1. Introduction

2. Electronic Area Protection

3. Network of Mobile Sensory Units

4. Digital Model of the Area of Interest

4.2. Digital Terrain Model

We suppose that input significant points $S=\left\{\left[u_l,v_l,h_l\right];l=0,1,\cdots ,k\right\}$ to the area $\left[u_l,v_l\right]$ are given. The $\left[u_l,v_l\right]$ 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 $50\times 50cm$ 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 $50\times 50cm$. For the value of d, it is true that $0<d$. The set $M_S$ contains points on which the sensory units must be placed. The set $V_S$ will mean internal significant points. Therefore, $S-V_S$ 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 [1,4].

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

• Algorithm for 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:

$$S=\left\{\left[u_l,v_l,h_l\right];l=0,1,\cdots ,k\right\}=\left\{\left[x_l,y_l,z_l\right];l=0,1,\cdots ,k\right\}$$

(3)

2. Marking of $minx={\displaystyle \underset{l=0,\cdots k}{min}}{x}_l$, $miny={\displaystyle \underset{l=0,\cdots k}{min}}{y}_l$, $minz={\displaystyle \underset{l=0,\cdots k}{min}}{z}_l$ a

$$S^{\prime }=\left\{\left[d\ast (x_l-minx),d\ast (y_l-miny),d\ast (z_l-minz)\right];l=0,1,\cdots ,k\right\}=\left\{\left[x_l^{\prime },y_l^{\prime },z_l^{\prime }\right];l=0,1,\cdots ,k\right\}.$$

(4)

3. 3. Let’s mark $mx={\displaystyle \underset{l=0,\cdots k}{max}}{x}_l^{\prime }$, $my={\displaystyle \underset{l=0,\cdots k}{max}}{y}_l^{\prime }$ a $mz={\displaystyle \underset{l=0,\cdots k}{max}}{z}_l^{\prime }$, and point $O=\left[0,0,0\right]$.

4. In this way a Euclidean coordinate system with the origin O will be created, see [2]. Dimensions of the digital terrain model will be $mx\times my\times mz$. Points of the set $S^{\prime },M_{S^{\prime }},V_{S^{\prime }}$, are created based on relation (3).

5. The digital model of the terrain is going to be an $A=\left(a_{ij}\right)$ matrix of type $mx\times my$. The significant points $\left[x_l^{\prime },y_l^{\prime },z_l^{\prime }\right]\in S^{\prime }$ are going to be elements $a_{x_l^{\prime }{y}_l^{\prime }}$ of the matrixA and their values are going to be the height of the terrain $a_{x_l^{\prime }{y}_l^{\prime }}=z_l^{\prime }$.

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 $U=\left[x,y,z\right]\in S^{\prime }$ it holds that:

$$0\le x\le mx,0\le y\le my,0\le z\le mz.$$

(5)

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 $O=\left[0,0,0\right]$ (steps of the algorithm 1. 2. and 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 ($M_{S^{\prime }}$).

A digital image of the terrain (matrix A of type $mx\times my$ in units of d) can be created by using the point $O^{\prime }=\left[0,0,h_0\right]$, with dimensions $mx\times my\times mz$. It is necessary to find point $O^{\prime }$ in the field too (it is a point with GPS coordinates (min x, min y, $h_0$). It is necessary to digitally scan the terrain so the point $O^{\prime }$ will be located in the lower-left corner of the digital model. The x -axis is defined by a straight line between point O and point $O_x$ (this point has GPS coordinates (min x, y) and points $O,O^{\prime },O_x$ form one plane) and y - axis is placed between point O and point $O_y$ (this point has GPS coordinates (x, min y) and the points $O,O^{\prime },O_y$ form one plane). The scanned area will be a rectangle of size $mx\times my$ in d units.

Figure 1. Digital terrain model and the origin O.

Figure 1. Digital terrain model and the origin O.

A digital image of the terrain must be created to obtain a matrix A. In the matrix A it is necessary to mark significant points $S^{\prime }$ according to the assignment (4) and significant points that must be included in the area of interests $M_{S^{\prime }}$ and internal significant points $V_{S^{\prime }}$. Conversion of GPS coordinates to the Cartesian system (3) is possible, e.g., in MATLAB using the command:

$$[X,Y,Z]=geodetic2ecef(spheroid,lat,lon,h)$$

(6)

This command is used to convert geodetic coordinates to the Earth-centred, Earth-fixed coordinate system (ECEF). In this case, the height h is the ellipsoidal height of the point,$(lat,lon)$ are the GPS coordinates of the point and the parameter spheroid is a parameter of the MATLAB system, which stands for the fact that we are examining in our considerations globe.

