## 2 Materials and Methods  
In this study, the gear tooth meshing problem was described with surface tangency conditions. The two meshing teeth surfaces \(\Sigma_{1}\) and \(\Sigma_{2}\) are assumed to be a continuous \(C^{1}\) and rotating around a fixed axis \(\overrightarrow{w}_{i}\):  
\[\overrightarrow{n}_{i}=\frac{\partial\overrightarrow{r}_{i}}{\partial \overrightarrow{u}_{i}}\times\frac{\partial\overrightarrow{r}_{i}}{\partial \overrightarrow{v}_{i}},i=1,2 \tag{1}\]  
where \(\overrightarrow{r}_{i}\left(\overrightarrow{u}_{i},\overrightarrow{v}_{i} \right)\in C^{1}\), \(\left(\overrightarrow{u}_{i},\overrightarrow{v}_{i}\right)\in E_{i}\) (Figure 1). The parameterization for \(\overrightarrow{u}_{i}\) corresponds to the radial distance of a contact point from the surface axis of rotation, and \(\overrightarrow{v}_{i}\) is the algebraic projection on the same axis, so:  
\[\overrightarrow{u}_{i}=\left|\left|\left|\overrightarrow{r}_{i}-proj_{ \overrightarrow{w}_{i}}\overrightarrow{r}_{i}\right|\right|\right. \tag{2}\]  
\[\overrightarrow{v}_{i}=\overrightarrow{w}_{i}\cdot proj_{\overrightarrow{w}_ {i}}\overrightarrow{r}_{i}, \tag{3}\]  
Thus, \(\overrightarrow{u}_{i}\) and \(\overrightarrow{v}_{i}\) form a right angle, where \(\overrightarrow{u}_{i}\) is the length of the vertical leg, and \(\overrightarrow{v}_{i}\) is the horizontal leg aligned along the axis of rotation \(\overrightarrow{w}_{i}\), with the hypotenuse \(\overrightarrow{r}_{i}\) is the position vector of any point on the surface \(\Sigma_{i}\). The surface equations \((\overrightarrow{f}_{i})\) can be expressed in terms of the vectors mentioned above and by introducing a rotation matrix \(R_{i}(\phi_{i})\), corresponding to the rotation around the axis \(\overrightarrow{w}_{i}\) to the angle \(\phi_{i}\). Hence, vector \(\overrightarrow{r}_{i}\) can be written as expressed as:  
\[\overrightarrow{r}_{i}=R_{i}(\phi_{i})\overrightarrow{f}_{i}\left( \overrightarrow{u}_{i},\overrightarrow{v}_{i}\right),\ \ \ \ \ i=1,2 \tag{4}\]  
The surface tangency condition formulates the surface contact equations:  
\[\overrightarrow{n}_{1}(u_{1},v_{1})\times\overrightarrow{n}_{2}(u_{2},v_{2})= \overrightarrow{0}, \tag{5}\]  
When substituting Equation (1) to Equations (4) and (5) to express surface normal, the tangency condition yields:  
\[R_{1}(\phi_{1})\left(\frac{\partial\overrightarrow{f}_{1}(u_{1},v_{1})}{ \partial u_{1}}\times\frac{\partial\overrightarrow{f}_{1}(u_{1},v_{1})}{ \partial v_{1}}\right)\times R_{2}(\phi_{2})\left(\frac{\partial\overrightarrow {f}_{2}(u_{2},v_{2})}{\partial u_{2}}\times\frac{\partial\overrightarrow{f}_ {2}(u_{2},v_{2})}{\partial v_{2}}\right)=\overrightarrow{0}, \tag{6}\]  
Figure 1: The contact schematic of the gear teeth surfaces, where point A is the contact point between the surfaces \(\Sigma_{1}\) and \(\Sigma_{2}\), which rotate around the fixed axes \(\mathrm{w}_{1}\) and \(\mathrm{w}_{2}\), respectively. Furthermore, a common point of contact can be expressed as a zero-vector sum:  
\[\overset{\rightarrow}{r}_{1}(u_{1},v_{1})-\overset{\rightarrow}{a}_{12}- \overset{\rightarrow}{r}_{2}(u_{2},v_{2})=\overset{\rightarrow}{0}, \tag{7}\]  
The angle of rotation of the touching surface \(\phi_{2}\) can be determined by the function \(u_{2}\) and \(v_{2}\), which can be calculated directly in terms of \(u_{1}\) and \(v_{1}\) from specific geometric relationships:  
\[ \begin{array}{c}\overset{\rightarrow}{v}_{2}=proj_{\overset{\rightarrow} {w}_{2}}\overset{\rightarrow}{r}_{2}(u_{2},v_{2})=proj_{\overset{\rightarrow} {w}_{2}}\overset{\rightarrow}{f}_{2}(u_{2},v_{2})=\overset{\rightarrow}{w}_ {2}\cdot\left(\overset{\rightarrow}{r}_{1}(u_{1},v_{1})-\overset{ \rightarrow}{a}_{12}\right)=\\ \overset{\rightarrow}{w}_{2}\cdot\left(R_{1}(\phi_{1})f_{1}(u_{1},v_{1})- \overset{\rightarrow}{a}_{12}\right)=\mathcal{F}_{\text{\tiny$p$}}(\phi_{1},u_{1},v_{1}),\end{array}  \tag{8}\]  
where \(\mathcal{F}_{\text{\tiny$p$}}(\phi_{1},u_{1},v_{1})\) is the function of \(\phi_{1}\), \(u_{1}\), and \(v_{1}\).