## 2 Quadratic Contact Surface Analysis and Force Modeling  
Specifically, studying the second-order surface of gear contact is of paramount significance in optimization, enhancing performance, and preventing malfunctions. Initially, this method aids in comprehending the shape variation and load distribution on the gear contact surface in real-time engagement, facilitating the optimization of mechanical properties like enhanced load-bearing capacity, noise reduction, and wear diminution. Furthermore, insights from the second-order surface can drive design optimizations, for instance, reshaping the gear to minimize sliding friction. Comprehending the characteristics of the gear contact surface also paves the way for designing intricate and efficient gear systems, like helical and level gears. Ultimately, understanding the gear contact via the second-order surface can illuminate the wear and failure mechanisms, which are critical for proactive fault prevention and gear maintenance. In analyzing the contact of cycloid and pin gears, a simplified model that equivalently describes the contact as two cylindrical bodies interacting can be envisioned, as illustrated in Figure 1. The contact between the two cylindrical shapes can be characterized by a second-order equation, represented by the general form as Equation (1):  
\[ax^{2}+by^{2}+cz^{2}+2 fyz+2 gzx+2 hxy+2 ux+2 vy+2 wz+d=0 \tag{1}\]  
Given that this equation passes through the origin, by substituting \(x=0\), \(y=0\), and \(z=0\) into Equation (1), it can be deduced that \(d=0\). The point of origin is a mutual tangent plane for both cylindrical bodies, characterized by \(x=0\), \(y=0\), \(z=0\), and \(\frac{9z}{\alpha x}=0\). The partial differentiation of Equation (1), with respect to \(x\) yields, is \(\mu=0\). Similarly, by differentiating with respect to \(y\), we deduce that \(v=0\). With these outcomes, Equation (1) can be condensed into its simplified representation, labeled Equation (2). \[ax^{2}+by^{2}+cz^{2}+2 fyz+2 gzx+2 hxy+2 wz=0 \tag{2}\]Having obtained the first-order derivatives of the contact surface, and drawing on [PERSON]'s theorem, it becomes imperative to compute the second-order derivatives to fully specify the equation representing the contact surface. Thus, the second-order partial derivatives of Equation (2) with respect to both \(x\) and \(y\) need to be calculated, which lead to Equations (3) and (4), respectively. \[2a+2c(\frac{\partial z}{\partial x})^{2}+2 cz\frac{\partial^{2}z}{\partial x^{2} }+2 fy\frac{\partial^{2}z}{\partial x^{2}}+2 gx\frac{\partial^{2}z}{\partial x ^{2}}+2g\frac{\partial z}{\partial x}+2g\frac{\partial z}{\partial x}+2w \frac{\partial^{2}z}{\partial x^{2}}=0 \tag{3}\]  
\[2b+2c(\frac{\partial z}{\partial y})^{2}+2 cz\frac{\partial^{2}z}{\partial y^{ 2}}+2 fgy\frac{\partial^{2}z}{\partial y^{2}}+2f\frac{\partial z}{\partial y}+ 2f\frac{\partial z}{\partial y}+2 gx\frac{\partial^{2}z}{\partial y^{2}}+2w \frac{\partial^{2}z}{\partial y^{2}}=0 \tag{4}\]  
Upon inserting \(x=0\), \(y=0\), \(z=0\), \(\frac{\partial z}{\partial x}=0\), and \(\frac{\partial z}{\partial y}=0\) into Equations (3) and (4), the resultant computation manifests as Equation (5). \[ \begin{cases}\frac{\partial^{2}z}{\partial x^{2}}=-\frac{a}{w}\\ \frac{\partial^{2}z}{\partial y^{2}}=-\frac{b}{w}\end{cases}  \tag{5}\]  
Proceeding in a similar fashion, by differentiating Equation (2) first with respect to \(x\) and then \(y\), we obtain Equation (6). \[\frac{\partial^{2}z}{\partial x\partial y}=-\frac{h}{w} \tag{6}\]  
By coupling Equations (5) and (6) with the first two terms of the Taylor series expansion, we derive Equation (7). Ultimately, the culmination of these calculations and derivations yields the result depicted in Equation (8).