## 4 Methodology
### The 7-Parameter Transformation  
The 7-parameter transformation, or Helmert's transformation, is the most widespread and most used coordinate transformation in geodesy and geoinformatics and belongs to the group of similar transformations. It is based on three translations along and three rotations around three (X, Y, Z) axes as well as unique scale for all axes. In practice, two approximations of [PERSON]'s transformation are used: Bursa-Wolf (Figure 4a) and Molodensky-Badekas (Figure 4b). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Standard Deviation** \\ \hline \(t_{x}\) (m) & 651.902 & 2.333 \\ \(t_{y}\) (m) & \(-\)210.792 & 0.695 \\ _da_ (m) & 497.803 & 5.390 \\ _df_ & 767.897 & 2.467 \\ _df_ & 4.828 \(\times\) 10\({}^{-6}\) & 7.535 \(\times\) 10\({}^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of standard Molodensky 5-parameter transformation from HDKS to HTRS96. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Standard Deviation** \\ \hline \(t_{x}\) (m) & 652.010 & 2.332 \\ \(t_{y}\) (m) & \(-\)210.746 & 0.695 \\ \(t_{z}\) (m) & 497.354 & 5.432 \\ _da_ (m) & 767.889 & 2.466 \\ _df_ & 4.890 \(\times\) 10\({}^{-6}\) & 7.583 \(\times\) 10\({}^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of abridged Molodensky 5-parameter transformation from HDKS to HTRS96. Bursa-Wolf transformation rotation is performed around the origin of the coordinate system, and the general expression for transformation is [43]  
\[\mathbf{X}_{o}=\mathbf{t}+\mu\cdot\mathbf{R}\cdot\mathbf{X}_{i} \tag{29}\]  
or, in matrix form,  
\[ \begin{bmatrix}X\\ Y\\ Z\end{bmatrix} _{o}= \begin{bmatrix}t_{x}\\ t_{y}\\ t_{z}\end{bmatrix} +\mu\cdot \begin{bmatrix}r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\end{bmatrix} \cdot \begin{bmatrix}X\\ Y\\ Z\end{bmatrix} _{i} \tag{30}\]  
where \(\mathbf{X}_{o}\) is the position vector in the output coordinate system, \(\mathbf{t}\) is the translation vector from the input to the output coordinate system, \(\mu\) is scale, \(\mathbf{R}\) is the rotation matrix from the input to the output coordinate system, and \(\mathbf{X}_{i}\) is the position vector in the input coordinate system. The translation vector as well as the position vectors are expressed in meter units, rotation matrix elements in radians, and scale can be expressed as  
\[\mu=1+d\mu \tag{31}\]  
where \(d\mu\) is a scale change in ppm (parts per million, \(10^{-6}\)) or ppb (parts per billion, \(10^{-9}\)). Molodensky-Badekas transformation represents a generalization of the Bursa-Wolf transformation by defining the point around which the rotation is performed, called the centroid of the identical points and is given by expression [43]  
\[ \begin{bmatrix}X\\ Y\\ Z\end{bmatrix} _{o}= \begin{bmatrix}X_{0}\\ Y_{0}\\ Z_{0}\end{bmatrix} _{i}+ \begin{bmatrix}t_{x}\\ t_{y}\\ t_{z}\end{bmatrix} +\mu\cdot \begin{bmatrix}r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\end{bmatrix} \cdot \begin{bmatrix}X-X_{0}\\ Y-Y_{0}\\ Z-Z_{0}\end{bmatrix}  \tag{32}\]  
where the \(X_{0}\), \(Y_{0}\), and \(Z_{0}\) coordinates are defined by the expressions  
\[ \begin{split} X_{0}&=\frac{\sum_{i=1}^{k}X_{i}^{i}}{k}\\ Y_{0}&=\frac{\sum_{i=1}^{k}Y_{i}^{i}}{k}\\ Z_{0}&=\frac{\sum_{i=1}^{k}Z_{i}^{i}}{k}\end{split}  \tag{33}\]  
Figure 4: The 7-parameter transformation (according to [4]): (**a**) Bursa–Wolf, (**b**) Molodensky–Badekas. where \(X_{i}\), \(Y_{i}\), \(Z_{i}\) are coordinates in the input coordinate system, and \(k\) is the number of points. Translation and scale change in the 7-parameter transformation are unambiguously defined, whereas that is not the case with rotations. Knowing the rotation angles is not enough to carry out the rotation of the coordinate system. It is necessary to know the convention that defines the rotation matrix (direction and type of rotations) and the order of matrix multiplication.