## 3 Analytical Engagement Angle Model
### Calculation of Cutter-Workpiece Engagement Angle
\begin{array}{l}-\arccos\frac{\mathbf{j}\cdot\overset{ \rightarrow}{C\mathcal{P}_{1}}}{\left|\left|\right|\overset{\rightarrow}{C \mathcal{P}_{2}}\right|}=-\arccos\frac{y_{2}(\theta_{pre2})-y_{C}(\theta)}{R_{ c}},\;\text{when}\;\mathbf{i}\cdot\overset{\rightarrow}{C\mathcal{P}_{2}} <0\;\text{and}\;\mathbf{j}\cdot\overset{\rightarrow}{C\mathcal{P}_{2}}>0\\ \\ \arccos\frac{\mathbf{j}\cdot\overset{\rightarrow}{C\mathcal{P}_{2}}}{\left| \left|\right|\overset{\rightarrow}{C\mathcal{P}_{2}}\right|}=\arccos\frac{y_ {2}(\theta_{pre3})-y_{C}(\theta)}{R_{c}},\;\text{when}\;\mathbf{i}\cdot \overset{\rightarrow}{C\mathcal{P}_{2}}\geq 0\\ \\ 2\pi-\arccos\frac{\mathbf{j}\cdot\overset{\rightarrow}{C\mathcal{P}_{2}}}{ \left|\left|\right|\right|\overset{\rightarrow}{C\mathcal{P}_{2}}\right|}=2\pi -\arccos\frac{y_{2}(\theta_{pre3})-y_{C}(\theta)}{R_{c}},\;\text{when}\; \mathbf{i}\cdot\overset{\rightarrow}{C\mathcal{P}_{2}}<0\;\text{and}\; \mathbf{j}\cdot\overset{\rightarrow}{C\mathcal{P}_{2}}<0\end{array} \right. \tag{17}\]  
where \(\mathbf{i}\) is the unit vector of the \(x_{c}\)-axis. \(\mathbf{j}\) is the unit vector of the \(y_{c}\)-axis. Similarly, the exit angle \(\varphi_{ex}\) can be calculated as follows:  
\[\varphi_{ex}=\varphi_{1}=\left\{ \begin{array}{l}-\arccos\frac{\mathbf{j}\cdot \overset{\rightarrow}{C\mathcal{P}_{1}}}{\left|\left|\right|\overset{ \rightarrow}{C\mathcal{P}_{1}}\right|}=-\arccos\frac{y_{1}(\theta)-y_{C}( \theta)}{R_{c}},\;\text{when}\;\mathbf{i}\cdot\overset{\rightarrow}{C \mathcal{P}_{1}}<0\;\text{and}\;\mathbf{j}\cdot\overset{\rightarrow}{C \mathcal{P}_{1}}>0\\ \\ \arccos\frac{\mathbf{j}\cdot\overset{\rightarrow}{C\mathcal{P}_{1}}}{\left| \left|\right|\overset{\rightarrow}{C\mathcal{P}_{1}}\right|}=\arccos\frac{y_ {1}(\theta)-y_{C}(\theta)}{R_{c}},\;\text{when}\;\mathbf{i}\cdot\overset{ \rightarrow}{C\mathcal{P}_{1}}\geq 0\\ \\ 2\pi-\arccos\frac{\mathbf{j}\cdot\overset{\rightarrow}{C\mathcal{P}_{1}}}{ \left|\left|\right|\overset{\rightarrow}{C\mathcal{P}_{1}}\right|}=2\pi- \arccos\frac{y_{1}(\theta)-y_{C}(\theta)}{R_{c}},\;\text{when}\;\mathbf{i} \cdot\overset{\rightarrow}{C\mathcal{P}_{1}}<0\;\text{and}\;\mathbf{j}\cdot \overset{\rightarrow}{C\mathcal{P}_{1}}<0\end{array} \right. \tag{18}\]  
For segment \(A^{\prime}S^{\prime}\) of the toolpath, as shown in Figure 7, the two intersection points of the cutting-edge circle and the previous milling contour are involved in the engagement angle calculation. Point \(P_{2}\) is used to calculate the start angle \(\varphi_{st}\) by means of Equation (17). Point \(P_{3}\) is used to calculate the exit angle \(\varphi_{ex}\), which can be expressed as follows:  
\[\varphi_{ex}=\left\{ \begin{array}{l}\arccos\frac{\mathbf{j}\cdot\overset{ \rightarrow}{C\mathcal{P}_{1}}}{\left|\left|\right|\overset{\rightarrow}{C \mathcal{P}_{3}}\right|}=\arccos\frac{y_{3}(\theta_{pre3})-y_{C}(\theta)}{R_{c }},\;\text{when}\;\mathbf{i}\cdot\overset{\rightarrow}{C\mathcal{P}_{3}}\geq 0\\ \\ 2\pi-\arccos\frac{\mathbf{j}\cdot\overset{\rightarrow}{C\mathcal{P}_{3}}}{ \left|\left|\right|\overset{\rightarrow}{C\mathcal{P}_{3}}\right|}=2\pi- \arccos\frac{y_{3}(\theta_{pre3})-y_{C}(\theta)}{R_{c}},\;\text{when}\; \mathbf{i}\cdot\overset{\rightarrow}{C\mathcal{P}_{3}}<0\end{array} \right. \tag{19}\]  
Figure 7: Engagement angle calculation for the \(A^{\prime}S^{\prime}\) segment of the toolpath. Point \(P_{1}\) is used to calculate the angle \(\varphi_{1}\), which can be expressed as follows:  
\[\varphi_{1}=\left\{ \begin{array}{l}\arccos\frac{\mathbf{j}\cdot\overset{ \rightarrow}{C\!P_{1}}}{\left|\left|\left|\overset{\rightarrow}{C\!P_{1}} \right|\right|}=\arccos\frac{\mathbf{\
u}_{1}(\theta)-\mathbf{\
u}_{C}(\theta) }{R_{c}},\text{ when }\mathbf{i}\cdot\overset{\rightarrow}{C\!P_{1}}\geq 0\\ \\ 2\pi-\arccos\frac{\mathbf{j}\cdot\overset{\rightarrow}{C\!P_{1}}}{\left|\left| \left|\overset{\rightarrow}{C\!P_{1}}\right|\right|}=2\pi-\arccos\frac{ \mathbf{\
u}_{1}(\theta)-\mathbf{\
u}_{C}(\theta)}{R_{c}},\text{ when }\mathbf{i}\cdot \overset{\rightarrow}{C\!P_{1}}<0\end{array} \right.