# The Influence of Pillar Displacements on Geodetic Measurements
## 1 Introduction
### Law of Propagation of Variances and Covariances
The two points are related by \(y_{B^{\prime}}=y_{B}+\delta_{y_{B}}\) and \(x_{B^{\prime}}=x_{B}+\delta_{x_{B}}\), where \(\delta_{y_{B}}\) and \(\delta_{x_{B}}\) represent the displacements of the position of point \(B\) (Figure 4). Consequently, we measure the horizontal angle \(\alpha^{\prime}=\alpha+\delta_{\alpha}\), where \(\delta_{\alpha}\) is the error of the horizontal angle \(\alpha\), instead of measuring horizontal angle \(\alpha\). The horizontal angle \(\alpha^{\prime}\) is calculated as the difference between the measured horizontal direction at control point \(C\) and the (displaced) measured horizontal direction at orientation point \(B^{\prime}\)[4]:  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]  
Due to the displacement of the orientation point \(B\), the horizontal angle \(\alpha\) shows an error. Considering the approximations \(y_{B^{\prime}}=y_{B}\) and \(x_{B^{\prime}}=x_{B}\), we can calculate the position of the control point \(C\):  
\[y_{C}=y_{A}+d_{AC}\sin(y_{A}^{B}+\alpha^{\prime})\text{ and } \tag{15}\]  
Figure 4: Displacement at orientation point \(B\): geometry. Assuming \(\sigma_{y_{b}}^{2}=\sigma_{x_{B}}^{2}=\sigma_{B}^{2}\) and \(\sigma_{xy_{B}}=0\), and \(\
u_{A}^{C}=\
u_{A}^{B}+\alpha^{\prime}\), the result further simplifies to:  
\[\sigma_{y_{C}}^{2}=\sigma_{B}^{2}.\frac{d_{AC}^{2}}{d_{AB}^{2}}\cos^{2}\
u_{A} ^{C}, \tag{27}\]  
\[\sigma_{x_{C}}^{2}=\sigma_{B}^{2}.\frac{d_{AC}^{2}}{d_{AB}^{2}}\cdot\sin^{2}\
u _{A}^{C},\text{ and} \tag{28}\]  
\[\sigma_{xy_{C}}=-\sigma_{B}^{2}\frac{1}{2}\frac{d_{AC}^{2}}{d_{AB}^{2}}\cdot \sin 2\
u_{A}^{C}. \tag{29}\]  
We can now calculate the values of the semi-axis of the error ellipse and the bearing of the longer semi-axis [13]:  
\[a^{2}=\frac{\sigma_{x_{C}}^{2}+\sigma_{y_{C}}^{2}+\sqrt{\left(\sigma_{x_{C}}^ {2}-\sigma_{y_{C}}^{2}\right)^{2}+4\sigma_{xy_{C}}^{2}}}{2}=\sigma_{B}^{2}. \frac{d_{AC}^{2}}{d_{AB}^{2}}\text{ and} \tag{30}\]  
\[a=\sigma_{B}.\frac{d_{AC}}{d_{AB}}, \tag{31}\]  
\[b^{2}=\frac{\sigma_{x_{C}}^{2}+\sigma_{y_{C}}^{2}-\sqrt{\left(\sigma_{x_{C}}^ {2}-\sigma_{y_{C}}^{2}\right)^{2}+4\sigma_{xy_{C}}^{2}}}{2}=\frac{1}{2}\Bigg{(} \sigma_{B}^{2}.\frac{d_{AC}^{2}}{d_{AB}^{2}}-\sigma_{B}^{2}.\frac{d_{AC}^{2}} {d_{AB}^{2}}\Bigg{)}\text{ and} \tag{32}\]  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]0  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]1  
\[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]2  
From Equations (31) and (33), we can observe that the error ellipse becomes a line. More detailed numerical results will be presented in Section 3.1. The Influence of the Temperature Displacements at Control Point C on the Position of the Control Point C  
In case the control point \(C\) has been displaced, we aim at the point \(C^{\prime}(y_{C^{\prime}},x_{C^{\prime}})\) instead of point \(C(y_{C},x_{c})\), where \(y_{C}=y_{C}+\delta_{y_{C}}\) and \(x_{C^{\prime}}=x_{C}+\delta_{x_{C}}\) and \(\delta_{y_{C}}\) and \(\delta_{x_{C}}\) are the displacement of the position of control point \(C\) (Figure 5). \[\alpha^{\prime}=\sigma_{A}^{C}-\sigma_{A}^{B^{\prime}}. \tag{14}\]3 Displacement at control point C: geometry. As expected, if the control point \(C\) has been displaced as a result of the temperature load on the pillar, its position has moved for exactly the same value as the displacements by temperature at the top of the pillar. The Influence of the Temperature Displacements at Survey Point \(A\) on the Position of the Control Point \(C\)  
If the temperature load has changed the position of the survey point \(A\), we stem from the point \(A^{\prime}(y_{A^{\prime}},x_{A^{\prime}})\), instead of the point \(A(y_{A},x_{A})\), where \(y_{A^{\prime}}=y_{A}+\delta_{y_{A}}\) and \(x_{A^{\prime}}=x_{A}+\delta_{x_{A}}\) and \(\delta_{y_{A}}\) and \(\delta_{x_{A}}\) represent the displacement of the position of point \(A\) (Figure 6).