## 2 Materials and Methods
### Design of PNAM Corrector Based on Feature Point
The general form of the Gaussian function is shown as Formula (7):  
\[f(\alpha)=A_{1}\times e^{-(\alpha-B_{1})^{2}/2c^{2}} \tag{7}\]  
Where \(\alpha\) is the angle between OM and OT; and \(A_{1}\), \(B_{1}\), and \(c\) are real constants, which represent the peak, symmetry axis, and standard deviation of the Gaussian function curve, respectively. These three parameters control the position and shape of the Gaussian function curve; On the interval \((-\infty,B_{1}-3c)\), it is approximately equal to 0, on the interval \((B_{1}-3c,B_{1})\), it is increasing; and on the interval \((B_{1},+\infty)\), it is not increasing. The above characteristics fully meet the requirements of _f(\(\alpha\))_ in Formula (4). Then, according to the Gaussian function, the angle \(\alpha\) can control the change of the rotation \(\theta\). The key problem is realizing the mapping from the mathematical model to the physical model and determining the three parameters in the Gaussian function to realize the rotation of the healthy side reasonably and effectively. The following rules should be followed when performing contralateral rotation on a unilateral complete cleft lip and palate sample:  
1. The back end of the healthy side below point C should remain unchanged. According to the properties of the Gaussian function and \(3\sigma\) principle, the critical condition is \(\alpha=B_{1}-2c\); that is, let \(B_{1}-2c=\alpha_{o}\);
2. Point I is rotated to the median vector plane, and the maximum rotation of point I is \(\theta_{m}=\frac{\pi}{2}-\alpha_{i}\). Then, let \(B_{1}=\alpha_{i}\), \(A_{1}=\theta_{m}\). Through the above mapping, the physical model parameters \(\alpha_{o}\), \(\alpha_{i}\) and \(\theta_{m}\) of the healthy side rotation are calculated by the coordinates of the calibrated point T, point C and point I in the XOY plane, and then, according to Formula (8), the three parameters of the Gaussian function \(A_{1}\), \(B_{1}\), and \(c\) are as follows: \[\left\{ \begin{aligned} A_{1}&=\theta_{m}=\pi/2- \alpha_{i}\\ B_{1}&=\alpha_{i}\\ c&=(B_{1}-\alpha_{o})/2\end{aligned} \right.\] (8) According to the design requirements and the trend of inward modification coefficient \(k\), \(g(\alpha)\) should satisfy the following characteristics: 1. When \(0\leq\alpha\leq\alpha_{o}\), \(g(\alpha)=1\) or \(g(\alpha)\approx 1\), and in order to make the reconstruction in the healthy lateral still continuous at point C, \(g(\alpha)\) should be derivative at \(\alpha=\alpha_{o}\);
2. When \(\alpha_{o}<\alpha<\alpha_{p}\), the magnitude of the decrease should be as large as possible to restore normal dental arch morphology as a reference. According to the above characteristics, the piecewise function is constructed, and the mathematical model is shown in Formula (9):  
\[g(\alpha)=\left\{ \begin{aligned} 1,0\leq\alpha\leq\alpha_{o} \\ a\times\alpha+b,\alpha_{o}<\alpha<\alpha_{p}\\ c,\alpha\geq\alpha_{p}\end{aligned} \right. \tag{9}\]  
Since this model needs to be continuous in the defined domain, the parameters \(a\), \(b\) during the period of \(\alpha_{o}\leq\alpha\leq\alpha_{p}\) are determined. It is known that \(g(\alpha)\) passes through the two points \((\alpha_{o},1)\) and \((\alpha_{p},c)\). To restore the alveolar bone to normal dental arch shape as much as possible as a reference, OP and OP should be as circular as possible, namely, OP = OP'; thus the parameters \(a\), \(b\), and \(c\) of this model can be calculated, as shown in Formula (10):  
\[\left\{ \begin{aligned} a=\left(OP/OP^{\prime}-1\right)/\left( \alpha_{p}-\alpha_{o}\right)\\ b=\alpha_{p}/\left(\alpha_{p}-\alpha_{o}\right)\\ c=OP/OP^{\prime}\end{aligned} \right. \tag{10}\]  
According to the above method, with the angle \(\alpha\) as the variable, the rotation \(\theta\) of the healthy side and the inward reconstruction coefficient \(k\) are obtained, which can be used to operate the healthy side.