## 2 Materials and Methods
### Methodology
#### 2.2.1 Single-Step Transformation--Separate  
The transformation from the local system (Sucha Gora) to the national system PL-2000, separately for each mining area in the subject area, was performed using a single-stage, first-degree conformal transformation. The transformation parameters were determined based on the coordinates of points (shafts) in both systems (local and national), in accordance with the procedure developed as a result of the research performed from 2017 and presented in the references [10; 11]:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Name of Coal** & **Name of** & **Number of** & **Number of** \\  & **Mine (CM)** & **Mining Area** & **Boundary Points** & **Adjustment Points** \\  & & & **(Tie Points)** & **(Shafts)** \\ \hline \multirow{2}{*}{1} & CM Boleslaw & \multirow{2}{*}{Laziska II} & \multirow{2}{*}{18} & \multirow{2}{*}{3} \\  & Smialy & & & \\ \hline
2 & CM Budryk & Ornontowice I & 11 & 4 \\ \hline
3 & CM Budryk & Ornontowice II & 7 & 1 \\ \hline
4 & CM Debiersko & Debiersko 1 & 19 & 2 \\ \hline \multirow{2}{*}{5} & CM Knurów- & \multirow{2}{*}{Knurów} & \multirow{2}{*}{19} & \multirow{2}{*}{8} \\  & Szczyglowice & & & \\ \hline \multirow{2}{*}{6} & CM Knurów- & \multirow{2}{*}{Szczyglowice} & \multirow{2}{*}{13} & \multirow{2}{*}{5} \\  & Szczyglowice & & & \\ \hline
7 & CM Sösnica & Makoszowy II & 28 & 5 \\ \hline
8 & CM Ruda & Bielszowice III & 18 & 6 \\ \hline
9 & CM Ruda & Zabrze I & 30 & 3 \\ \hline
10 & CM Ruda & Halemba I & 15 & 5 \\ \hline
11 & CM Ruda & Halemba II & 9 & - \\ \hline \multirow{2}{*}{12} & CM Ruda & Halemba-East & \multirow{2}{*}{10} & \multirow{2}{*}{1} \\  & Shaft & & & \\ \hline
13 & CM Ruda & Wirek II & 24 & 3 \\ \hline
14 & [PERSON] & Sösnica III & 27 & 6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the analyzed mining areas.  
The selection and analysis of adjustment points obtained from the mine or obtained as a result of direct field measurements.
* The determination of the degree and type of transformation based on the adjustment points obtained from the mines. The first-degree conformal transformation (Helmert) was adopted in this research--due to the possibility to obtain transformation coefficients consistent with those currently used by the mine.  
The equations in the matrix notation are expressed as follows (1):  
\[\begin{bmatrix}X\\ Y\end{bmatrix}=\begin{bmatrix}t_{X}\\ t_{Y}\end{bmatrix}+m\begin{bmatrix}cos\phi&-sin\phi\\ sin\phi&cos\phi\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} \tag{1}\]  
where  
\(X\)--coordinate \(x\) in the secondary (national) system;  
\(Y\)--coordinate \(y\) in the secondary (national) system;  
\(x\)--coordinate \(x\) in the primary (local) system;  
\(y\)--coordinate \(y\) in the primary (local) system;  
\(t_{X}\)--the shift (translation) vector in the \(x\) direction;  
\(t_{Y}\)--the shift (translation) vector in the \(y\) direction;  
\(m\)--the transformation scale change factor;  
\(\phi\)--the rotation angle of the coordinate system axes;  
\(\begin{bmatrix}cos\phi&-sin\phi\\ sin\phi&cos\phi\end{bmatrix}\)--the rotation matrix.  
Comparison of the coordinates of the adjustment points with the coordinates obtained from the transformation. The verification process should include the calculation of the differences in the \(x\) and \(y\) coordinates and the resultant difference. If the deviation exceeds the assumed maximum error values, then the transformation parameters should be re-determined.