## 5 Cam Mechanism
Additionally, Point C moves along a known trajectory with the origin of the coordinate system as its center and \(r_{0}\) as its radius. Now, there exists a curve \(\Gamma\) attached to Point A, which moves in conjunction with the rotation of Point A. During the motion of Points A and C, this curve satisfies both the tangential relationship in the formula and the constraint of the constant total length. Point B is the intersection point of the \(\Gamma\) curve and the line BC, thereby serving as both a point on the \(\Gamma\) curve and a point where the tangent of curve \(\Gamma\) intersects with BC. The objective was to determine the expression for the \(\Gamma\) curve. Given that Point B lies on the tangent of the curve, and that the system is continuous, calculating the trajectory of Point B is equivalent to obtaining the expression for the \(\Gamma\) curve. Based on the analysis in the previous section, the motion trajectories of Points A and C are known to be functions of \(\alpha\). By using infinitesimal analysis, \(\alpha\) can be divided into n segments, and the motion trajectories of Points A and C can be represented by n + 1 points, which can be expressed as Equation (20). \[ \begin{split}(x_{Ai},y_{Ai})&=(x_{a}(\alpha(i)),y_{ a}(\alpha(i))),i\in[0,n]\\ (x_{Ci},y_{Ci})&=(x_{c}(\alpha(i)),y_{c}(\alpha(i)) ),i\in[0,n]\end{split}  \tag{20}\]  
At the same time, there were n + 1 tangent points on the \(\Gamma\) trajectory, which can be denoted as Equation (21). \[(x_{Bi},y_{Bi})=(x_{b}(\alpha(i)),y_{b}(\alpha(i))),i\in[0,n] \tag{21}\]  
It should be noted that Point B is a tangent point on the \(\Gamma\) trajectory, which is attached to Point A, and that Point A is a point on the circle. Therefore, it is convenient to convert each Point B into a point in polar coordinates with Point D as the center and DA as the positive direction, i.e., to perform a conversion from Cartesian coordinates to the polar coordinates of the \(\Gamma\) trajectory as system of Equations (22). \[ \begin{split}\rho_{Bi}&=\sqrt{(x_{Bi}+d)^{2}+{y_{Bi }}^{2}}\\ \theta_{Bi}&=\theta-arctan\frac{y_{Bi}}{x_{Bi}+d} \end{split}  \tag{22}\]  
When considering the infinitesimal method, the constraint condition of the unchanged string length was no longer expressed as a continuous integral, but rather represented as the sum of lengths between the different points remaining invariant. Thus, the following constraint condition applies:  
\[m_{0}=BC+\hat{A}B, \tag{23}\]  
Figure 17: Simplified mathematical model of the cam trajectory. which can be transformed into Equation (24). \[m_{0}=CB_{k}+\sum_{i=1}^{k}B_{i-1}B_{i},i\leq k\leq n, \tag{24}\]  
where \(k\) represents the position wherein Points A and C have moved to the \(k\)th position, and the tangent point at this moment is located at \(B_{k}\). \(B_{i}\) indicates the position of the previous tangent point on the \(\Gamma\) trajectory in the current state. Transforming it into a point in polar coordinates for calculation has two advantages. Firstly, because the \(\Gamma\) trajectory itself was moving, converting the tangent point \(B_{i}\) to a point attached to its own polar coordinate system facilitates the elimination of this rotation. Secondly, when representing the tangent point in Cartesian rectangular coordinates, its value in the x direction was one that was non-increasing. Meanwhile, in polar coordinates, its angle was found to be increasing. Furthermore, using polar coordinates for representation can prevent the \"tangent point retreat\" phenomenon and facilitate the calculation of the distance between tangent points. In an infinitesimal analysis, the constraint condition of the equal tangent angles was no longer applicable as the curved term of the cam trajectory was replaced by extremely small line segments. It should be noted that, after \(k\) iterations, the value of the \(k-1\) steps is a known value in the formula, and the formula becomes an elliptic Equation (25).