When we have available the ellipsoid heights of the points, we can apply the command (6). Then assignment

$$a_{xy}=Z$$

(7)

defines the elements of the matrix $A=\left(a_{xy}\right)$ of type $mx\times my$ in d units. Matrix A represents the digital model of the terrain. This model using raster data structures.

When the altitude or orthometric heightH is available, we need to calculate the ellipsoidal height h. The relation ships between heights are shown in Figure 2. Methods for calculating altitude (orthometric height) can be found in publications such as [4]. Another option is to apply newer versions of the program MATLAB (newer than R2022b), where theoretical calculations are already implemented and there are functions for this calculation [7].

Of course, other mathematical systems such as SageMath or Geogebra also enable geographic calculations too. Publication [1] contains the basic mathematical principles of geography. Detailed theoretical information about geodetic coordinate systems can be found in the publication [5], and about plane geometry of the type we use in the design [3].

4.5. Digital Model of the Area of Interest

We now summarize the procedure for defining a digital model of the area of interest.

The method of defining the area of interest in the terrain model

• A method of defining the area of interest in the terrain model

Input: A - digital terrain model,

$S^{\prime }$ - set of significant points (+) , $M_S$- set of significant points which are necessary (+, yellow color), $V_{S^{\prime }}$ set of internal significant points (+, blue color)

(+, red color in Figure 3 alternative boundary significant points, it is a set $(S^{\prime }-V_{S^{\prime }})-M_{S^{\prime }}$)

Output:

P - polygon of the area of interest,$p\left(P\right)$ - minimum number of sensory units.

• Determination of the sequence of significant points $r_0,$$r_1,\cdots ,r_n=r_0$ in matrix A so that all necessary boundary significant points have to be included (+, yellow color). These points are completed with boundary points (+ red color) to create a simple polygon P of the area of interest.
• For a given polygon P, we calculate the value $p\left(P\right)$ - it is an estimate of the minimum number of sensory units which are needed.

We note that this method cannot be algorithmized. This is the reason why the application is needed, which with the data inputs and outputs will help solve this problem. When the polygon P is defined, the length of the polygon is calculated based on (8) and according to (2) we can estimate the minimum number of sensory units in the network. The next task will be to create a method to deploy the sensory units on the polygon P.

Figure 3. Polygon of the area of interest

Figure 3. Polygon of the area of interest

4.6. Algorithm for Creating a Network of Mobile Sensory Units

Suppose that the polygon P of the area of interest is given. Polygon P contains significant points $r_0,r_1,\cdots ,r_n$ a $r_0=r_n$. All the necessary significant points that belong to the set $M_S$ are vertices of the polygon P. In a digital model of the terrain is given point $O=[0,0,0]$. According to this point, the coordinates of the points are given $r_i=\left[x_i,y_i,z_i\right]$.

• The algorithm for creating a network of sensory units

Input: P - polygon of the area of interest is given with significant points $r_0,r_1,\cdots ,r_n$ where $r_0=r_n$. The set of significant points $S^{\prime }$, the set of necessary significant points $M_{S^{\prime }}$ and the set of internal significant points $V_{S^{\prime }}$. Digital model of terrain A, $range\left(s\right)$ – the reach of the sensory element, $\tau$ - sensitivity of the sensory element.

Output: Q - polygon of the area of interest, which contains points $q_0,q_1,\cdots ,q_k$ and $q_0=q_k$ which form the network of sensory units.

1. for $i=1,\cdots ,n$

2. $j=i-1$ , point selection $r_j\in P$, $Q_j=\varnothing$

3. $l=1,k=1$, $q_k=r_j$.

