## 3 Results  
In order to individually demonstrate the three-dimensional displacements of derotation and tipping within the hemimaxillary, possible case histories are presented below. Figure 1 represents the pre-treatment situation of the maxillary upper jaw and palatine bones, and the reference system coordinates, specifically the occlusal view (in transverse section, 1a) and the posterior view (in frontal section, 1b) of these bones. In this case, no \begin{table}
\begin{tabular}{c c} \hline \hline
**Landmarks** & \\ \hline Anterior Nasal Spine (Ans) & Most anterior point of the Ans (right and left) \\ \hline Posterior Nasal Spine (Pns) & Most posterior point of the Pns \\ \hline Nasion (N) & Most anterior point of the fronto-nasal suture \\ \hline Mesio-Vestibular Cusp (MVC) & Apex of the MV cusp of the upper first molar (right and left) \\ \hline Mesio-Palatal Cusp (MPC) & Apex of the MP cusp of the upper first molar (right and left) \\ \hline Disto-Buccal Cusp (DBC) & Apex of the DB cusp of the upper first molar (right and left) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Three-dimensional landmarks’ position of analysis. \begin{table}
\begin{tabular}{c c} \hline \hline
**Panes** & \\ \hline Hemi Sagittall Plane (HSP) & Plane passing through the Ans (right and left), Pns, and N \\ \hline Hemi Coronal Plane (HCP) & Plane perpendicular to the HSP (right and left) and HAP, passing through the Pns \\ \hline Hemi Axial Plane (HAP) & Plane perpendicular to the HSP (right and left) and HCP, passing through the Pns \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Three-dimensional reconstruction of analysis. \begin{table}
\begin{tabular}{c c} \hline \hline  
sign of either rotation or molar tipping was observable. In fact, it was evident that angles \(\gamma\)L and \(\gamma\)R were equal not only to each other but also to their respective alternate interior angles (marked by *) according to the geometric theorem of parallel lines, considering the Pns-Ans segment and any line parallel to that segment. The same principle applies to the \(\beta\)L and \(\beta\)R angles, considering the centerline and any line parallel to that segment. The opening of the medial palatine suture by RME results in the division of the maxillary upper jaw into the two hemimaxillary teeth through a pyramid-like splaying process. This phenomenon is partly responsible for the increased buccal tipping and rotation values of the first molars. Therefore, the situations that can result from such expansion are different. This study identifies and explores five example cases that may occur post-expansion, taking a stable intra-hemimaxillary plane as the reference; by analyzing them, it is possible to evaluate the angles that describe the three-dimensional movement of the upper permanent first molars. The first case can be seen in Figure 2 in occlusal (2a) and posterior (2b) views and represents the case history in which orthopedic expansion of the maxillary upper jaw is not associated with any molar rotation or tipping (in fact, the \(\gamma\) and \(\beta\) angles remained unchanged). An orthopedic effect, however, may be associated with a dental effect as well, either unilaterally or bilaterally, of tipping or rotation. Figure 3 shows the case where both the \(\gamma\) and \(\beta\) angles decrease (\({}^{-}\)) monolaterally due to distal molar rotation and buccal tipping on that side. If, in contrast, the disto-rotation is bilateral, as in Figure 4a, the \(\gamma\) angle would decrease (\({}^{-}\)) further on both sides due to both dental and skeletal effects. Figure 4b, on the other hand, shows, in addition to the skeletal effect, bilateral buccal tipping for which the \(\beta\) angle decreases (\({}^{-}\)) compared with that of the pre-expansion condition. Although difficult to find clinically, an additional case may be represented by a subject in which the expander results in disto-rotation/buccal tipping on one side and meso-rotation/palatal tipping on the other (Figure 5).