## 3 Results  
Figure 2b presents a selected STL model of a lower dental arch obtained by an intra-oral scanner; further details are presented in [34]. The contour plot of the evaluated digital model in 3D space is presented in Figure 9a. Figure 9b presents a 3D plot of its selected body (a molar tooth) with the associated contour plot in Figure 9c, which is used for detail visualisation of its shape and to specify the regions of interest. Figure 10 presents digital models of the left and the right canine teeth. The details of their contours presented in Figure 10c,d were used to precisely locate the teeth tops coordinates \(x_{k},z_{k}\) associated with the value  
\[y_{k}=\max(y(i,j))\ \text{for}\ i,j\in U_{k}, \tag{5}\]  
Figure 9: A 3D analysis of the lower dental arch presenting (**a**) the contour plot of the selected STL model, (**b**) the 3D plot of the molar tooth, and (**c**) the corresponding contour plot, which allows detection of specific surface areas. where \(U_{k}\) specifies the rectangular region around the \(k\)-th tooth. This semiautomatic specification of the teeth tops coordinates allows us to estimate the distances between the left and the right canine teeth (for \(k=L3\) and \(k=R3\), respectively) by relation (2), which defines a valuable coefficient used to specify the dental arch and to evaluate the treatment. An alternative approach based upon object rotation into the horizontal plane is presented in [22; 34]. The evaluation of distances between canine teeth was done for 20 observations of the same individual in [34]. The difference in standard deviation to process the lower and upper dental arches was explained by the less convenient scanning of the upper dental arch, which was controlled on the monitor of the scanner itself. Figure 11 presents the process of the rigid registration [48] of the input and base images (in Figure 11a,b, respectively) of the same individual. The fixed points are selected from the 3D model and the tops of the individual teeth. The image registration includes scaling, rotation and translation of all data points \(\{x(i,j),z(i,j)\}\) into their new locations \(\{X(i,j),Z(i,j)\}\), which can be performed by matrix multiplication  
\[[X(i,j),Z(i,j)]=[x(i,j),z(i,j),1]\ \left[ \begin{array}{cc}a(3)&b(3)\\ a(2)&b(2)\\ a(1)&b(1)\\ \end{array} \right] \tag{6}\]  
with optimized values of the transform matrix. The distances between symmetrical values evaluated for the base and registered images by Equation (4), as described in [34], are lower than 4% in this case. The results of the registration presented in Figure 11c can be further used to evaluate the treatment and time comparison of dental arch changes. Figure 10: Digital model of (**a**,**c**) left, and (**b**,**d**) the right canine teeth used to evaluate the dental arch parameters. Intraoral scanners provide stereographic files that can be used for both digital analysis of real objects and for their 3D printing [49; 50; 51; 52]. The accuracy depends on the scanner and printer technology used, as presented in Figure 12.  
AR in stimulatory is closely related to real data acquired as a plaster cast or as a digital model. In the computational environment, these data can be analysed by numerical methods and computational intelligence tools can be used to evaluate the information that is necessary for diagnostics and the treatment of dental disorders. In this way, the real world can be combined with virtual reality, while the computational environment extends direct observations and allows for a more detailed understanding of real situations and processes. Digitization has become a regular part of dentistry, orthodontics and maxillofacial surgery. The 3D reconstructive imagery based on a cone beam computed tomography (CBCT), intraoral and facial scans and computer-aided design techniques and technologies is more precise and can replace 2D X-rays, photographs, making impressions, models, splints and fabricating prostheses.