# Algorithms for Delivery of Data by Drones in an Isolated Area Divided into Squares
## 2 Materials and Methods  
### Drone Network Architecture and Communication  
A network map is essential for both drone flight control and simultaneous localization and mission tasks. As noted in [36], three forms of map generation exist: metric, topological, and hybrid maps. A metric map is represented as a grid, geometric, or feature map. Topological maps are represented by graphs comprised of nodes and edges, where nodes represent places, and the edges represent the paths between the nodes [36]. A hybrid map consists of small metric map locations in nodes. These nodes are connected by edges, which are the paths between the metric maps [36]. In this work, a topological map was proposed. The 2D network surface is intended to be covered with equidistantly distributed points. These points form regular polygons covering the 2D surface. Below, we prove that the only possible means of covering the 2D surface in this manner is by using equilateral triangles or squares. First, we prove the following lemma:  
**Lemma 1**. : _There are 3 ways to cover a 2D surface using regular polygons: equilateral triangles, squares, or regular hexagons._  
Proof of Lemma 1.: The sum of the degrees of the angles of a polygon with \(n\) vertices (\(n\geq 3\)) is \(180^{\circ}\cdot(n-2)\). Each angle of a regular polygon has \([360^{\circ}\cdot(n-2)]/n\) degrees. If a point of the surface is a vertex of a polygon, then it is a vertex for \(m\) polygons (\(m\geq 3\)) around this point (Figure 1). Thus, at such a point we have:  
\[m\cdot\frac{180^{\circ}\cdot(n-2)}{n}=360^{\circ}\Leftrightarrow\frac{2\cdot n }{n-2}=m \tag{1}\]  
It results that \(2\cdot n/(n-2)=2+4/(n-2)\) is an integer value greater or equal to 3. This implies that \(n-2\) is a divisor of 4 and, because \(n\geq 3\), it follows that \(n-2\) is one of the values 1, 2, or 4, which is equivalent to the fact that \(n\) is 3, 4, or 6, and m is 6, 4 or, respectively, 3. This means that the 2D surface can only be covered with equilateral triangles, squares, or regular hexagons. **Theorem 1**. : _There are two ways to equidistantly cover a 2D surface with points._  
Proof of Theorem 1.: Using the result from Lemma 1 and the fact that the points are considered equidistant on the 2D surface, it follows that there are only two ways to cover the 2D surface: with equilateral triangles or with squares, because using regular hexagons is impossible (the vertices of the hexagons are not equidistantly positioned). \(\square\)  
A network to cover a surface with hexagon cells containing three equilateral triangles was proposed in [41]. In the current paper, a network is proposed to cover a similar surface with square cells containing two operational squares (Figure 2). Each operational cell is covered by a drone. A battery charging/changing dock, which is shared by two drones, is placed in the center of each cell. The docks are able to automatically change/recharge drones, without manual intervention, allowing fully autonomous drone management. The mission profile of each drone consists of several phases or steps: engines start, take off, climb at the cruise altitude, cruise, hovering and data exchange, descent, landing, and engines shut-down. The cruise phase of the flight mission profile consists of four segments (Figure 3). Charging or changing the drone battery is a necessary step at the end of the flight mission. Figure 1: Regular polygons sharing the same vertex. Figure 2: Square cell with two drones. A drone with a quadcopter configuration [42], i.e., a DJI Mavic 2 Pro (DJI, Shenzhen, China) with a size of 214 \(\times\) 91 \(\times\) 84 mm (length \(\times\) width \(\times\) height), and takeoff weight of 905 g, was considered in this study. The performance characteristics of this drone are presented in Table 1[43]. All of the drones of the proposed squares network operate in a pre-programmed manner. The drone network communicates with data exchange points via wireless links.