## 3 Curvature Interference Limit Function of Meshing
\tag{22}\]  
By employing trigonometric identities, it is possible to obtain as  
\[[LB+(y_{\text{o1}}+a_{\text{d}})C]^{2}+(x_{\text{o1}}C+LA)^{2}-[x_{\text{o1}} B-(y_{\text{o1}}+a_{\text{d}})A]^{2}=p^{2}A_{1}\varphi^{2}+B_{1}p\varphi+C_{1}=0, \tag{23}\]  
where the coefficients are \(A_{1}=\left(L^{2}-x_{\text{o1}}^{2}\right)n_{x}^{2}+\left[L^{2}-(y_{\text{o1} }+a_{\text{d}})^{2}\right]n_{y}^{2}-2n_{x}n_{y}x_{\text{o1}}(y_{\text{o1}}+a_{ \text{d}})\), \(B_{1}=2\left\{n_{x}C\left[L^{2}-x_{\text{o1}}^{2}\right]-n_{y}C_{\text{d}} \left[L^{2}-(y_{\text{o1}}+a_{\text{d}})^{2}\right]-x_{\text{o1}}(y_{\text{o1 }}+a_{\text{d}})\left(n_{y}C_{B}-n_{x}C_{A}\right)\right.\)  
\(\left.+CL\left[n_{3x}(y_{\text{o3}}+a_{\text{31}})-n_{3y}x_{\text{o3}}\right]\right\}\), \(C_{1}=C_{B}^{2}\left(L^{2}-x_{\text{o1}}^{2}\right)+C_{A}^{2}\left[L^{2}-(y_{ \text{o1}}+a_{\text{d}})^{2}\right]+2C_{A}C_{B}x_{\text{o1}}\)  
\((y_{\text{o1}}+a_{\text{d}})\)\(\ \ +2C_{12}L\left[n_{3y}x_{\text{o1}}-n_{3x}(y_{\text{o1}}+a_{ \text{d}})\right]\left(z_{\text{o1}}+z_{\text{i2}}+\frac{L_{\text{i}}^{(1)}}{2 }\right)+\left[(y_{\text{o1}}+a_{\text{d}})^{2}+x_{\text{o1}}^{2}\right]C_{12}^ {2},\)  
\(C_{A}=n_{y}\left(z_{\text{o1}}+z_{\text{i2}}+\frac{L_{\text{i3}}^{(1)}}{2} \right)-n_{z}(y_{\text{o1}}+a_{\text{d}})\), and \(C_{B}=-n_{x}\left(z_{\text{o1}}+z_{\text{i2}}+\frac{L_{\text{i3}}^{(1)}}{2} \right)+n_{z}x_{\text{o1}}.\)  
By utilizing Equation (23), the curvilinear coordinate \(\varphi\) of key points \(\text{P}_{j}\) on the curvature interference limit line can be expressed as  
\[\varphi_{P_{j}}^{(S)}(\theta)=\frac{-B_{1}+\left(-1\right)^{S}\sqrt{B_{1}^{2}-A _{1}C_{1}}}{pA_{1}}. \tag{24}\]  
Figure 7: Projection in axial section: (**a**) projection of worm helical surface in the axial coordinate system; (**b**) projection of worm gear tooth surface in its axial coordinate system. The meshing equation in Equation (19) can be rephrased as \(\sin(\varphi_{1}-\varphi+\varphi_{0})=-C/\sqrt{A^{2}+B^{2}}\) concerning \(\varphi_{1}\). Consequently, within one meshing cycle, by utilizing this trigonometric equation \(\sin(\varphi_{1}-\varphi+\varphi_{0})\), one of the solutions for \(\varphi_{1}\) can be expressed as a function of \(\theta\) and \(\varphi\) as  
\[\varphi_{1}(\theta,\varphi)=\varphi-\varphi_{0}-\arcsin\frac{C}{\sqrt{A^{2}+ B^{2}}}, \tag{25}\]  
where \(\sin\varphi_{0}=\frac{B}{\sqrt{A^{2}+B^{2}}}\), and \(\cos\varphi_{0}=\frac{A}{\sqrt{A^{2}+B^{2}}}\). Finally, through elimination, the function \(u(\theta)\) from Equation (22), \(\varphi_{P_{j}}^{(S)}(\theta)\) in Equation (24), and \(\varphi_{1}(\theta,\varphi)\) in Equation (25) are substituted into the curvature interference limit function in Equation (19). As a result, a univariate equation \(\psi(\theta)=0\) with \(\theta\) can be derived. The necessary and sufficient condition for the equation \(\psi(\theta)=0\) to possess valid solutions within a given solving region is the existence of intersections between the curve of \(\psi(\theta)=0\) and the abscissa axis within the solving interval. Therefore, by employing geometric construction methods [31], both the existence of solutions and appropriate initial values for numerical iteration can be determined through graphical inspection. By observing the number of intersections between the curve of \(\psi(\theta)\) and the abscissa axis within the solving interval, the number of equation solutions of \(\psi(\theta)=0\) can be determined, which directly corresponds to the number of curvature interference limit lines. After determining the number of solutions, the curvature interference function \(\psi(\theta)=0\) is combined with boundary conditions to yield the loss function as  
\[L_{\text{Total}}=\lambda_{C}\psi(\theta)+\lambda_{GBC}L_{GBC}+\lambda_{KBC}L_{ KBC}, \tag{26}\]  
where \(L_{BC}\) are boundary conditions, and dynamically adjusting the weights \(\lambda_{C}\), \(\lambda_{GBC}\), and \(\lambda_{KBC}\) in \(L_{\text{Total}}\) can ensure that each condition can be satisfied in a balanced way during the model training process.