## 3 Analytical Engagement Angle Model
### Effective Cutting Interval of Milling Contour
The point on the toolpath corresponding to the revolution angle of \(\theta_{S,2}\) is defined as \(S^{\prime}\), whose coordinates can be expressed as follows:  
\[\overset{\rightarrow}{OS^{\prime}}= \begin{bmatrix}x_{C}(\theta_{S,2})&y_{C}( \theta_{S,2})\end{bmatrix}  \tag{10}\]  
Point \(E\) is determined within the first and second milling cycles, i.e., within the revolution interval \([0,4\pi]\). During the first two milling cycles, the milling contour itself has four intersection points. As shown in Figure 2, one intersection point is close to the upper boundary of the slot; three intersection points are close to the lower boundary of the slot. The intersection point near the upper boundary is the final point \(E\), whose coordinates can be expressed as follows:  
\[\overset{\rightarrow}{OE}= \begin{bmatrix}x_{E}&y_{E}\end{bmatrix} =\left\{ \begin{array}{cc}\left[ \begin{array}{cc}x_{C}(\theta_{E,1})&y_{C}(\theta_ {E,1})\\ x_{C}(\theta_{E,2})&y_{C}(\theta_{E,2})\end{array} \right]+R_{C}\Big{[} \begin{array} []{cc}r_{1}(\theta_{E,1})&r_{2}(\theta_{E,1})\\ r_{1}(\theta_{E,2})&r_{2}(\theta_{E,2})\end{array} \right]\right.\text{, and }\theta_{E,1}<\theta_{E,2} \tag{11}\]  
where \(x_{E}\) and \(y_{E}\) are the coordinates of point \(E\). \(\theta_{E,1}\) and \(\theta_{E,2}\) denote two revolution angles in the first and second milling cycles while satisfying \(\theta_{E,1}<\theta_{E,2}\). More specifically, \(\theta_{E,1}\) is in the revolution interval \([\pi/2,3\pi/2]\) and \(\theta_{E,2}\) is in the revolution interval \([5\pi/2,7\pi/2]\). Figure 4: Boundary points of the transition surface of the workpiece. Equation (11) is also a transcendental equation and must be solved by using the numerical method. The point on the toolpath corresponding to the revolution angle of \(\theta_{E,2}\) is defined as \(E^{\prime}\), whose coordinates can be expressed as follows:  
\[\overset{\rightarrow}{OE^{\prime}}=\left[x_{C}(\theta_{E,2})\quad y_{C}(\theta _{E,2})\right] \tag{12}\]  
It should be noticed that points \(S\) and \(E\) are the boundary points of the newly machined surface, i.e., the new transition surface of the workpiece. Point \(E\) is also the cut-out point within a milling cycle. However, point \(S\) is not the cut-in point within a milling cycle. As shown in Figure 5, when the cutter center is located at point \(S^{\prime}\), point \(S\) is on the cutting-edge circle. Point \(P_{1}\) coincides with point \(S\). At the same cutter location, there is another intersection point between the cutting-edge circle and the previous milling contour, namely point \(P_{2}\). At this moment, the cutter has cut into the material. The actual cut-in point is point \(A\), where the cutter is tangent to the previous milling contour. The cutter location corresponding to the tangent point \(A\) is the intersection point of the toolpath itself within one milling cycle, which is denoted as point \(A^{\prime}\). Its coordinates can be expressed as follows:  
\[\overset{\rightarrow}{OA^{\prime}}=\left[x_{A^{\prime}}\quad y_{A^{\prime}} \right]=\left\{ \begin{array}{cc}\left[\begin{array}{cc}x_{C}(\theta_{A,1} )&y_{C}(\theta_{A,1})\\ x_{C}(\theta_{A,2})&y_{C}(\theta_{A,2})\end{array}\right]\\ \end{array} \right.,\text{ and }\theta_{A,1}<\theta_{A,2} \tag{13}\]  
where \(x_{A^{\prime}}\) and \(y_{A^{\prime}}\) are the coordinates of point \(A^{\prime}\). \(\theta_{A,1}\) and \(\theta_{A,2}\) denote two revolution angles in the first milling cycle while satisfying \(0<\theta_{A,1}<\theta_{A,2}<\pi/2\). Then, for each milling cycle, the effective revolution interval corresponding to the actual cutting process can be expressed as follows:  
\[\left[\theta_{A,2}+2\pi(m-1)\right]\leq\theta\leq\left[\theta_{E,2}+2\pi(m-1)\right] \tag{14}\]  
where  
\[m=\text{int}\left(\frac{\theta}{2\pi}\right) \tag{15}\]  
where \(m\) is the number of complete milling cycles. Notably, int is the function that rounds positive numbers towards zero (e.g., int(1.25) = 1, int(2.75) = 2). Figure 5: Cut-in point of the effective cutting interval.