## 2 Cutting Engagement of Worm
### Equation of Worm Tooth Surface  
From Figure 4, different coordinate system settings of \(\sigma_{\text{a}}\) yield distinct equations of \(\Sigma_{\text{g}}^{(S)}\). Therefore, the equations of \(\Sigma_{\text{g}}^{(S)}\) can be uniformly expressed within \(\sigma_{\text{d}}\) as:  
\[\left(\overset{\rightarrow}{\boldsymbol{r}}_{\text{d}}\right)_{\text{d}}=u \sin\delta_{\text{g}}^{(S)}\sin\Bigl{(}\frac{\pi}{S}-\theta\Bigr{)}\overset{ \rightarrow}{\boldsymbol{i}}_{\text{d}}+(-1)^{S}u\sin\delta_{\text{g}}^{(S)} \cos\Bigl{(}\frac{\pi}{S}-\theta\Bigr{)}\overset{\rightarrow}{\boldsymbol{j }}_{\text{d}}+(-1)^{S}\biggl{(}u\cos\delta_{\text{g}}^{(S)}-\frac{r_{\text{g} }}{\tan\delta_{\text{g}}^{S}}\Bigr{)}\overset{\rightarrow}{\boldsymbol{k}}_{ \text{d}},\,\,(S=1,2), \tag{3}\]  
Then, through coordinate rotation \(\sigma_{\text{l}}\rightarrow\sigma_{\text{o},\text{l}-2}\), the radius vector \(\overset{\rightarrow}{\boldsymbol{r}}_{\text{l}}\) can be acquired in \(\sigma_{\text{o},\text{l}-2}\) as:  
\[\left(\overset{\rightarrow}{\boldsymbol{r}}_{\text{d}}\right)_{\text{o1}}=R \left[\overset{\rightarrow}{\boldsymbol{j}}_{\text{o1}},\gamma\right]R \left[\overset{\rightarrow}{\boldsymbol{i}}_{\text{o4}},\varepsilon_{S} \right]\left(\overset{\rightarrow}{\boldsymbol{r}}_{\text{d}}\right)_{\text{d }}=x_{\text{o1}}\overset{\rightarrow}{\boldsymbol{i}}_{\text{o1}}+y_{\text{ o1}}\overset{\rightarrow}{\boldsymbol{j}}_{\text{o1}}+z_{\text{o1}}\overset{ \rightarrow}{\boldsymbol{k}}_{\text{o1}} \tag{4}\]  
whose normal vector is \(\left(\overset{\rightarrow}{\boldsymbol{n}}\right)_{\text{o1}}=n_{x}\overset{ \rightarrow}{\boldsymbol{i}}_{\text{o1}}+n_{y}\overset{\rightarrow}{ \boldsymbol{j}}_{\text{o1}}+n_{z}\overset{\rightarrow}{\boldsymbol{k}}_{ \text{o1}}\). The symbols \(R\left[\overset{\rightarrow}{\boldsymbol{j}}_{\text{o1}},\gamma\right]\) and \(R\left[\overset{\rightarrow}{\boldsymbol{i}}_{\text{o4}},\varepsilon_{S}\right]\) represent the rotating matrix [3]. After deduction, the components of \(\left(\overset{\rightarrow}{\boldsymbol{r}}_{\text{d}}\right)_{\text{o1}}\) and \(\left(\overset{\rightarrow}{\boldsymbol{n}}\right)_{\text{o1}}\) are \(x_{\text{o1}}=A_{x}u+B_{x},y_{\text{o1}}=A_{y}u+B_{y}\), \(z_{\text{o1}}=A_{z}u+B_{z}\), and \(n_{x}=A_{x}\cot\delta_{\text{g}}^{(S)}-\frac{(-1)^{S}\cos\varepsilon_{S}\cos \gamma}{\sin\delta_{\text{g}}^{(S)}}\), \(n_{y}=A_{y}\cot\delta_{\text{g}}^{(S)}+\frac{(-1)^{S}\sin\varepsilon_{S}}{\sin \delta_{\text{g}}^{(S)}}\), and \(n_{z}=A_{z}\cot\delta_{\text{g}}^{(S)}-\frac{(-1)^{S}\cos\varepsilon_{S}\cos \gamma}{\sin\delta_{\text{g}}^{(S)}}\). The coef  
Figure 4: Relative positions of coordinate systems during grinding \(i\)- and \(e\)-flanks. \[ \begin{array}{l}\text{ficients are }A_{x}=\sin\delta_{\text{g}}^{(S)}\cos\gamma\Big{[} \sin\big{(}\frac{\pi}{S}-\theta\big{)}+(-1)^{5}\sin\varepsilon_{S}\tan\gamma \cos\big{(}\frac{\pi}{S}-\theta\big{)}\Big{]}+(-1)^{5}\sin\gamma\\ \cos\varepsilon_{S}\cos\delta_{\text{g}}^{(S)},\ A_{y}\ =\ (-1)^{S}\sin\delta_{\text{g}}^{(S)} \Big{[}\cos\varepsilon_{S}\cos\big{(}\frac{\pi}{S}-\theta\big{)}+\frac{(-1)^{5 }\sin\delta_{\text{g}}^{(S)}\sin\varepsilon_{S}-\sin\varepsilon_{S}}{\cos \delta_{\text{g}}^{(S)}}\Big{]},\ A_{z}=-\sin\delta_{\text{g}}^{(S)}\sin\gamma \Big{[}\sin\big{(}\frac{\pi}{S}-\theta\big{)}+(-1)^{5+1}\sin\varepsilon_{S} \cot\gamma\cos\big{(}\frac{\pi}{S}-\theta\big{)}\Big{]}+(-1)^{5}\cos\varepsilon _{S}\cos\gamma\cos\delta_{\text{g}}^{(S)},\\ B_{x}=(-1)^{S+1}\frac{\
u_{\text{g}}^{(S)}\cos\varepsilon_{S}\sin\gamma}{ \tan\delta_{\text{g}}^{(S)}},\ B_{y}=(-1)^{S}\frac{\
u_{\text{g}}^{(S)}\sin \varepsilon_{S}}{\tan\delta_{\text{g}}^{(S)}},\text{ and }B_{z}=(-1)^{S+1}\frac{\
u_{\text{g}}^{(S)}\cos \varepsilon_{S}\cos\gamma}{\tan\delta_{\text{g}}^{(S)}}.\end{array} \]  
Significantly, the aforementioned coefficients \(x_{01}\), \(y_{01}\), \(z_{01}\), \(n_{x}\), \(n_{y}\), and \(n_{z}\) are functions of \(\theta\). In contrast, the coefficients \(B_{x}\), \(B_{y}\), and \(B_{z}\) are all constants.