unit normal vector \(\overset{\rightarrow}{n}_{s}^{(f)}\) to shaper tooth surface at any candidate point \(M\) in mesh. The relative velocity between two gear teeth in direct contact can be found in \(S\)  
\[\overset{\rightarrow}{v}_{s2}^{(f)}=\left[\begin{array}{c}-(x_{s}\sin \theta_{s}+y_{s}\cos\theta_{s})\omega_{s}+z_{s}\omega_{2}\\ (x_{s}\cos\theta_{s}-y_{s}\sin\theta_{s})\omega_{s}+v_{0s}\\ -(x_{s}\cos\theta_{s}-y_{s}\sin\theta_{s})\omega_{2}\end{array}\right] \tag{8}\]  
The equation of meshing between shaper \(\Sigma_{s}\) and curve-face gear \(\Sigma_{2}\)  
\[\begin{array}{ll}f(\theta_{s},\phi_{s},u_{s})&=\overset{\rightarrow}{n}_{s }^{(f)}\cdot\overset{\rightarrow}{v}_{s2}^{(f)}\\ &=-r_{s}\cos\alpha_{0s}\omega_{s}+u_{s}\omega_{2}\sin(\theta_{s}+\phi_{s}\pm \alpha_{0s})-v_{0s}\cos(\theta_{s}+\phi_{s}\pm\alpha_{0s})\\ &=0\end{array} \tag{9}\]  
The curve-face gear tooth surface \(\Sigma_{2}\) can be obtained by the envelope to the family of surfaces \(\Sigma_{s}\). Consequently, the equations of the generated curve-face gear surface can be expressed by vector function \(\overset{\rightarrow}{r}_{2}^{(2)}(\theta_{s},\phi_{s})\) and the equation of meshing \(f(\theta_{s},\phi_{s},u_{s})=0\), that is:  
\[\overset{\rightarrow}{r}_{2}^{(2)}(\theta_{s},\phi_{s})=\left[\begin{array}[] {c}r_{s}\cos\theta_{2}[\cos(\theta_{s}+\phi_{s})+(\phi_{s}-\phi_{0s})\cos \alpha_{0s}\sin(\theta_{s}+\phi_{s}\pm\alpha_{0s})]+\sin\theta_{2}u_{s}(\theta _{s},\phi_{s})\\ -r_{s}\sin\theta_{2}[\cos(\theta_{s}+\phi_{s})+(\phi_{s}-\phi_{0s})\cos\alpha_ {0s}\sin(\theta_{s}+\phi_{s}\pm\alpha_{0s})]+\cos\theta_As depicted in the partial enlarged version of Figure 5, the contact curves on the curve-face gear tooth surface \(\Sigma_{2}\) correspond to the meshing of the shaper with curve-face gear. In addition, \(B_{2}\), \(c_{2}\) and \(h_{2}\) are correspondingly tooth width, clearance and whole depth of the curve-face gear, respectively.  
It is worth noting that, the surfaces of the curve-face gear and cylindrical gear with the same specification as the shaper are also in line contact in the process of meshing. Besides, according to the conditions of undercutting and pointing, the determination of tooth width of curve-face gears has been discussed in detail in our previous research [42], so those won't be covered again here.