## 3 Results
### Distance and Angle between MSP and Each Mandibular Midsagittal Plane before and after Surgery  
The distance (\(1.77\pm 1.19\) mm) and angle (\(2.87\pm 1.49^{\circ}\)) between the MSP and cmAMP before surgery were significantly smaller than those with MSP, AMP, and mAMP as the reference plane (\(p=0.001\)) (Table 4). This distance and angle between the MSP and cmAMP after surgery decreased to \(1.15\pm 0.74\) mm and \(2.02\pm 0.82^{\circ}\), respectively. These values were also significantly smaller than those between \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**MSP**} & \multicolumn{2}{c}{**AMP**} & \multicolumn{2}{c}{**mAMP**} & \multicolumn{2}{c}{**cmAMP**} & \multirow{2}{*}{\(p\)**-Value**} \\ \cline{2-2} \cline{5-8}  & **Mean** & & & & & & & & \\ \hline Left segment & & & & & & & & \\ Non-overlapping & 4567.87 \({}^{\text{a}}\) & 1047.01 & 5044.53 \({}^{\text{b}}\) & 1117.22 & 4603.61 \({}^{\text{a}}\) & 949.71 & 4112.34 \({}^{\text{a}}\) & 990.93 & 0.01 \\ Non overlapping & 8212.43 \({}^{\text{ab}}\) & 3045.53 & 14,093.04 \({}^{\text{c}}\) & 6689.53 & 10,641.74 \({}^{\text{b}}\) & 4926.48 & 5423.47 \({}^{\text{a}}\) & 2467.45 \(<\)0.001 \\ SI & 0.74 \({}^{\text{b}}\) & 0.07 & 0.55 \({}^{\text{a}}\) & 0.17 & 0.67 \({}^{\text{b}}\) & 0.11 & 0.83 \({}^{\text{c}}\) & 0.04 & \(<\)0.001 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Before Surgery**} & \multicolumn{2}{c}{**After Surgery**} & \multicolumn{2}{c}{**Difference**} & \multirow{2}{*}{\(p\)**-Value**} \\ \cline{2-2} \cline{5-8}  & **Mean** & & & & & & & & \\ \hline Right segment & & & & & & & & \\ Non-overlapping surface (mm\({}^{2}\)) & 4347.03 & 775.29 & 3899.65 & 842.71 & 447.37 & 684.56 & 0.001 \\ Non-overlapping volume (mm\({}^{3}\)) & 7942.44 & 2790.26 & 6036.33 & 2357.24 & 1906.11 & 2567.01 & \(<\)0.001 \\ Left segment & & & & & & & & \\ Non-overlapping surface (mm\({}^{2}\)) & 4567.87 & 1047.01 & 4240.63 & 1008.61 & 327.23 & 623.12 & 0.009 \\ Non-overlapping volume (mm\({}^{3}\)) & 8212.43 & 3045.53 & 6469.82 & 2703.88 & 1742.60 & 2944.47 & 0.004 \\ SI & 0.74 & 0.07 & 0.80 & 0.05 \(-\)0.05 & 0.07 & \(<\)0.001 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Differences in the non-overlapping surface, volume, and similarity index (SI) using the mirroring technique based on the MSP before and 1 year after surgery. MSP and AMP and between MSP and mAMP. As shown in Figure 3, the smaller the angle and distance difference between the two planes in the three-dimensional space, the closer and more consistent are the two planes. Since the difference in the angle and distance between the cmAMP and the postoperative MSP is the smallest, which is thought to reflect an improvement in the asymmetry of the anterior segment of the hemi-mandible, it indicates that that cmAMP is the closest plane to the MSP 1 year after surgery. Comparison of SI between MSP 1 Year after Surgery and Each Mandibular Midsagittal Plane before Surgery  
Upon analysis using the one-way ANOVA followed by [PERSON]'s test, the SI before surgery, which was \(0.55\pm 0.17\) using AMP and \(0.67\pm 0.11\) using mAMP, showed a statistically significant difference with the SI using MSP 1 year after surgery. This finding indicates that, in the patient with mandibular asymmetry, if the pre-surgical AMP or mAMP is assumed to be the midsagittal plane of the mandible and these mandible anterior segment planes move in a direction overlapping with the MSP, then the mandible symmetry may not be greatly recovered (\(p<0.001\)) (Figure 4). In other words, if pre-surgical AMP and mAMP are considered midsagittal planes of the mandible and postoperative MSP matches while planning and during the surgery, other additive surgeries, including cortical osteotomy, geniolplasty, or bone grafting, will be required to obtain an SI value as high as the SI value calculated based on MSP 1 year after surgery.