4. $k=k+1$

5. if $\left|r_j-r_{j-1}\right|\leqq \tau \ast range\left(s\right)$ then $q_k=r_j$, $j=j+1$, $l=l+1$ (see, point I. on the Figure 4)

6. if $\left|r_i-r_{i-1}\right|>\tau \ast range\left(s\right)$ then

  if $r_i\in M_{S^{\prime }}$ then $q_k=r_j$, $j=j+1$, $l=l+1$ (see, point II. on the Figure 4)

  if $r_i\notin M_{S^{\prime }}$ then it is possible (there is no obstacle between the points $r_j$ and D)

      $q_k=D$, $j=j+1$, $l=l+1$

     if there is an obstacle between the points $r_j$and D, then

      $q_k=r_j$, $j=j+1$, $l=l+1$ (see, point III. on the Figure 4)

7. if $i<n$ then go to 2 (in this step the polygon $Q_j$ with vertices $q_1,q_2,\cdots ,q_l$ was created)

8. if $i=n$ then $Q=Q_k$, $\left|Q_k\right|={\displaystyle \underset{i=1,\cdots ,n}{min}}\left|Q_i\right|.$

Figure 4. Selecting the next point

Figure 4. Selecting the next point

Note: Point D lies on the line $r_i$ and $r_{i+1}$ at a distance of $2\ast range\left(s\right)$. Point D is not a significant point.

Brief description of the algorithm: We have a cycle considering the variablej in which we select the vertices of the polygon P. According to points 5-7. we create a polygon $Q_j$. with vertex $r_{j-1}$. Polygons $Q_j$ contain a set of necessary significant points and other points where sensory units are located. In point 8, we determine the polygon, which contains the smallest number vertices from that set of polygons $Q_1,\cdots ,Q_n$.

This method finds a polygon Q such that the length of the polygon is $p\left(Q\right)\leqq p\left(P\right)$ and the number of vertices of Q is less than or equal to the number of vertices of P. The polygon Q is a digital model of the boundary of the area of interest, the vertices of the polygon define the location of the sensor units in the area of interest.

5. Conclusions

References

1. Harvey, F. A Primer of GIS: Fundamental Geographic and Cartographic Concepts; GUILFORD PUBN, 2015.
2. Maling, D.H. Coordinate Systems and Maps Projections; Pergamon Press, 1992.
3. Chiossi, S.G. Elementary Plane Geometry. In Essential Mathematics for Undergraduates; Springer International Publishing, 2021; pp. 401–437. [CrossRef]
4. Sickle, J.V. Basic GIS Coordinates; CRC Press. Taylor & Francis Group, 2017.
5. Lawrence, J.L. Celestial Calculations - a Gentle Introduction to Computational Astronomy; The MIT Press, Cambridge, Massachusetts, London England, 2018.
6. Xiao, N. GIS Algorithms; SAGE Publications Ltd, 2015.
7. Mathworks. Find Ellipsoidal Height from Orthometric Height, 2023. https://www.mathworks.com/help/map/ellipsoid-geoid-and-orthometric-height.html.
8. Conley, J. A Geographer’s Guide to Computing Fundamentals; Springer International Publishing, 2022. [CrossRef]
9. Kurdel, P.; Češkovič, M.; Gecejová, N.; Labun, J.; Gamec, J. The Method of Evaluation of Radio Altimeter Methodological Error in Laboratory Environment. Sensors 2022, 22, 5394. [Google Scholar] [CrossRef] [PubMed]
10. Novotňák, J.; Fiľko, M.; Lipovský, P.; Šmelko, M. Design of the System for Measuring UAV Parameters. Drones 2022, 6, 213. [Google Scholar] [CrossRef]
11. Fiľko, M.; Kessler, J.; Semrád, K.; Novotňák, J. Manufacturing of the Positioning Fixtures for the Security Sensors Using 3D Printing. Transportation Research Procedia 2022, 65, 98–105. [Google Scholar] [CrossRef]
12. Szabó, P.; Ferencová, M.; Blišťanová, M. A Spatially Bounded Airspace Axiom. Axioms 2022, 11, 244. [Google Scholar] [CrossRef]
13. Kasper, P.; Lipovsky, P.; Smelko, M. Application Software of Modular System for Magnetic Characteristics Measurement. In Proceedings of the 2022 New Trends in Aviation Development (NTAD). IEEE; 2022. [Google Scholar] [CrossRef]
14. Szabó, P.; Ferencová, M.; Železník, V. Cloud Computing in Free Route Airspace Research. Algorithms 2022, 14, 123. [Google Scholar] [CrossRef]
15. Nayak, S.; Zlatanova, S. (Eds.) Remote Sensing and GIS Technologies for Monitoring and Prediction of Disasters; Springer Berlin Heidelberg, 2008.