# The Influence of Pillar Displacements on Geodetic Measurements
## 3 Results and Discussion  
The Influence of the Temperature Displacements at Orientation Point \(B\) on the Position of Control Point \(C\)  
We will evaluate the precision of the coordinates of point \(C\) for various coordinate differences \(\Delta y_{A}^{C}\) and \(\Delta x_{A}^{C}\) between points \(A\) and \(C\) using Equations (27)-(29). We used the variances \(\sigma_{y_{C}}^{2}\) and \(\sigma_{x_{C}}^{2}\) and the covariance \(\sigma_{xy_{C}}\) to calculate the standard error ellipses (31), (33), and (35). Figure 7 and Table 1 show the direction and shape of the standard error ellipses resulting from the displacement of the orientation point \(B\) for different positions of the control point \(C\). We have shown ellipses for different positions of the simulated points, different bearings between points \(A\) and \(C\) (\(\
u_{A}^{C}=0^{\circ},\ 30^{\circ},\ 60^{\circ},\ 90^{\circ},\ 120^{\circ},\ 150^{\circ},\ 180^{\circ},\ 210^{\circ},\ 240^{\circ},\ 270^{\circ},\ 300^{\circ},\ 330^{\circ}\)) and different distances between the points (\(d_{AC}=50,100,150\) m) and for various coordinate differences between points \(A\) and \(C\). The bearing between survey point \(A\) and the orientation point \(B\) is \(\
u_{A}^{B}=45^{\circ}\), while the distance between survey point \(A\) and orientation point \(B\) is measured at \(d_{AB}=150\) m.  
Table 1 shows the precision of the position of the control point \(C\) and the elements of the standard error ellipse due to the influence of the displacement of the orientation point \(B\) for \(\sigma_{y_{B}}^{2}=\sigma_{x_{B}}^{2}=1\) mm\({}^{2}\) and \(\sigma_{yx_{B}}=0\) mm\({}^{2}\) at different angles and distances between survey point \(A\) and control point \(C\) and \(\
u_{A}^{B}=45^{\circ}\), \(d_{AB}=150\) m.  
From Figure 7 and the values in Table 1, we can see that:  
* we have the limit case of the error ellipses with one principal axis equal to zero,
* the direction of the major semi-axis is orthogonal to the line connecting points \(A\) and \(C\),
* the semi-major axis increases with the distance from the control point \(C\). The displacements of the orientation point \(B\) therefore not only causes an error in the measuring direction between survey point \(A\) and orientation point \(B\), but also lead to errors in the horizontal angle with the tip in survey point \(A\) and branches in the direction of orientation point \(B\) and control point \(C\)The Influence of the Temperature Displacements at Control Point C on the Position of Control Point C  
If the control point \(C\) has been displaced due to the temperature load on the pillar, the error ellipses are of course the same regardless of the angles of orientation and the distances between the points, which is clearly shown in Figure 8. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\
u_{A}^{C}\) & \(d_{AC}\) & \(\sigma_{xy_{C}}\) & \(\sigma_{y_{C}}\) & \(\sigma_{x_{C}}\) & \(a\) & \(b\) & \(\theta\) \\ \([^{\circ}]\) & **[m]** & **[mm2]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[\({}^{\circ}]\)** \\ \hline
0 & 50 & 0.00 & 0.33 & 0.00 & 0.33 & 0.00 & 90.00 \\
0 & 100 & 0.00 & 0.67 & 0.00 & 0.67 & 0.00 & 90.00 \\
0 & 150 & 0.00 & 1.00 & 0.00 & 1.00 & 0.00 & 90.00 \\
30 & 50 & 0.05 & 0.29 & 0.17 & 0.33 & 0.00 & 120.00 \\
30 & 100 & 0.19 & 0.58 & 0.33 & 0.67 & 0.00 & 120.00 \\
30 & 150 & 0.43 & 0.87 & 0.50 & 1.00 & 0.00 & 120.00 \\
60 & 50 & 0.05 & 0.17 & 0.29 & 0.33 & 0.00 & 150.00 \\
60 & 100 & 0.19 & 0.33 & 0.58 & 0.67 & 0.00 & 150.00 \\
60 & 150 & 0.43 & 0.50 & 0.87 & 1.00 & 0.00 & 150.00 \\
90 & 50 & 0.00 & 0.00 & 0.33 & 0.33 & 0.00 & 180.00 \\
90 & 100 & 0.00 & 0.00 & 0.67 & 0.67 & 0.00 & 180.00 \\
90 & 150 & 0.00 & 0.00 & 1.00 & 1.00 & 0.00 & 180.00 \\ \hline \end{tabular}
\end{table} 
Table 1: The precision of the position of the control point \(C\) and the elements of the standard error ellipse due to the influence of the displacement as a result of the temperature difference at orientation point \(B\).