## 2 Rack Cutters and Gear Surfaces  
Among the various methods for manufacturing gears, we assume that a rack cutter is used to manufacture the gears and analyze them using an analytical approach [22; 23; 24]. Figure 1 shows the profile of a basic rack cutter and the relationship between the coordinate systems \(\mathbf{C}_{c}[X_{c},Y_{c},Z_{c}]\) and \(\mathbf{C}_{g}[X_{g},Y_{g},Z_{g}]\) of the rack cutter and the spur gear, respectively. The rack design parameters \(h_{ap}\), \(h_{fp}\), \(p_{fp}\) are addendum, dedendum and fillet radius, respectively. The coordinate systems \(\mathbf{C}_{1}[X_{1},Y_{1},Z_{1}]\) and \(\mathbf{C}_{2}[X_{2},Y_{2},Z_{2}]\) are the reference coordinate systems of the manufacturing process. Here, \(u_{c}\) is the \(x\)-coordinate of the position vector \(r_{c}\), \(d_{c}\) is the distance between the rack cutter and the gear, and \(\alpha\) is the pressure angle of the rack cutter profile. In the coordinate system \(\mathbf{C}_{c}\) of the rack cutter, the position vector \(\vec{r}_{c}\) of the rack cutter surface can be represented as  
\[\vec{r}_{c}=\begin{bmatrix}u_{c}\\ u_{c}\tan(\frac{\pi}{2}-\alpha)\\ \theta_{c}\\ 1\end{bmatrix}, \tag{1}\]  
where \(\theta_{c}\) denotes the \(z\)-coordinate of \(\vec{r}_{c}\). When the rack cutter translates along \(\overline{O_{c}x_{c}}\) by \(-\phi_{c}\), the gear blank rotates about the axis \(\overline{O_{g}z_{g}}\) by \(\phi_{g}\) at the same time. The relationship between these movements is given by  
\[\phi_{g}=\frac{\phi_{c}}{d_{c}}. \tag{2}\]  
Figure 1: Relationship between the coordinate systems \(C_{c}\) and \(C_{g}\) of the rack cutter and the spur gear, respectively.  
Figure 2 shows the trace of the rack cutter profile in the coordinate system \(\mathbf{C}_{g}\) of the spur gear. The position vector \(\vec{r}_{g}\) of the gear in \(\mathbf{C}_{g}[X_{g},Y_{g},Z_{g}]\) can be written in terms of the family of rack cutter surfaces \(\vec{r}_{c}\) as  
\[\mathbf{M}_{gc}=\begin{bmatrix}\cos\phi_{g}&\sin\phi_{g}&0&-\phi_{c}\cos\phi_ {g}+d_{c}\sin\phi_{g}\\ -\sin\phi_{g}&\cos\phi_{g}&0&-\phi_{c}\sin\phi_{g}+d_{c}\cos\phi_{g}\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix},\quad\vec{r}_{g}(u_{c},\phi_{c},\theta_{c})=\quad M_{gc}\vec{r}_ {c}. \tag{3}\]  
The position \(\vec{r}_{g}\) of the gear is described in a three-dimensional volume about \(u_{c},\phi_{c}\), and \(\theta_{c}\). Meanwhile, the gear surface is an envelope of the family of rack cutter surfaces, and its profile is an involute curve. The equation of meshing satisfied by the envelope of the family of rack cutter surfaces is given by  
\[f_{c}(u_{c},\phi_{c},\theta_{c})= \left(\frac{\partial\vec{r}_{g}^{*}}{\partial u_{c}}\times \frac{\partial\vec{r}_{g}^{*}}{\partial\theta_{c}}\right)\cdot\frac{\partial \vec{r}_{g}^{*}}{\partial\phi_{c}}=0, \tag{4}\]  
where \(\vec{r}_{g}^{*}\) is a vector obtained by eliminating the last element of \(\vec{r}_{g}\). It should be noted that Equation (4) can be rewritten as an explicit function of \(u_{c}\) in terms of \(\phi_{c}\):  
\[u_{c}= \frac{\phi_{c}\tan^{2}\alpha}{1+\tan^{2}\alpha}\quad, \tag{5}\]  
which allows us to represent the position \(\vec{r}_{g}\) of the gear in a two-dimensional space in terms of \(\phi_{c}\) and \(\theta_{c}\), which is the manufactured gear surface that we can analyze.