## 2 Materials and Methods
### Three-Dimensional Facial Scans Analysis
The initial alignment process, where two 3D models are brought into a roughly corresponding spatial configuration without relying on prior position data, is known as global registration [22]. This procedure is a key step in shape matching and 3D model alignment. Global registration does not depend on known starting positions; it tends to produce less accurate results and is generally used to initialize more precise, local alignment techniques. The Iterative Closest Point (ICP) algorithm is one of the most commonly used [22]. The ICP algorithm refines alignment by reducing the squared distances between comparable points in two point clouds, known as the target and reference models. To ensure accuracy, it is essential to register these scans using anatomically stable regions. Anatomically stable regions include portions of the forehead, nasal bridge and infraorbital region. Although centrally located, this region lies within the upper third of the face and demonstrates minimal morphological variation due to growth. Its stability makes it particularly suitable for assessing changes in the middle and lower facial thirds, which are more susceptible to developmental and treatment-induced modifications [12; 23; 24]. Once these key landmarks are identified for overlap, the alignment process becomes more straightforward and helps minimize fitting errors specifically at those locations. Points outside of these chosen areas are excluded from the alignment function and therefore do not influence the minimization of differences. The scans were then matched with the ICP algorithm to generate a unified coordinate space in which spatial differences between the models could be measured. This approach facilitated the assessment of morphological changes, particularly in regions influenced by the Twin Block appliance, such as the chin, jawline, and mandible. As this process was conducted, visual and metric inspection was applied to validate the registration accuracy prior to measurement extraction. A distance map was generated by computing point-to-point distances across the facial surface, which were then visualized using a color-coded heat map. This visualization technique assigns colors to specific ranges of distances, facilitating the interpretation of areas where soft tissue protrusion or retrusion occurred. For instance, regions with minimal displacement were displayed in cooler tones (blue/gray), while regions with greater soft tissue movement--such as the mandible, lower lip, cinnad hair--were shown in red tones. Changes in hair were significant mainly because of a change in haircut. For distance measurements we used Hausdorff distance which is a commonly used metric for comparing two 3D surfaces or point clouds [25]. It measures the greatest of all the shortest distances from a point on one surface to the closest point on the other surface, capturing the maximum deviation between the two shapes. This makes it particularly useful in evaluating geometric accuracy and detecting local misalignments in applications such as model registration, surface reconstruction, and shape comparison. A lower Hausdorff distance indicates better alignment and greater similarity between the compared models. These visual results were not only useful for confirming the quantitative measurements but also provided a valuable visual tool for communicating changes. The combination of model superimposition and color-based analysis offers an accessible and effective way to interpret complex 3D facial changes resulting from orthodontic treatment. A statistical analysis was conducted using the SPSS software package (version 26.0; IBM Corp., Armonk, NY, USA). Changes pre- and post-treatment were evaluated using paired \(t\)-tests, with statistical significance defined as \(p\) < 0.05. Descriptive statistics, including means and standard deviations, were calculated to summarize sample characteristics. Measurement reliability was verified through the Intraclass Correlation Coefficient (ICC), ensuring precision and reproducibility.