## 4 Digital Model of the Area of Interest
### Digital Terrain Model  
We suppose that input significant points \(S=\{[u_{l},v_{l},h_{l}];l=0,1,\cdots,k\}\) to the area \([u_{l},v_{l}]\) are given. The \([u_{l},v_{l}]\) are GPS coordinates of individual significant points, \(h_{l}\) is their altitude for \(l=0,1,\cdots,k\). By using these points, we will create a 3D model of the terrain in which signaling sensors will be placed. There will be created a network of sensory units. It is also necessary to consider the spatial resolution of the technology that will create the digital model of the terrain. This parameter is denoted as \(d\)--detail of terrain modeling. Dimensions of the sensory element (width and depth) are approximately \(50\times 50\)  cm so in this case, we can set the detail to \(d=2\) by default. This parameter means that a rectangle of size \(100\times 50\) cm can be covered with two frames that have size \(50\times 50\)  cm. For the value of \(d\), it is true that \(0<d\). The set \(M_{5}\) contains points on which the sensory units must be placed. The set \(V_{5}\) will mean internal significant points. Therefore, \(S-V_{5}\) is a set of boundary significant points. To model the mobile sensory units network in the Euclidean space, we need to define a coordinate system, the center of the coordinate system and \(x,y,z\)-axis. The center of the coordinate system will be the center of the Earth, the \(x\)-axis will be vernal equinox in the equator plane, the \(y\)-axis will be perpendicular to the \(x\)-axis in the equator plane, and the \(z\)-axis will be perpendicular to \(x\) and \(y\)-axes. This coordinate system is called Earth centered inertial. The more general name of this system is the Ecliptic coordinate system. The coordinate system can be implemented in Cartesian and spherical coordinates, see [6; 9]. For the sake of simplicity, we move the origin of the coordinate system to a point defined based on entry significant points. Calculation of the origin of the new coordinate system and the digital terrain model contains these steps. Algorithms in article will be entered in the English natural language. Notes: Different heights can be distinguished by color during the modeling of the terrain (see Figure 1). Height gradient of the terrain will be \(mz\). For each point \(U=[x,y,z]\in S^{\prime}\) it holds that:  
\[0\leq x\leq mx,0\leq y\leq my,0\leq z\leq mz. \tag{3}\]  
Our digital terrain model is raster (grid) and contains a regular distribution of points. We can see in the Figure 1 the shift of the origin of the coordinate system to the point \(O=[0,0,0]\) (steps of the Algorithms 1-3). This shift is necessary to calculate distances easier and locally with the help of the digital terrain model in applications. In the Digital terrain model section, a \(90\times 130\) matrix \(A\) is displayed. Significant points are marked as signs: +. Red points can be border and blue points internal significant points of the area (\(V_{S^{\prime}}\)) and the significant points marked with yellow must be included in the network of sensory units (\(M_{S^{\prime}}\)). **Algorithm 1:** Creating a digital model of the terrain around the area of interest. ```
input:\(S\), \(d\), \(M_{S},V_{S}\) output:\(S^{\prime}\), \(mx\), \(mz\), \(my\), \(O\), \(M_{S^{\prime}}\), \(V_{S^{\prime}}\)
```  
1. Declaration of coordinates of \(S\) in the Cartesian system:  
\[S=\{[u_{l},v_{l},h_{l}];l=0,1,\cdots,k\}=\{[x_{l},y_{l},z_{l}];l=0,1,\cdots,k\} \tag{4}\]
2. Marking of \(\min x=\min\limits_{l=0,\cdots k}x_{l}\), \(\min y=\min\limits_{l=0,\cdots k}y_{l}\), \(\min z=\min\limits_{l=0,\cdots k}z_{l}\) and  
\[S^{\prime}=\{[d\times(x_{l}-\min x),d\times(y_{l}-\min y),d\times(z_{l}-\min z )];l=0,1,\cdots,k\} \tag{5}\]
3. Let's mark \(mx=\max\limits_{l=0,\cdots k}x^{\prime}_{l}\), \(my=\max\limits_{l=0,\cdots k}y^{\prime}_{l}\) a \(mz=\max\limits_{l=0,\cdots k}z^{\prime}_{l}\), and point \(O=[0,0,0]\). 4. In this way a Euclidean coordinate system with the origin \(O\) will be created, see [7]. Dimensions of the digital terrain model will be \(mx\times my\times mz\).