## 2 Materials and Methods
### Three-Dimensional Facial Scans Analysis  
Three-dimensional facial scans were captured using the Arc scanner (Bellus3D, Bellus3D, Inc., Campbell, CA, USA) along with the Bellus3D software, version 1.6.2. The scanning procedure involved four distinct head movements: rotating 90\({}^{\circ}\) to the left and right, and tilting approximately 45\({}^{\circ}\) upward and downward. Facial scanning was performed under illumination provided by a ring light with a correlated color temperature (CCT) of 3000 K. Natural head position (NHP) was used to ensure consistency in image capture as it is proven to be reproducible [18; 19]. Participants were seated on a height-adjustable stool and instructed to align their eyes with the level of the front-facing camera. Seating height was modified as needed to facilitate proper NHP alignment. Prior to scanning, subjects were asked to maintain a relaxed jaw posture with their mouths gently closed. Scans were repeated in cases where any irregularities or deviations from the required positioning were observed. Measurements were taken and recorded in millimeters, specifically the distance between the tragus and the pogonion, to assess soft tissue changes related to the length of the mandible. Mandibular skeletal measurements are typically assessed using the linear distance between condylion (Co) and gnathion (Gn), which represents the effective mandibular length [17]. In soft tissue, the tragus is the anatomical landmark most analogous to the condylion. Therefore, the distance between the tragus and pogonion is commonly used to evaluate soft tissue changes in the lower jaw region [18]. The discrete geodesic problem refers to the shortest path connecting a source and a target across a spherical surface. In the current study, we used the \"single source, all destinations\" technique proposed by [PERSON] and [PERSON] [20]. Their method is based on a specialized variant of [PERSON]'s algorithm, known as the \"continuous Dijkstra,\" which \begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Description** \\ \hline SNA & Sagittal relationship of the maxilla to the anterior cranial base \\ SNB & Sagittal relationship of the mandible to the anterior cranial base \\ ANB & Sagittal relationship of the maxilla to the mandible \\ Wits & Linear measurement of skeletal class \\ ANPG & Angle of convexity of the facial bony structures \\ SN:NL & Maximly plane angle \\ MaxMandAngle & Intermaximillary angle \\ N-S-Ar & Cranial base angle \\ SArGo & Articular angle \\ MeGoAr & Mandibular angle \\ NSGn & Y-axis angle \\ SN:GoGn & Mandibular plane angle \\ CoGoMe & Condylar angle \\ U1:Max & Proclination of the upper incisors to the maxillary base \\ L1:Mand & Proclination of the lower incisors to the mandibular base \\ U1:L1 & Interincisal angle \\ U1 MxPlane & Protrusion of the upper incisors to the apical base of the maxilla \\ L1 MdPlane & Protrusion of the lower incisors to the apical base of \\ L1 MdPlane & the mandible \\ OJ & Overjet \\ OB & Overbite \\ Ls:E1 & Distance of the upper lip from the Ricketts’ esthetic line \\  & (Prm-Pgn) \\ Li:E1 & Distance of the lower lip from the Ricketts’ esthetic line \\ CoGn & Mandibular length \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables measured in the cephalometric analysis. computes the shortest paths along the edges of a subdivided mesh. For our implementation on triangular mesh structures, we utilized [PERSON]'s geodesic algorithm [21], accessed through a Cython wrapper for the original C++ code. The overlapping of pre-treatment and post-treatment 3D facial models was performed to analyze soft tissue changes resulting from Twin Block appliance therapy. The 3D scans were obtained separately before and after treatment, comprising approximately a period of one year, and then processed for comparative analysis. Generating detailed and accurate visualizations is crucial for detecting differences between the reconstructed 3D models. However, this requires the precise alignment of the models.