## 3 Curvature Interference Limit Function of Meshing
\begin{split}\left(\overset{\rightarrow}{\boldsymbol{\alpha}}_{ \xi}\right)_{01}&=R\left[\overset{\rightarrow}{\boldsymbol{ \kappa}}_{01},\varphi_{1}-\varphi\right]\left(\overset{\rightarrow}{ \boldsymbol{\beta}}_{1}\right)_{01}&=\overset{\rightarrow}{ \boldsymbol{\alpha}}_{\xi x}\overset{\rightarrow}{\boldsymbol{i}}_{01}+ \overset{\rightarrow}{\boldsymbol{\alpha}}_{\xi y}\overset{\rightarrow}{ \boldsymbol{j}}_{01}+A_{z}\overset{\rightarrow}{\boldsymbol{k}}_{01},\\ \left(\overset{\rightarrow}{\boldsymbol{\alpha}}_{\eta}\right)_{01 }&=R\left[\overset{\rightarrow}{\boldsymbol{k}}_{01},\varphi_{1}- \varphi\right]\left(\overset{\rightarrow}{\boldsymbol{\beta}}_{2}\right)_{01 }&=\overset{\rightarrow}{\boldsymbol{\alpha}}_{\eta x}\overset{ \rightarrow}{\boldsymbol{i}}_{01}+\overset{\rightarrow}{\boldsymbol{\alpha}}_ {\eta y}\overset{\rightarrow}{\boldsymbol{j}}_{01}+\delta z_{2z}\overset{ \rightarrow}{\boldsymbol{k}}_{01},\end{split}  \tag{16}\]  
where \(\alpha_{\xi x}=A_{x}\cos(\varphi_{1}-\varphi)-A_{y}\sin(\varphi_{1}-\varphi)\), \(\alpha_{\xi y}=A_{x}\sin(\varphi_{1}-\varphi)+A_{y}\cos(\varphi_{1}-\varphi)\), \(\alpha_{\eta x}=g_{2x}\cos(\varphi_{1}-\varphi)-g_{2y}\sin(\varphi_{1}-\varphi)\), and \(\alpha_{\eta y}=g_{2x}\sin(\varphi_{1}-\varphi)+g_{2y}\cos(\varphi_{1}-\varphi)\). Based on this, using Equation (16), the normal vector \(\overset{\rightarrow}{\boldsymbol{N}}\) of the contact line can be acquired in \(\sigma_{01}^{*}\) as:  
\[\left(\overset{\rightarrow}{\boldsymbol{N}}\right)_{01}=N_{\xi}\left(\overset {\rightarrow}{\boldsymbol{\alpha}}_{\xi}\right)_{01}+N_{\eta}\left(\overset{ \rightarrow}{\boldsymbol{\alpha}}_{\eta}\right)_{01}, \tag{17}\]  
where \(N_{\xi}=\left\{\frac{g_{2x}}{\Psi_{4}}\left[-g_{2z}\left(A_{x}x_{01}^{*}-A_{y}V _{x}\right)+\mu\left(g_{2x}x_{01}^{*}-g_{2y}V_{x}\right)\right]-\frac{g_{2z}}{ i_{12}}\right\}\sin(\varphi_{1}-\varphi)\)  
\(+\left\{\frac{g_{2z}}{\Psi_{4}}\left[-g_{2z}\left(A_{y}x_{01}^{*}+A_{x}V_{x} \right)+\mu\left(g_{2y}x_{01}^{*}+g_{2x}V_{x}\right)\right]-\frac{g_{2z}}{i_{1 2}}\right\}\cos(\varphi_{1}-\varphi)\)  
\(+\left[\frac{g_{2z}\left(x_{01}^{*}+A_{y}\right)\left(-A_{z}\right)+i_{12} \Psi_{4}}{i_{12}\Psi_{4}}\right]g_{2z}\), and \(N_{\eta}=\left\{\frac{\mu}{\Psi_{4}}\left[g_{2z}\left(A_{x}x_{01}^{*}-A_{y}V _{x}\right)-\mu\left(g_{2x}x_{01}^{*}-g_{2y}V_{x}\right)\right]+\right.\)  
\(\left.k_{2}\left(g_{2x}x_{01}^{*}-g_{2y}V_{x}\right)+\frac{A_{x}}{i_{12}} \right\}\sin(\varphi_{1}-\varphi)\)\(+\left\{\frac{\mu}{\Psi_{4}}\left[g_{2z}\left(g_{2x}x_{01}^{*}-g_{2y}V_{x} \right)-\mu\left(g_{2y}x_{01}^{*}+g_{2x}V_{x}\right)\right]+\right.\)  
\(\left.k_{2}\left(g_{2y}x_{01}^{*}+g_{2x}V_{x}\right)+\frac{A_{y}}{i_{12}} \right\}\cos(\varphi_{1}-\varphi)\)\(+\frac{\left(x_{01}^{*}+a\right)g_{2z}}{i_{12}}\left[\frac{\mu}{\Psi_{4}} \left(A_{z}-\mu\right)+k_{2}\right]-A_{z}\). From \(\overset{\rightarrow}{\boldsymbol{V}}_{12}\) in Equation (12), \(\overset{\rightarrow}{\boldsymbol{N}}\) in Equation (17), and \(\frac{\partial\phi}{\partial\varphi_{1}^{*}}\) in Equation (15), the curvature interference limit function \(\psi\) can be attained as:  
\[\psi(u,\theta,\varphi,\varphi_{1}) =\left(\overset{\rightarrow}{\boldsymbol{N}}\right)_{01}\cdot \left(\overset{\rightarrow}{\boldsymbol{V}}_{12}\right)_{01}+\phi_{\varphi 1}=N_{\xi}\left(\overset{\rightarrow}{\boldsymbol{V}}_{12}\right)_{01}\cdot \left(\overset{\rightarrow}{\boldsymbol{\alphaEquation (20), together with the boundary conditions of the tooth surface, is used to construct the loss function. Subsequently, the variables to be solved are determined. A deep neural network is then established, comprising the initialization of the network, forward propagation of outputs, loss computation, and backpropagation for optimization. During the training process, if the loss does not meet the prescribed accuracy threshold, the model continues to iterate and update. Once the loss has converged, the numerical results of the key points on the curvature interference boundary are obtained.