## 7 Results and Discussion
### Mechanical Rotation Angles  
To understand how the two \(\alpha\) and \(\beta\) heliostat rotation angles change throughout the day, they were plotted in Figures 10-13 as a function of time for the SW, S, SE, NW, N, and NE representative heliostats, using the two most important mechanical models: azimuth-elevation and tilt-roll. The heliostats were simulated under the most favorable conditions: the longest day (20 June) at the lowest latitude (Protaras) for rectangular heliostats with a 10\({}^{\circ}\) sloped ground, to minimize shading and blocking effects, as reported previously [17]. The equivalent results for Julich are relatively similar, so they are not shown here. For the shortest day (21 December), the curves also present the same general trends, although the Sun's trajectories are much lower, causing the angles controlling altitude (angle \(\beta\) in the AE models, angle \(\alpha\) in the TR models) to be about 30\({}^{\circ}\) higher to compensate (the heliostat mirrors are more vertical). Figure 10: Primary axis rotation, \(\alpha\) (in degrees), for the 6 representative heliostats (SW, S, SE, NW, N, and NE), with an azimuth-elevation model, calculated on 20 June at Protaras, from 08:00 to 16:00 LCT. Figure 11: Secondary axis rotation, \(\beta\) (in degrees), for the 6 representative heliostats (SW, S, SE, NW, N, and NE), with an azimuth-elevation model, calculated on 20 June at Protaras, from 8:00 to 16:00 LCT. Figure 12: Primary axis rotation, \(\alpha\) (in degrees), for the 6 representative heliostats (SW, S, SE, NW, N, and NE), with a tilt-roll model, calculated on 20 June at Protaras, from 8:00 to 16:00 LCT. The heliostat curves represented in Figures 10-13 are surprisingly symmetrical for both mechanical models, particularly for the tilt-roll. The primary axis curves for the elevation-azimuth model (Figure 10) are similar to the secondary axis curves for the tilt-roll model (Figure 13), as both essentially control the sun's azimuth. Similarly, the secondary axis curves for the elevation-azimuth model (Figure 11) are quite similar to the primary axis curves for the tilt-roll model (Figure 12), as both essentially control the sun's altitude. However, even a qualitative analysis shows that the curves for the tilt-roll model are simpler, require fewer rotations, and a smaller angular range. This is evident when comparing the angular range in Figure 13 ([\(-50^{\circ}\), +50\({}^{\circ}\)]) with that in Figure 10 ([\(-70^{\circ}\), +70\({}^{\circ}\)]), and in Figure 12 ([25\({}^{\circ}\), 45\({}^{\circ}\)]) with that in Figure 11 ([25\({}^{\circ}\), 60\({}^{\circ}\)]). For each tilt-roll heliostat, the primary angle (controlling altitude, in Figure 12) is essentially constant during the day, while the secondary axis (controlling azimuth, in Figure 13) rotates essentially at a constant angular speed. This is the rationale behind the so-called polar-oriented single-axis heliostat. Close inspection of Figure 13 shows that this speed is about 7.5\({}^{\circ}\)/hour. This is the expected value, as the sun's angular speed changes approximately 360\({}^{\circ}\)/24 h = 15\({}^{\circ}\)/hour: when the input ray is fixed and the mirror rotates \(\delta\), the reflected ray rotates 2\(\delta\). Similarly, when the input ray rotates 2\(\delta\), for the reflected ray to remain fixed, the mirror must rotate \(\delta\). Actually, the sun's angular speed changes slightly throughout the year, as exemplified in Figure 9. For 20 June 2024, the angular speed is exactly 13.76\({}^{\circ}\)/hour [25], so the mirror's angular speed should be around 13.76\({}^{\circ}\)/2 = 6.88\({}^{\circ}\)/hour. Measuring, for example, the average slope for the two red curves in Figure 13 (S, N heliostats) gives approximately 7.32\({}^{\circ}\)/hour. These results show that for this model, the two rotation axes are almost decoupled, and the axis a remains almost fixed throughout most of the day.