## 3 Analytical Engagement Angle Model  
### Effective Cutting Interval of Milling Contour  
Figure 3 shows the milling contour corresponding to the ellipse-based trochoidal toolpath. According to Equation (2), the tangential vector \(\textbf{t}(\theta)\) at the current cutter location \(C\) can be expressed as follows:  
\[\textbf{t}(\theta)= \begin{bmatrix}t_{1}(\theta)&t_{2}(\theta)\end{bmatrix} = \begin{bmatrix}\frac{t_{x}(\theta)}{\sqrt{t_{x}^{2}(\theta)+t_{y}^{2}(\theta)} }&\frac{t_{y}(\theta)}{\sqrt{t_{x}^{2}(\theta)+t_{y}^{2}(\theta)}}\end{bmatrix}  \tag{5}\]  
where  
\[ \begin{bmatrix}t_{x}(\theta)&t_{y}(\theta)\end{bmatrix} = \begin{bmatrix}\frac{ dx_{C}(\theta)}{d\theta}&\frac{dy_{C}(\theta)}{d\theta}\end{bmatrix} = \begin{bmatrix} \frac{s}{2\pi t}+b\cos\theta&a\sin\theta\end{bmatrix}  \tag{6}\]  
where \(t_{1}(\theta)\) and \(t_{2}(\theta)\) are the normalized components of the tangential vector \(\textbf{t}(\theta)\) in the x-axis and y-axis directions, respectively. \(t_{x}(\theta)\) and \(t_{y}(\theta)\) are the partial derivatives of Equation (2). The radial vector \(\textbf{r}(\theta)\) at the current cutter location \(C\) is orthogonal to the tangential vector \(\textbf{t}(\theta)\) and can be obtained as follows:  
\[\textbf{r}(\theta)= \begin{bmatrix}r_{1}(\theta)&r_{2}(\theta)\end{bmatrix} = \begin{bmatrix}\frac{t_{y}(\theta)}{\sqrt{t_{x}^{2}(\theta)+t_{y}^{2}(\theta)} }&\frac{-t_{x}(\theta)}{\sqrt{t_{x}^{2}(\theta)+t_{y}^{2}(\theta)}}\end{bmatrix}  \tag{7}\]  
where \(r_{1}(\theta)\) and \(r_{2}(\theta)\) are the normalized components of the radial vector \(\textbf{r}(\theta)\) in the \(x\)-axis and \(y\)-axis directions, respectively. Point \(P_{1}\) is the furthest point from the center of the current fundamental ellipse to the cutting-edge circle and can be calculated as follows:  
\[\overrightarrow{OP_{1}}= \begin{bmatrix}x_{1}(\theta)&y_{1}(\theta)\end{bmatrix} = \overrightarrow{OC}+R_{c}\textbf{r}(\theta)= \begin{bmatrix}x_{C}(\theta)&y_{ C}(\theta)\end{bmatrix} +R_{c} \begin{bmatrix}r_{1}(\theta)&r_{2}(\theta)\end{bmatrix}  \tag{8}\]  
where \(x_{1}(\theta)\) and \(y_{1}(\theta)\) are the coordinates of point \(P_{1}\). Equation (8) is the model of the milling contour left by the cutter, on which \(P_{1}\) represents an arbitrary point. The transition surface of the workpiece is formed by the milling contour. It is part of the milling contour. Therefore, it is necessary to first determine the  
Figure 3: Milling contour corresponding to the ellipse-based trochoidal toolpath. effective revolution interval within one milling cycle. The cutter-workpiece engagement is also calculated in this revolution interval. As shown in Figure 4, points \(S\) and \(E\) represent the initial and final points of the transition surface of the workpiece, respectively. Since the ellipse-based trochoidal toolpath is periodic, the calculation of points \(S\) and \(E\) are discussed within the first two cycles. Point \(S\) is determined within the first milling cycle, i.e., within the revolution interval \([0,2\pi]\). During the first milling cycle, the milling contour itself has one intersection point, whose coordinates can be expressed as follows:  
\[\overset{\rightarrow}{OS}= \begin{bmatrix}x_{S}&y_{S}\end{bmatrix} =\left\{ \begin{array}{cc}\left[ \begin{array}{cc}x_{C}(\theta_{S,1})&y_{C}(\theta_ {S,1})\\ x_{C}(\theta_{S,2})&y_{C}(\theta_{S,2})\end{array} \right]+R_{C}\Big{[} \begin{array} []{cc}r_{1}(\theta_{S,1})&r_{2}(\theta_{S,1})\\ r_{1}(\theta_{S,2})&r_{2}(\theta_{S,2})\end{array} \right]\right.\text{, and }\theta_{S,1}<\theta_{S,2} \tag{9}\]  
where \(x_{S}\) and \(y_{S}\) are the coordinates of point \(S\). \(\theta_{S,1}\) and \(\theta_{S,2}\) denote two revolution angles in the first milling cycle, while satisfying \(\theta_{S,1}<\theta_{S,2}\). Equation (9) is a transcendental equation and cannot be solved directly by using the analytical method. The two solutions \(\theta_{S,1}\) and \(\theta_{S,2}\) are obtained by using the numerical method.