## 2 Materials and Methods
### Mathematical Description of the Method
\right]=P_{i}^{\prime} \tag{11}\]  
where \(R\) (\(\alpha\))--rotation matrix, \(\alpha\)--the angle of inclination of the ellipse, \(x_{i}\)--position coordinate on the horizontal axis of any point from the set, \(y_{i}\)--position coordinate on the vertical axis of any point from the set, and \(P_{i}^{\prime}\)--points after rotation by negative value by the angle of inclination of the ellipse \(\alpha\). Knowing the horizontal and vertical coordinates for the rotated points \(P_{i}^{\prime}\) is necessary to determine the dimensions of the semiaxis a and the semiaxis b of the ellipse (55). In order to determine the dimensions of the semiaxis of the ellipse, the method uses the standard deviation of the horizontal \(x_{i}^{\prime}\) and vertical \(y_{i}^{\prime}\) coordinates of the analyzed set of points \(P_{i}\). Depending on the type of distribution, this approach allows to control the percentage of points inside the ellipse by controlling a multiple of the standard deviation. For a set of points with a normal distribution, the dimension of the semiaxis a (12) corresponds to twice the standard deviation of the horizontal coordinate \(2\sigma_{x}\) and the dimension of the semiaxis \(b\) (13) corresponds to twice the standard deviation of the vertical coordinate \(2\sigma_{y}\). Assuming a double standard deviation for normal distributions results in 95.4% of all analyzed \(P_{i}\) points being inside the plotted ellipse. In order to implement the method for non-normal point distributions, it is advisable to express the lengths of the semiaxes \(a\) and \(b\) as the product of the conformity coefficient \(\varrho\) of the semiaxis dimensions to the point distribution and the standard deviation (14, 15). In this case, changes in the coefficient \(\varrho\) translate into the percentage of measurement points that will be included in the plotted ellipse, and thus into the degree of similarity in the representation of the space of presence for the analyzed points \(P_{i}\). \[a=2\sigma_{x}=2\sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}^{\prime}-\overline{x^{ \prime}}\right)^{2}}{n}} \tag{12}\]  
\[b=2\sigma_{y}=2\sqrt{\frac{\sum_{i=1}^{n}\left(y_{i}^{\prime}-\overline{y^{ \prime}}\right)^{2}}{n}} \tag{13}\]  
\[a=\rho\sigma_{x}=\rho\sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}^{\prime}-\overline{ x^{\prime}}\right)^{2}}{n}} \tag{14}\]  
\[b=\rho\sigma_{y}=\rho\sqrt{\frac{\sum_{i=1}^{n}\left(y_{i}^{\prime}-\overline{ y^{\prime}}\right)^{2}}{n}} \tag{15}\]  
where \(a\)--the longer semiaxis of the ellipse parallel to the trendline of the analyzed set of points \(P_{i}\), \(b\)--the shorter semiaxis of the ellipse perpendicular to the trendline of the analyzed set of points \(P_{i}\), \(\sigma_{x}\)--standard deviation of horizontal coordinates of the set of inverted points \(P_{i}^{\prime}\), \(\sigma_{y}\)--standard deviation of vertical coordinates of inverted set of points \(P_{i}^{\prime}\), \(\sigma_{i}^{\prime}\)--horizontal coordinate of any point from set of inverted points \(P_{i}^{\prime}\), \(y_{i}^{\prime}\)--vertical coordinate of any point from the set of inverted points \(P_{i}^{\prime}\), \(\overline{x^{\prime}}\)--mean of horizontal coordinates of the inverted set of points \(P_{i}^{\prime}\), \(\overline{y^{\prime}}\)--mean of vertical coordinates of the inverted set of points \(P_{i}^{\prime}\), and \(\varrho\)--conformity coefficient for fitting the semiaxis dimensions to the analyzed distribution. The last step of the presented algorithm is to plot the ellipse (E6). In order to do this, it is necessary to follow the previous steps, which will provide information about the parameters describing the ellipse. These parameters include the angle of inclination of the directional line of the ellipse \(\alpha\) (E2), the coordinates of the position of the center of the ellipse \(\mathbf{x}_{\text{COG}}\) and \(y_{\text{COG}}\) (E3), the dimension of the semiaxis a (E5), and the dimension of the semiaxis \(b\) (E5).