## 4 Methodology
### The 7-Parameter Transformation
 \tag{39}\]  
By applying the same procedure, from Figure 5b, the complete rotation matrix for the PVT convention can also be obtained by multiplying three rotation matrices:  
\[ \begin{split}\mathbf{R}_{x}&=\begin{bmatrix}1&0&0\\ 0&\cos\alpha_{x}&-\sin\alpha_{x}\\ 0&\sin\alpha_{x}&\cos\alpha_{x}\end{bmatrix}\end{split} \]  
\[ \begin{split}\mathbf{R}_{y}&=\begin{bmatrix}\cos\alpha_{y}&0&\sin\alpha_{y}\\ 0&1&0\\ -\sin\alpha_{y}&0&\cos\alpha_{y}\end{bmatrix}\end{split}  \tag{40}\]  
\[ \begin{split}\mathbf{R}_{z}&=\begin{bmatrix}\cos\alpha_{z}&-\sin\alpha_{z}&0\\ \sin\alpha_{z}&\cos\alpha_{z}&0\\ 0&0&1\end{bmatrix}\end{split} \]  
which results in [44]  
\[\begin{split}\mathbf{R}_{PVT}^{ZYX}=\begin{bmatrix}\cos\alpha_{y} \cos\alpha_{z}&\sin\alpha_{x}\sin\alpha_{y}\cos\alpha_{z}-\cos\alpha_{x}\sin \alpha_{z}&\cos\alpha_{x}\sin\alpha_{y}\cos\alpha_{z}+\sin\alpha_{x}\sin \alpha_{z}\\ \cos\alpha_{y}\sin\alpha_{z}&\sin\alpha_{x}\sin\alpha_{y}\sin\alpha_{z}+ \cos\alpha_{x}\cos\alpha_{z}&\cos\alpha_{x}\sin\alpha_{y}\sin\alpha_{z}---- \sin\alpha_{x}\cos\alpha_{z}\\ -\sin\alpha_{y}&the overall rotation matrix can be simplified by approximating its elements (\(\sin\alpha_{i}=\alpha_{i}\), \(\cos\alpha_{i}=1\), \(\sin\alpha_{i}\sin\alpha_{j}=0\)), so Equations (39) and (41) have a simpler form:  
\[\mathbf{R}_{CFR}^{ZYX}= \begin{bmatrix}1&\alpha_{z}&-\alpha_{y}\\ -\alpha_{z}&1&\alpha_{x}\\ \alpha_{y}&-\alpha_{x}&1\end{bmatrix}  \tag{42}\]  
\[\mathbf{R}_{PVT}^{ZYX}= \begin{bmatrix}1&-\alpha_{z}&\alpha_{y}\\ \alpha_{z}&1&-\alpha_{x}\\ -\alpha_{y}&\alpha_{x}&1\end{bmatrix}  \tag{43}\]  
To illustrate the importance of the rotation matrix form, a numerical transformation of a point from HDKS to HTRS96 based on a set of identical transformation parameters in three different cases for CFR convention--using a simple rotation matrix (Equation (42)), using the rotation matrix \(R^{XYZ}\), and using the rotation matrix \(R^{ZYX}\) (Equation (39))--will be presented below. Based on transformation parameters from HDKS to HTRS96: \begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(t_{x}\) (m) & 546.509 \\ \(t_{y}\) (m) & 162.269 \\ \(t_{z}\) (m) & 469.395 \\ \(\mu\) (ppm) & \(-4.417\) \\ \(r_{x}\) (\(r^{\prime\prime}\)) & \(-5.906\) \\ \(r_{y}\) (\({}^{\prime\prime}\)) & \(-2.075\) \\ \(r_{z}\) (\({}^{\prime\prime}\)) & 11.507 \\ \hline \hline \end{tabular}   
point \(P\) in the HDKS coordinate system with the coordinates  
\[P_{HDKS}= \begin{bmatrix}X\\ Y\\ Z\end{bmatrix} = \begin{bmatrix}4,485,995.037\\ 1,296,375.198\\ 4,329,893.947\end{bmatrix} \]  
can be transformed in the HTRS96 coordinate system for three different cases:  
* Simple rotation matrix: \(P_{HTRS96}= \begin{bmatrix}4,486,637.611\\ 1,296,157.502\\ 4,330,336.208\end{bmatrix} \)
* Rotation matrix \(R^{XYZ}\): \(P_{HTRS96}= \begin{bmatrix}4,486,637.603\\ 1,296,157.501\\ 4,330,336.198\end{bmatrix} \)
* Rotation matrix \(R^{ZYX}\): \(P_{HTRS96}= \begin{bmatrix}4,486,637.597\\ 1,296,157.497\\ 4,330,336.206\end{bmatrix} \)  
From the results above, it is evident that the choice of the rotation matrix during the transformation between global and local coordinate systems has an impact on the transformed coordinate, with a value of around 0.5-1 cm. Therefore, when acquiring transformation parameters, it is essential to clearly define which rotation matrix was used in the computation process so that the same matrix can be used in the transformation procedure. When it comes to transformation between two global coordinate systems, the choice of the rotation matrix is not critical because the rotation angles are so small that they will not have a significant impact on the transformed coordinate.