## 4 Discussion  
In order to compare the two proposed methods (from Sections 2.1 and 2.2) for determining the points P\({}_{1}\) and P\({}_{2}\) 100 million experiments were performed. For each experiment, two random lines d\({}_{1}\) and d\({}_{2}\) were selected and the point Q was also randomly chosen. \begin{table}
\begin{tabular}{l l c c c} \hline \hline  & **Points** & **E (m)** & **N (m)** & **Z(m)** & **Calculated Z(m)** \\ \hline Q\({}_{1}\) & & 552,749.2478 & 456,578.3797 & & 707.8851 \\  & A\({}_{11}^{1}\) & 552,749.7024 & 456,578.7371 & 707.6000 & \\  & A\({}_{12}^{1}\) & 552,749.4725 & 456,578.9520 & 707.6000 & \\  & A\({}_{21}^{2}\) & 552,748.8265 & 456,578.0230 & 708.0000 & \\  & A\({}_{22}^{2}\) & 552,749.2769 & 456,577.9832 & 708.0000 & \\ Q\({}_{2}\) & & 552,754.2700 & 456,577.8350 & 709.3279 & \\  & A\({}_{21}^{2}\) & 552,754.0980 & 456,572.9276 & 709.2000 & \\  & A\({}_{22}^{1}\) & 552,754.4526 & 456,572.8445 & 709.2000 & \\  & A\({}_{21}^{2}\) & 552,754.0176 & 456,571.5626 & 709.6000 & \\  & A\({}_{22}^{2}\) & 552,754.3366 & 456,571.5767 & 709.6000 & \\ Q\({}_{3}\) & & 552,766.2100 & 456,580.9970 & 704.2017 & \\  & A\({}_{31}^{3}\) & 552,768.2179 & 456,583.3340 & 704.0000 & \\  & A\({}_{12}^{12}\) & 552,768.7357 & 456,582.6679 & 704.0000 & \\  & A\({}_{31}^{3}\) & 552,760.8755 & 456,578.3463 & 704.4000 & \\  & A\({}_{22}^{3}\) & 552,760.9848 & 456,577.8984 & 704.4000 & \\ Q\({}_{4}\) & & 552,760.2440 & 456,588.7720 & 704.2330 & \\  & A\({}_{11}^{4}\) & 552,762.5489 & 456,590.7208 & 704.0000 & \\  & A\({}_{12}^{4}\) & 552,762.9178 & 456,590.3418 & 704.0000 & \\  & A\({}_{21}^{4}\) & 552,755.2345 & 456,585.1132 & 704.4000 & \\ Q\({}_{5}\) & & 552,754.9360 & 456,585.3872 & 704.4000 & \\  & A\({}_{11}^{5}\) & 552,755.2345 & 456,585.4520 & 704.7360 & \\  & A\({}_{11}^{5}\) & 552,755.2345 & 456,585.1132 & 704.4000 & \\  & A\({}_{2}^{3}\) & 552,755.1438 & 456,584.4878 & 704.4000 & \\  & A\({}_{21}^{3}\) & 552,752.7819 & 456,582.6377 & 704.8000 & \\  & A\({}_{22}^{4}\) & 552,752.3362 & 456,582.9349 & 704.8000 & \\ Q\({}_{6}\) & & 552,753.3850 & 456,581.5540 & 705.0855 & \\  & A\({}_{11}^{6}\) & 552,753.3671 & 456,581.7908 & 704.8000 & \\  & A\({}_{2}^{1}\) & 552,753.7000 & 456,581.5900 & 704.8000 & \\  & A\({}_{21}^{2}\) & 552,753.2125 & 456,581.6078 & 705.2000 & \\  & A\({}_{22}^{5}\) & 552,753.5536 & 456,581.3981 & 705.2000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: 3D coordinates of the Q points. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Estimated Z (m)** & **Measured Z (m)** \\ \hline Q\({}_{1}\) & 707.8921 & 707.8990 \\ Q\({}_{2}\) & 709.3279 & 709.3260 \\ Q\({}_{3}\) & 704.2017 & 704.2140 \\ Q\({}_{4}\) & 704.2330 & 704.2210 \\ Q\({}_{5}\) & 704.7360 & 704.7500 \\ Q\({}_{6}\) & 705.0855 & 70 The tests were performed using an ASUS ROG GL752 VW-T4015D laptop with Intel(r) Core(tm) i7-6700 HQ 2.60 GHz processor, and 8 GB of RAM. First, we analyzed the speed of the proposed methods. The total running time of the first method was 5.82 s. The total running time of the second method was 40.14 s. So, the first method is almost 7 times faster than the second one (see Figure 13). The first method is considerably faster because it requires elementary calculations, while the second one runs an iterative algorithm to solve the Equation (18). We also calculated the accuracy of the first method against the second method since the error for the second method can be set as low as desired. An average error of 0.61% was obtained, the lowest error being 0 and the highest 1.17% (Table 4). So, we concluded that the accuracy of the approximate first method is good enough. The steps of positioning the house will be next discussed, and the utility of the methods proposed in this paper will be shown. The designer receives a topographic map in electronic format. On this topographic support, the designer will place the designed building on the scale, having to comply with a series of rigors imposed by law, among which: the minimum distance from the property limit and/or from the road axis, etc.