## 2 Materials and Methods
By utilizing vector calculations, the positions of the entities in three-dimensional space can be determined in terms of distance. These distances can then be assessed relative to one another, such as through angle measurements. Additionally, the distance from individual measurement points to constructed reference planes can be calculated. The area of a specified region can also be computed by aggregating the surface areas of the known polygons within that defined area. In collaboration with OnyxCeph\({}^{3\text{TM}}\), we implemented patches that were adapted to our specific Frankfurt approach requirements for measuring models with one-sided cleft lip and palate in the evaluation software. The respective built-in patches to analyze our key anatomical landmarks are available for use in the Measurement module of the software. Critical anatomical and constructed landmarks on cleft lip and palate models are defined below to ensure consistent measurements. The selection of the measuring points detailed in Table 1 was based on a number of factors, including the publications of [PERSON] et al. [18], [PERSON] [19], [PERSON] [20], and [PERSON] [21]. If a point appears on both jaw segments, a \({}^{\prime}\) indicates that the point is  
Figure 3: Socketed model of a left-sided cleft lip, jaw, and palate: (**a**) tailed view with the focus on the smaller segment; (**b**) occlusal view of the small and large segment; (**c**) tailed view with the focus on the large segment. Figure 2: Intraoral 3D scan of a left-sided cleft lip, jaw, and palate: (**a**) tilted view with the focus on the smaller segment; (**b**) occlusal view of the small and large segment; (**c**) tilted view with the focus on the large segment. located on the smaller jaw segment. Figure 4 and Table 1 show the measuring points on a dental left-sided cleft lip and palate model. To identify the transversal landmarks, the large and small segments were divided into the subsegments (T-C2, C2-Inz, and Inz-P1 for the large segment and P2-C2\({}^{\prime}\) and C2\({}^{\prime}\)-T\({}^{\prime}\) for the small segment). The most vestibular and most palatal points in each section were determined (Table 2, Figure 5). \begin{table}
\begin{tabular}{c c} \hline \hline
**Measuring Points** & **Description** \\ \hline T-C2 vest & The most vestibular point perpendicular to the line T-C2 \\  & (large segment) \\ \hline T-C2 pal & The most palatal point perpendicular to the line T-C2 \\  & (large segment) \\ \hline C2-Inz vest & The most vestibular point perpendicular to the line \\  & C2-Inz (large segment) \\ \hline C2-Inz pal & The most palatal point perpendicular to the line \\  & (large segment) \\ \hline Inz-P1 vest & The most vestibular point perpendicular to the line \\  & Inz-P1 (large segment) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Width measuring points of the partial jaw segments and description. Figure 4: Illustration of the anatomical measuring points. The anatomical measuring points and their description are presented in Table 1.  
The following constructed measuring points are based on the measurement points listed in Table 1 and are shown in Table 3 and Figure 6. These and the measurement points form the basis for the subsequent distance and angle measurements. The measurement points of the alveolar cleft width are shown in Table 4 and Figure 7. The segment length measurements are shown in detail in Table 4 and Figure 8.  
The large and small segments were divided into subsegments to identify the transversal landmarks (T-C2, C2-Inz, and Inz-P1 for the large segment and P2-C2\({}^{\prime}\) and C2\({}^{\prime}\)-T\({}^{\prime}\)for the small segment). The most vestibular and most palatal points in each section were determined. For the segmental width measurements, the greatest distance per