## 3 Curvature Interference Limit Function of Meshing  
To enhance meshing efficiency, the worm gear is supposed to be configured with a left-hand rotational direction when paired with a right-hand worm. As the mesh of the right-hand worm gears with left-hand worms follows an analogous meshing theory, this configuration will not be further discussed. In this section, the portion of the face worm gear proximal to the inner diameter is designated as the inner end, while the opposite side is termed the outer end, as illustrated in Figure 5. The tooth surface of the face worm gear is denoted as \(\Sigma_{2}^{(S)}\). Figure 5: Coordinate system settings while machining face worm gear. During machining \(\Sigma_{2}^{(S)}\), a stationary frame \(\sigma_{02}\bigg{\{}O_{2};\stackrel{{\rightarrow}}{{i}}_{02}, \stackrel{{\rightarrow}}{{j}}_{02},\stackrel{{ \rightarrow}}{{k}}_{02}\bigg{\}}\) and a rotating frame \(\sigma_{2}\bigg{\{}O_{2};\stackrel{{\rightarrow}}{{i}}_{2}, \stackrel{{\rightarrow}}{{j}}_{2},\stackrel{{ \rightarrow}}{{k}}_{2}\bigg{\}}\) reflecting the original and present positions of the face worm gear are created. \(a_{12}\) is the center distance. While the worm turns around \(\stackrel{{\rightarrow}}{{k}}_{1}\) via a certain angle \(\varphi_{1}^{*}\), the turning angle of the worm gear is \(\varphi_{2}\), where \(\varphi_{2}=\varphi_{1}^{*}/i_{12}\) and \(i_{12}\) indicate the gear ratio. Detailed steps for creating \(\sigma_{02}\) and \(\sigma_{2}\) are given in ref. [31]. When the worm turns around \(\stackrel{{\rightarrow}}{{k}}_{1}\), the worm helicoid \(\Sigma_{1}^{(S)}\) can form a family of one-parameter surfaces in \(\sigma_{01}\). Thereupon, the equation for \(\Sigma_{1}^{(S)}\) can be obtained in \(\sigma_{01}\) from Equation (8) as:  
\[\left(\stackrel{{\rightarrow*}}{{r}}_{1}\right)_{01}=R\left[ \stackrel{{\rightarrow}}{{k}}_{01},\varphi_{1}\right]\left( \stackrel{{\rightarrow}}{{r}}_{1}\right)_{1}=x_{01}^{*} \stackrel{{\rightarrow}}{{i}}_{01}+y_{01}^{*}\stackrel{{ \rightarrow}}{{j}}_{01}+(z_{01}-p\varphi)\stackrel{{ \rightarrow}}{{k}}_{01}, \tag{11}\]  
whose normal vector is \(\left(\stackrel{{\rightarrow*}}{{n}}\right)_{01}=n_{x}^{*} \stackrel{{\rightarrow}}{{i}}_{01}+n_{y}^{*}\stackrel{{ \rightarrow}}{{j}}_{01}+n_{z}\stackrel{{\rightarrow}}{{k}}_{01}\), and coefficients are \(x_{01}^{*}=x_{01}\cos(\varphi_{1}-\varphi)-(y_{01}+a_{\rm d})\sin(\varphi_{1} -\varphi)\), \(y_{01}^{*}=x_{01}\sin(\varphi_{1}-\varphi)+(y_{01}+a_{\rm d})\cos(\varphi_{1} -\varphi)\), \(n_{x}^{*}=n_{x}\cos(\varphi_{1}-\varphi)-n_{y}\sin(\varphi_{1}-\varphi)\), and \(n_{y}^{*}=n_{x}\sin(\varphi_{1}-\varphi)+n_{y}\cos(\varphi_{1}-\varphi)\). During face worm gear machining operations, the worm angular velocity is conventionally designated as \(\left|\stackrel{{\rightarrow}}{{\omega}}_{1}\right|=1\)rad/s [3]. Accordingly, the angular velocity of the worm gear can be given as \(\left|\stackrel{{\rightarrow}}{{\omega}}_{2}\right|=1/i_{12}\)rad/s, where \(i_{12}\) represents the transmission ratio. As the face worm gear is left handed, the orientation of \(\stackrel{{\rightarrow}}{{\omega}}_{2}\) is along the negative direction of \(\stackrel{{\rightarrow}}{{k}}_{02}\). Hence, the angular velocity vector \(\stackrel{{\rightarrow}}{{\omega}}_{12}\) and the relative velocity vector \(\stackrel{{\rightarrow}}{{V}}_{12}\) can be attained in \(\sigma_{01}\) as:  
\[\left\{