## 4 Results for Reflectarray Synthesis  
Figure 3a illustrates the outline of coverage for South America and Florida. Figure 3b shows the minimum gain requirements within the defined coverage area [5]. Figure 3: (**a**) Outline of coverage of South America and Florida. (**b**) Minimum gain requirements in the defined coverage [5]. Figure 2: Magnitude of the \(x\) component of the incident electric field (dB) normalized relative to the maximum illumination obtained with (**a**) \(q=50\), (**b**) \(q=40\), (**c**) \(q=25\), and (**d**) \(q=15\). The edge illumination level are marked with red boxes. Two primary gain regions were required: 25.5 dBi (within the solid line contour) and 22.5 dBi (within the area bounded by the dashed and solid line contours). Additionally, a transition gain region between coverage and non-coverage areas was set to 14 dBi. Therefore, the minimum required gain was 25.5 dBi, 22.5 dBi, and 14 dBi for each respective region. In the dark blue region, a maximum gain of 0 dBi was required. All of these specifications must be met at an operating frequency of 13.5 GHz [5]. To meet these requirements, the following geometry properties and illumination proposed in [5] were considered. The reflectarray consisted of an elliptical flat panel with axes of \(832\times 806\) mm\({}^{2}\). The total reflectarray cell count was 3124, and they were arranged in a rectangular grid with \(64\times 62\) cells, each of which were \(13\times 13\) mm\({}^{2}\) in size. The center phase of the feeder antenna was located at \((-325,0,101)\) mm, referring to the reflectarray coordinate system (see Figure 1). The feeder antenna was designed to generate \(-10\) dB level illumination at the edge of the antenna aperture, referring to the peak illumination (located at the center of the reflectarray). To compute the tangential components of the incident electric field, the radiated electric field of the feeder antenna was modeled using the \(\cos^{q}(\theta)\) model given in Equation (19), with \(q_{\rm E}=q_{\rm H}=q=15\) at 13.5 GHz. We implemented the four algorithms used in [9] (open access), starting with the minimum required gain as the initial point and an algorithm tolerance \(\epsilon=0.1\). These algorithms were as referred to in [9]. Firstly, [PERSON]'s algorithm was Algorithm 1, which consists of applying successive orthogonal projections until convergence is reached. In the case of a non-empty intersection, convergence to the intersection point is only guaranteed by [PERSON]'s algorithm if the involved sets are convex. According to [9], the involved sets \(\mathcal{R}\) and \(\mathcal{M}\) are generally non-convex. However, this algorithm was commonly used in previous works [3; 4; 5; 6; 7; 8], and it is interesting to use as a reference algorithm. The separating hyperplanes algorithm was referred to as Algorithm 2, and it consists of building straight lines which contain the average points between successive projections when [PERSON]'s algorithm is stuck in a trap point. Thus, a sequence of intersection points between the straight lines and the \(\mathcal{M}\) set was created. It was expected that, on the one hand, the sequence would leave the trap point and, on the other hand, this sequence would converge to the intersection point. The decomposition method in polar cones was referred to as Algorithm 3, and it consists of building a straight line which contains the average points between successive projections when the [PERSON] algorithm is stuck in a trap point (similar to the previous algorithm), but in this case, projections in the polar cones are carried out. It was expected that the sequence generated by this algorithm would leave the trap point (Note that convergence to the intersection point was not expected for this algorithm). Finally, the hybridized algorithm combined Algorithm 3 and Algorithm 2. When the von Neumann algorithm is stuck in a trap point, during the first iterations, Algorithm 3 works to leave the trap point.