## 2 Materials and Methods
### Design of PNAM Corrector Based on Feature Point  
This section mainly uses the feature points extracted from the previous section to correct unilateral complete samples, mainly divided into the rotation of the healthy side and the internal construction of the healthy side. In the design of the PNAM appliance by the cooperating agency, the back end of the healthy side below point C is required to remain unchanged during the operation of the healthy side. The point I is required to be located on the median sagittal plane after rotation through the healthy side. At the same time, the front end of the healthy side above the point C has a relatively large degree of rotation and inward translation near the crack. Based on the above rules, this study achieved rotation and inward reconstruction of the healthy side. The design scheme shown in Figure 4A takes the origin O in the XOY plane as the rotation center and the connecting line OC as the critical condition. The back end of the healthy side remains unchanged, and the front end of the healthy side rotates to the affected side and rebuilds inward. Letting M be the upper point of the healthy side, the length of the connection OM with the rotation center O is 1, and the angle between OM and OT is \(\alpha\). It is assumed that the rotation of point M is \(\theta\), and the corresponding point of point M after rotation and inward reconstruction is M'. The length of the connection OM' with the rotation center O is I', and 1 and I' satisfy Formula (4):  
\[l^{\prime}=k\times l \tag{4}\]  
where \(k\) denotes the degree of inward modification of point M, called inward modification coefficient, and \(k\leq 1\). Due to the different design requirements between the front end of the healthy side and the back part and the closer the front end of the healthy side is to the crack, the greater the rotation strength and the degree of inward reconstruction--this study uses the angle \(\alpha\) between OM and OT to control the rotation of angle \(\theta\) point M and the inward reconstruction coefficient \(k\), which follow the Formulas (5) and (6):  
\[\theta=f(\alpha) \tag{5}\]If we let the angle between OC and OT be \(\alpha_{o}\), then the critical condition is \(\alpha_{o}\). We also let the angle between OI and OT be \(\alpha_{i}\), and the angle between OP and OT be \(\alpha_{p}\). According to the design requirements and the changing trend of rotation \(\theta\),\(f(\alpha)\) should meet the following characteristics:  
1. When \(0\leq\alpha\leq\alpha_{o}\),\(f(\alpha)=0\) or \(f(\alpha)\approx 0\), and in order to make the healthy side continuous at point C after rotation, \(f(\alpha)\) should be derivative at \(\alpha=\alpha_{o}\);
2. When \(\alpha_{o}<\alpha\leq\alpha_{i}\),\(f(\alpha)\) should be linearly or approximately linearly increasing, and at \(\alpha=\alpha_{i}\),\(f(\alpha)\) reaches the maximum \(\frac{\pi}{2}-\alpha_{i}\); that is, the rotation of point I is the largest, \(\theta_{m}=\frac{\pi}{2}-\alpha_{i}\) so that I point is rotated and falls on the median sagittal plane. 3. When \(\alpha>\alpha_{i}\),\(f(\alpha)\) should remain non-increasing. Figure 4: **(A)** Schematic diagram of the sound side design of the PNAM appliance. **(B)** Three cases of unilateral complete sample before and after PNAM design. **(a)**NO. 5 (healthy side on the left) PNAM appliance before design. **(b)** NO. 7 (healthy side on the left) PNAM appliance before design. **(c)** NO. 11 (healthy side on the right) PNAM appliance before design. **(d)** NO. 5 (healthy side on the left) PNAM appliance after design. **(e)** NO. 7 (healthy side on the left) PNAM appliance after design. **(f)** NO. 11 (healthy side on the left) PNAM appliance after design. **(c)** Final PNAM appliance model of sample 7. Based on this feature, this paper proposes a robust side rotation model based on the Gaussian function.