## 3 Results
### CA Deformation Analysis  
In Group 0 the CA exhibited notable deformation, particularly along the lingual and buccal surfaces of the maxillary anterior teeth (Figure 5.). The magnitude and pattern of deformation evolved from Stage 1 to Stage 50, reflecting progressive changes in the distribution of forces and areas of stress concentration over time. Similar deformation patterns were observed in Groups 1 and 2 (Figures 6 and 7). However, a comparison of the aligner shapes between Stage 1 and Stage 50 revealed marked differences in both the overall geometry of the appliance and the spatial distribution of deformation. Notably, the gingival margin of the maxillary central incisor showed a pronounced increase in  
Figure 3: Changes in canine tipping angle according to beam width variations during treatment progression. The graph shows the relationship between beam width (0, 1, 2, and 3 mm) and canine tipping patterns over 50 treatment stages, illustrating tipping control effectiveness.  
deformation, suggesting cumulative mechanical stress in that region during simulated treatment progression.  
Figure 4: Progression of canine rotation angle according to beam width variations during treatment stages. The graph demonstrates how different beam widths influence canine rotational movement, with the 3 mm beam showing distinctly different rotational control compared to the other configurations.  
Figure 5: Clear aligner deformation patterns for a 0 mm beam width configuration. Deformation visualization at the initial (Stage 1, upper panel) and final (Stage 50, lower panel) treatment stages from three viewing perspectives: lateral view with displacement vectors (**left**), occlusal view (middle), and color-coded deformation magnitude mapping (**right**).  
Compared to the other experimental groups, Group 3 demonstrated significantly less deformation of the gingival margin at the maxillary central incisor, indicating enhanced structural rigidity and superior deformation resistance in this region (Figure 8). The 3 mm beam width configuration provided substantially greater structural support within the extraction space relative to the alternative conditions, resulting in a distinctive deformation pattern around both the canine area and extraction sites. At Stage 50, deformation in the posterior canine region exhibited a more uniform distribution, directly correlating with the superior control of canine rotation in Group 3 as demonstrated in previous analyses. This  
Figure 6: Clear aligner deformation patterns for a 1 mm beam width configuration. Deformation visualization at the initial (Stage 1, upper panel) and final (Stage 50, lower panel) treatment stages from three perspectives: lateral view with displacement vectors, occlusal view, and quantitative color-coded mapping.  
Figure 7: Clear aligner deformation patterns for a 2 mm beam width configuration. Deformation visualization at the initial (Stage 1, upper panel) and final (Stage 50, lower panel) treatment stages from three perspectives: lateral view with displacement vectors, occlusal view, and quantitative color-coded mapping.  
enhanced performance can be attributed to the thicker beam structure, which effectively distributes the orthodontic forces across a broader surface area while simultaneously increasing the overall appliance rigidity. Consequently, this configuration facilitated a more stable force and moment transmission, promoting optimal bodily movement of the canine rather than uncontrolled tipping or rotation.