## 2 Materials and Methods
### The Method of the Geometric Interpretation of Facial Areas  
A method of the geometric interpretation of facial areas is proposed to obtain quantitative characteristics of qualitative features according to the given model. The method is used for the automated determination of facial expressions of human emotions in the form \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**\#** & **Quality Feature** & **Anger** & **Change Criterion** & **Fear** & **Change Criterion** \\ \hline \(x_{1}\) & Mouth & closed & [0.0.3] & open & [0.3;0.7] \\ \(x_{2}\) & Corners of the lips & raised & [0.3;0.7] & omitted & [0;0.3] \\ \(x_{3}\) & Eyes & squinted & [0.0.2] & wide open & [0.5;1] \\  & Eyebrows & reduced to the & [0.0.3] & divorced & [0.7;1] \\ \(x_{4}\) & (bridge of the nose) & bridge of the nose & [0.0; 0.3] & lifted up & [0.7;1] \\ \(x_{5}\) & Eyebrows & omitted & [0.0; 0.3] & lifted up & [0.7;1] \\ \(x_{6}\) & The corners of the & omitted & [0.0; 0.3] & lifted up & [0.6;1] \\  & eyebrows are external & omitted & [0.0; 0.3] & lifted up & [0.6;1] \\ \(x_{7}\) & The corners of the & eyebrows are internal & omitted & [0.0; 0.3] & lifted up & [0.6;1] \\ \hline \hline
**\#** & **Quality feature** & **Joy** & **Change criterion** & **Sadness** & **Change criterion** \\ \hline \(x_{1}\) & Mouth & open or closed & [0.6;1] & closed & [0;0.3] \\ \(x_{2}\) & Corners of the lips & raised & [0.7;1] & omitted & [0.0;3] \\ \(x_{3}\) & Eyes & squinted or wide & [0.2;0.5] & squinted & [0;0.3] \\  & & open & & reduced to the & \\ \(x_{4}\) & Eyebrows & normal & [0.3;0.7] & bridge of the nose & [0.1;0.5] \\  & & & or normal & \\ \(x_{5}\) & Eyebrows & elevated or normal & [0.3;0.7] & omitted or normal & [0.2;0.6] \\ \(x_{6}\) & The corners of the & elevated or normal & [0.3;0.7] & omitted or normal & [0.2;0.6] \\  & eyebrows are external & elevated or normal & [0.3;0.7] & omitted or normal & [0.2;0.6] \\ \(x_{7}\) & The corners of the & eyebrows are internal & [0.3;0.7] & omitted or normal & [0.2;0.6] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Qualitative characteristics of the features of facial expressions by emotional states. of quantitative characteristics of geometric shapes on a person's face. The scheme of the method is shown in Figure 3.  
The input information for the method is a vector \(P_{u}\) containing automatically marked landmarks on the face image. Such landmarks can be obtained in various known ways. For example, in thus work, this task was completed using the open-source toolkit called MediaPipe Face Mesh [42]. Figure 4a shows an example of face geometry description using MediaPipe Face Mesh. In Figure 4a, 486 markers \(P_{u}\), \(u=\overline{0,467}\), are represented by green circles. In step 1 of the method, the quantitative characteristics of the areas of the face are calculated. In step 1.1, the vector of features \(\mathbf{X}\) from model (1) is formalized by geometric shapes \(\{\alpha\}_{i=1}^{7}\), whose ends lie at points \(P_{u}\). The representation of facial expressions of emotions by qualitative characteristics is presented in Table 5.  
Figure 4: Human face with automatically marked landmarks: (**a**) all landmarks; (**b**) “Mouth” face area with geometric shape. Figure 3: Scheme of the method of the geometric interpretation of facial areas. Arrows here represent consistent transition from one method block to another. In step 1.2, the areas of geometric shapes \(\{\alpha\}_{i=1}^{7}\) are calculated. The type of shapes indicated in Table 5 is determined empirically for each part of the face. The distance of the segments forming the shapes \(\{\alpha\}_{i=1}^{7}\), is calculated according to the Euclidian distance formula. Next, we provide a detailed description of the shapes \(\{\alpha\}_{i=1}^{7}\), for each part of the face. The area of the face \"Mouth\" is described by a triangle, the ends of which lie in points \(P_{17}\), \(P_{37}\), and \(P_{267}\) (Figure 4b).