## 2 Materials and Methods
### Methodology
#### 2.2.2 Transformation Using the Authorial Procedure Based on the Method of Independent Models
The stage of linear flat-height transformation, constituting the first approximate values of unknowns. In the case of using the method of independent models for approximate fitting, height transformation is not required. It is sufficient to perform linear flat transformation. - In the case of points with known (adjustments) coordinates: \[ \begin{array}{c}a_{i}+u_{i}\cdot x_{k}^{i}+v_{i}\cdot y_{k}^{i}=X_{k}+\delta x _{k}\\ b_{i}+u_{i}\cdot y_{k}^{i}-v_{i}\cdot x_{k}^{i}=Y_{k}+\delta y_{k}\end{array} \] (6) - In case of tie points: \[ \begin{array}{c}a_{i}+u_{i}\cdot x_{k}^{i}+v_{i}\cdot y_{k}^{i}-a_{j}-u_{j} \cdot x_{k}^{j}-v_{j}\cdot y_{k}^{i}=\delta x_{k}^{i,j}\\ b_{i}+u_{i}\cdot y_{k}^{i}-v_{i}\cdot x_{k}^{i}-b_{j}-u_{j}\cdot x_{k}^{j}+v_{ j}\cdot y_{k}^{j}=\delta y_{k}^{i,j}\end{array} \] (7) where \(a_{i}\), \(b_{i}\) --the translation parameters of the i-th subset; \(k\)--point index (point number); \(i\), \(j\)--the index of transformation area; \(X_{k}\), \(Y_{k}\)--the flat coordinates of the adjustment point in the secondary system (with the index \(k\) after shifting to the center of gravity); \(x_{k}^{i}\), \(y_{k}^{i}\)--the flat coordinates of the point in the i-th primary system (with the index \(k\) after shifting to the center of gravity); \(u_{i}\), \(v_{i}\)--the transformation parameters that are a combination of rotation \(\phi\) and scale change. \[s\;(u_{i}=s_{i}\cdot cos\phi_{i},\;v_{i}=s_{i}\cdot sin\phi_{i})\] The above-mentioned parameters \(a\), \(b\), \(u\), and \(v\) are calculated from the weighted least squares principle. Height alignment can be performed using the following formulas: - In the case of points with known (adjustments) coordinates: \[c_{i}+s_{i}\cdot z_{k}^{i}=Z_{k}+\delta z_{k}^{i}\] (8) - In the case of tie points: \[c_{i}-c_{j}+s_{i}\cdot z_{k}^{i}-s_{j}\cdot z_{k}^{j}=\delta z_{k}^{i,j}\] (9) where \(c_{i}\), \(c_{i}\)--the height translation (shift) of the model; \(s_{i}\), \(s_{j}\)--the scale change coefficients determined from the flat alignment for a given model; \(Z_{k}\)--the height ordinate of the adjustment point (with the index \(k\)) in the secondary system after shifting to the center of gravity; \(z_{k}^{i}\), \(z_{k}^{i}\)--the height ordinates of the point (with index \(k\)) in the local systems of the models; \(k\)--point index (\(k\) = 1, 2, 3, \(\ldots\), \(m\), where \(m\)--the number of all points transformed in a given model); \(i\), \(j\)--the index of transformation area (\(i\) = 1, 2, 3, \(\ldots\), \(n\), \(j\) = 1, 2, 3, \(\ldots\), \(n\), where \(n\)--the number of transformation areas). The unknown parameters of the height transformation (translation) of the models can be determined in a similar way as in the case of flat alignment. The formulas used to calculate the approximate coordinates of points in the secondary system are as follows:  
\[ \begin{array}{c}X_{k}^{i}=a_{i}+u_{i}\cdot x_{k}^{i}+v_{i}\cdot y_{k}^{i}\\ Y_{k}^{i}=b_{i}+u_{i}\cdot y_{k}^{i}-v_{i}\cdot x_{k}^{i}\\ Z_{k}^{i}=c_{i}+s_{i}\ \cdot z_{k}^{i}\end{array}  \tag{10}\]  
\(s_{i}\) is calculated using the following formula:  
\[s_{i}=\sqrt{u_{i}^{2}+v_{i}^{2}} \tag{11}\]  
Explanations of parameters are analogous to those in Formulas (6)-(9). IV. After the execution of linear flat-height transformation, a three-dimensional transformation should be performed, based on the conformal transformation formulas. It is a transformation by similarity, so it does not cause the deformation of the model shape. * The matrix notation of the transformation equation is expressed as follows: \[X=\lambda\cdot A\cdot x+X_{0}\] (12) where \(\lambda\)--model scale change factor. The differential nature of these transformations (small rotations, displacements, and changes in scale) is possible through appropriate conditional equations. * Using the so-called small rotation matrix: \[A_{m}= \begin{vmatrix}1&-\delta k&\delta f\\ \delta k&1&-\delta w\\ -\delta f&\delta w&1\end{vmatrix} \] (13) where \(\delta k\), \(\delta f\), \(\delta w\)--the change in rotation angle.