## 3 Research Method
### Tooth Profile Precise Functions Solving
Suppose the coordinates of point K are (\(X_{K}\), \(Y_{K}\)), then the equation of the line PK is as follows:  
\[\xi+\frac{\eta}{\tan\psi}=-S \tag{10}\]  
The point (\(\xi\), \(\eta\)) is the center of curvature corresponding to the point (\(x\),\(y\)) on the curve \(y=\mathrm{f}(x)\), and its advanced mathematical definition is described in Equations (11) and (12):  
\[\xi=x-y^{\prime}\frac{1+y^{\prime 2}}{y^{\prime\prime}} \tag{11}\]  
\[\eta=y+\frac{1+y^{\prime 2}}{y^{\prime\prime}} \tag{12}\]  
The corresponding K point coordinates meet the following:  
\[\xi+\eta Y_{K}^{\prime}=X_{K}+Y_{K}Y_{K}^{\prime} \tag{13}\]  
With the following additional formula:  
\[\tan\psi=\frac{1}{Y_{K}^{\prime}} \tag{14}\]  
Combining the Equations (11)-(14) above, we can obtain  
\[\xi+\eta Y_{K}^{\prime}=-S \tag{15}\]  
With a series value of \(S\), the K point coordinates can be obtained. So far, the solution of the conjugate surface of the hob has been completed. Next, the corresponding positions of the cutting points B and K are determined on the processed gear blank. In order to facilitate further programming calculations, the method described in the literature ([PERSON], 2011) [27] needs to be used for coordinate transformation. Suppose the coordinate system on the gear to be processed is X\({}_{2}\)O\({}_{2}\)Y\({}_{2}\); make the transitional relationship of \(R\) (the gear pitch circle radius) as Equation (16):  
\[S=R\psi \tag{16}\]  
Then, the coordinate system X\({}_{1}\)-O\({}_{1}\)-Y\({}_{1}\) and the coordinate system X\({}_{2}\)-O\({}_{2}\)-Y\({}_{2}\) relationship will be as follows:  
\[ \begin{cases}x_{2}=(S+x_{1})\cos\varphi-(R+y_{1})\sin\varphi\\ y_{2}=(S+x_{1})\sin\varphi-(R+y_{1})\cos\varphi\end{cases}  \tag{17}\]  
According to the formulas above, the coordinate transformation between the coordinate system X\({}_{1}\)-O\({}_{1}\)-Y\({}_{1}\) and the coordinate system X\({}_{2}\)-O\({}_{2}\)-Y\({}_{2}\) can be realized, and the calculation of the end face tooth profile of the helical gear is completed. Finally, the tooth profile of each section parallel to the section needs to be solved. The offset angle \(\theta\) of the front and rear ends of the helical gear on the X-O-Y plane is described as follows:  
\[\theta=\frac{b}{R\cot\beta} \tag{18}\]  
In Equation (18), \(b\) is the tooth width, \(R\) is the index circle radius, and \(\beta\) is the helix angle. This is in addition to the following Their geometric relationship comes as following:  
\[R=\frac{mz}{2\cos\beta} \tag{19}\]  
Combining the Equations (18) and (19) above, we can obtain:  
\[\theta=\frac{2b\sin\beta}{mz} \tag{20}\]  
Suppose the coordinates of a point \(G\) on the tooth profile of the front face are (\(x_{G}\), \(y_{G}\)), and the coordinates of point G\({}^{\prime}\) G\({}^{\prime}\) on the corresponding back face are (\(x_{G^{\prime}},y_{G^{\prime}}\)), then we can ultimately obtain Equation (21):  
\[ \begin{cases}x_{G^{\prime}}=y_{G}\sin\theta+x_{G}\cos\theta\\ y_{G^{\prime}}=y_{G}\cos\theta-x_{G}\sin\theta\end{cases}  \tag{21}\]  
For different sections, the value of \(\theta\) changes according to the value of \(b\), and the corresponding tooth profile of all sections are obtained consecutively.