# Delivering Extended Cellular Coverage and Capacity Using High-Altitude Platforms
## 4 HAP Beam Pointing for Extended Coverage
### Beam-Pointing Algorithm
At the end of this first step, a set of tessellated cell centres \(\mathcal{C}_{T}=\{(0,d_{1},0),\ (0,d_{2},0),\ (0,d_{3},0),\cdots,\ (0,d_{n},0)\}\) is obtained, resulting in the structure shown in Figure 9a. Thus, \(\mathcal{C}_{T}\) is defined as follows,  
Figure 6.6  
Step 2:  
In the second step, all \(c_{i}\in\mathcal{C}_{T}\) is rotated by \(60^{\circ}\) as shown in Figure 9b to obtain another set of cell boresight coordinates \(\mathcal{C}_{R}\). The \(60^{\circ}\) rotation allows for the exploitation of the good tessellation properties of hexagonal geometry. The path vectors between the cells in \(\mathcal{C}_{T}(i)\) and their corresponding rotated copies in \(\mathcal{C}_{R}(i)\) are obtained using spherical linear interpolation defined by the following expression [36]. Figure 6.7  
where \(\theta\) is the angle between corresponding \(i\)th cells in \(\mathcal{C}_{T}\) and \(\mathcal{C}_{R}\), \(V_{i}\) and \(V_{f}\) are the start and final vectors of the corresponding cells with reference to the SPPC. Meanwhile, \(V_{I}\) represents the path vector between the cells with \(s\) determining the steps in the path. Steps 3-4:  
The solid curve between corresponding cell centre coordinates, as shown in Figure 9c, highlights the paths between the cells and their rotated copies. Ideally, an increasing number of cells will be formed azimuthally on the paths depicted by the solid lines between corresponding cells. These paths are divided into \(k+1\) equal path lengths. Here, \(k\) indicates the position of the paths, which are shown using the solid lines in Figure 9c, on the cellular arc/path starting from \(k=0\) at the SPPC. \(k\) new coordinates (equidistant from each other) are introduced starting from arc/path \(k=1\) to the last in the structure. However, introducing new cells azimuthally on the solid arcs/paths results in minimal or no overlap between neighbouring cells towards the SPPC. This is due to azim  
Figure 6.8 The cell tessellation processes. (**a**) The first step with cells pointing at increasing distance on the \(x\)-axis. (**b**) The rotation of the first cells from (**a**) yielding another set of cells in the second step. (**c,d**) The third and fourth steps where new cells are deployed between the structure in (**b**). distortion, which occurs particularly in aerial systems when beams are formed or steered electronically in azimuth using a planar phased array antenna [37; 38]. AAP distortion results in steering angle quantisation, grating lobes and main lobe gain reduction [37], which minimises the overlap between neighbouring cells as highlighted earlier. To mitigate against AAP distortion and compensate for the mainlobe gain loss, the new cells are pointed on the mirror images of the paths, which are the dashed convex paths, rather than the original solid concave paths. The mirror image approach, which works by packing the new cells formed azimuthally closer together, is a heuristic and less complex method that ensures proper overlap between neighbouring cells towards the SPPC. The optimality of this heuristic method has not been investigated on this occasion; however, it is shown to enhance the tessellation. A set of the new cell centre coordinates are assigned to \(\mathcal{C}_{P}\) to conclude the third and fourth steps with the resulting cellular structure when cells are pointed at these coordinates shown in Figure 9d. Step 5:  
Finally, in order to achieve full \(360^{\circ}\) coverage, the union of the set of cell boresight coordinates \(\mathcal{C}_{T}\), \(\mathcal{C}_{R}\) and \(\mathcal{C}_{P}\), which yields the tessellation in Figure 9d, is rotated 5 times. The resulting coordinates from the rotation are contained in another set \(\mathcal{C}_{F}\). In Figure 10, a set \(\mathcal{C}\) of the entire cell centre coordinates is therefore obtained as \(\mathcal{C}=\mathcal{C}_{T}\cup\mathcal{C}_{R}\cup\mathcal{C}_{P}\cup\mathcal{ C}_{F}\), and the cell footprint is shown. The resulting total number of cells \(N_{c}=|\mathcal{C}|\).