# The UWB Radar Application in the Aviation Security Systems
## 2 Materials and Methods
### Localization of the Target by the Straight-Line Method for Two Independent Radar Sets
#### 2.1.4 Application Procedure for the Straight-Line Method Usage  
In the case of using two radar sets in positions, as shown in the scenario (see Section 2.1.1 and Figure 1), there are also other real intersections created by combinations of ellipses of different sets. Possible combinations are as follows:  
For the intersections of the ellipses E\({}_{1}\)\({}^{\text{RS}}\)\({}^{1}\) and E\({}_{3}\)\({}^{\text{RS}}\)\({}^{2}\) in Figure 7a (the designation is the same as in Figures 2 and 3), it is possible to connect them by a straight line. The line drawn by the intersections of the ellipses will represent the 3 rd line of a straight-line method. Similarly, the intersections of the ellipses E\({}_{1}\)\({}^{\text{RS}}\)\({}^{1}\), E\({}_{4}\)\({}^{\text{RS}}\)\({}^{2}\) in Figure 7b, and the line drawn by them, are called the 4 th line of the straight-line method. By analogy, the lines drawn by intersections E\({}_{2}\)\({}^{\text{RS1}}\), E\({}_{3}\)\({}^{\text{RS}}\)\({}^{2}\) in Figure 8a and E\({}_{2}\)\({}^{\text{RS1}}\), E\({}_{4}\)\({}^{\text{RS2}}\) in Figure 8b are referred to as the 5 th and 6 th line of the straight-line method in that order. For calculation of the combined intersections of the ellipses (shown in Figures 7 and 8), it is possible to use the Bezout theorem, which does not impose restrictions on the location of the foci of ellipses. Algorithm for Determining the Intersections of Ellipses Using the Bezout Theorem  
Calculation of coefficients of matrix A, vector b, and constant c for ellipses i. Consider two ellipses E1 and E2 defined by focal points \(F_{1}^{(i)}=\left(x_{1}^{(i)},y_{1}^{(i)}\right)\) and \(F_{2}^{(i)}=\left(x_{2}^{(i)},y_{2}^{(i)}\right)\) major half-axes \(a1,a2\) determined by quadratic equations:  
\[ \begin{gathered} Q_{i}(\mathbf{p})=\mathbf{p}^{T}\mathbf{A}_{i}\mathbf{p}+\mathbf{b}_ {i}^{T}\mathbf{p}+c^{i}\\ =\left[\begin{array}{cc}x&y\end{array}\right]\times\left[ \begin{array}{cc}a_{00}^{(i)}&a_{01}^{(i)}\\ a_{01}^{(i)}&a_{11}^{(i)}\end{array}\right]\times\left[\begin{array}{cc}x \\ y\end{array}\right]+\left[\begin{array}{cc}b_{0}^{(i)}&b_{1}^{(i)}\end{array} \right]\times\left[\begin{array}{cc}x\\ y\end{array}\right]+c^{(i)}=0,\end{gathered}  \tag{8}\]  
where  
\[ \begin{gathered} a_{00}^{(i)}=\left(x_{1}^{(i)}-x_{2}^{(i)} \right)^{2}-4a_{i}^{2},\\ a_{01}^{(i)}=\left(x_{1}^{(i)}-x_{2}^{(i)}\right)^{*}\left(y_{1}^{(i)}-y_{2 }^{(i)}\right)\\ a_{11}^{(i)}=\left(y_{1}^{(i)}-y_{2}^{(i)}\right)^{2}-4a_{i}^{2},\\ b_{0}^{(i)}=\left(x_{1}^{(i)}-x_{2}^{(i)}\right)*\left(4a_{i}^{2}-\left(x_{1 }^{(i)}\right)^{2}-\left(y_{1}^{(i)}\right)^{2}+\left(x_{2}^{(i)}\right)^{2}+ \left(y_{2}^{(i)}\right)^{2}\right)+8a_{i}^{2}x_{2}^{(i)}\\ b_{1}^{(i)}=\left(y_{1}^{(i)}-y_{2}^{(i)}\right)*\left(4a_{i}^{2}-\left(x_{1 }^{(i)}\right)^{2}-\left(y_{1}^{(i)}\right)^{2}+\left(x_{2}^{(i)}\right)^{2}+ \left(y_{2}^{(i)}\right)^{2}+\left(y_{2}^{(i)}\right)^{2}\right)+8a_{i}^{2}y_{ 2}^{(i)}\\ c^{(i)}=\frac{1}{4}(4a_{i}^{2}-\left(x_{1}^{(i)}\right)^{2}-\left(y_{1}^{(i)} \right)^{2}+\left(x_{2}^{(i)}\right)^{2}+\left(y_{2}^{(i)}\right)^{2})-4a_{i}^ {2}{(x_{2}^{(i)})}^{2}-4a_{i}^{2}{(y_{2}^{(i)})}^{2}.\end{gathered} \]  
Calculation of auxiliary coefficients \(v_{0}\) to \(v_{10}\):  
\[ \begin{gathered} v_{0}=2\left(a_{00}^{(0)}a_{01}^{(1)}-a_{00}^{(1) }a_{01}^{(0)}\right),\\ v_{1}=a_{00}^{(0)}a_{11}^{(1)}-a_{00}^{(0)}a_{11}^{(1)},\\ v_{2}=a_{00}^{(0)}b_{0}^{(1)}-a_{00}^{(0)}b_{0},\\ v_{3}=a_{00}^{(0)}b_{1}^{(1)}-a_{00}^{(1)}b_{0}^{(1)},\\ v_{4}=a_{00}^{(0)}c^{(1)}-a_{00}^{(1)}c^{(0)},\\ v_{5}=2\left(a_{01}^{(0)}a_{11}^{(1)}-a_{01}^{(1)}a_{11}^{(1)}\right),\\ v_{6}=2\left(a_{01}^{(0)}b_{1}^{(1)}-a_{01}^{(1)}b_{0}^{(1)}\right),\\ v_{7}=2\left(a_{01}^{(0)}c^{(1)}-a_{01}^{(1)}c^{(0)}\right),\\ v_{8}=a_{01}^{(0)}b_{1}^{(1)}-a_{11}^{(1)}b_{0}^{(0)},\\ v_{9}=b_{0}^{(0)}b_{1}^{(1)}-b_{01}^{(1)}b_{0}^{(1)},v_{10}=b_{0}^{(0)}c^{ (1)}-b_{0}^{(1)}c^{(0)}.\end{gathered}  \tag{9}\]Calculation of coefficients \(u_{0}\) to \(u_{4}\) for Bezout's determinant  
\[