## 7 Results and Discussion
### Mechanical Rotation Angles
In both models, the westerly heliostats (SW, NW) require higher angles than the easterly heliostats (SE, NE) because the sun's altitude is always lower on the westerly side at 16:00 LCT than on the easterly side at 8:00 LCT, prompting a larger effort from the SW and NW heliostats to properly align. This occurs because both Julich (longitude = +6.3878\({}^{\circ}\)) and  
Figure 13: Secondary axis rotation, \(\beta\) (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. Protaras (longitude = +34.0583\({}^{\circ}\)) are located on the eastern side of the corresponding time zone (see [25] for details). As expected, in both models (Figures 11 and 12), the southern heliostats (SW, S, SE), which are closer to the target, require smaller rotations (smaller altitude-controlling angles) to align the altitude than the northern ones (NW, N, NE). Table 1 reports the most relevant data for the six mechanical models (see Figures 4 and 5) studied in this work: (1) azimuth-elevation matrix-aligned AE; (2) azimuth-elevation radial-aligned AE/TA; (3) tilt-roll matrix-aligned TR; (4) tilt-roll radial-aligned TR/TA; (5) target-aligned with azimuth-elevation rotations TA/AE; and (6) target-aligned with tilt-roll rotations TA/TR. The table includes minimum and maximum rotation angles and total yearly rotations, averaged for the six representative heliostats studied (SW, S, SE, NW, N, and NE), for both the primary (\(\alpha\)) and secondary (\(\beta\)) rotation axes for Julich and Protaras. According to these numbers, the three tilt-roll models (TR, TR/TA, and TA/TR) require less total rotation and smaller rotation ranges than the azimuth-elevation models (AE, AE/TA, and TA/AE). This is essentially due to the rotation axis controlling azimuth (the \(\alpha\) primary angle in AE models and the \(\beta\) secondary angle in TR models). The most striking difference is between the two target-aligned models, TA/AE and TA/TR, which rank respectively last and first in this mechanical rotation ranking. While TA/AE requires the whole \([-180^{\circ},+180^{\circ}]\) range to orient the heliostat, TA/TR requires only a much smaller \([-47^{\circ},+52^{\circ}]\) range. It should be mentioned that rotations are actually allowed to proceed beyond the fixed angle limits \([-180^{\circ},+180^{\circ}]\), so for example, if the angle goes from +179\({}^{\circ}\) to +181\({}^{\circ}\), the final angle is registered as \(-179^{\circ}\), but the actual rotation, which is added to the total rotation, is recorded as only 2\({}^{\circ}\). As a consequence, TA/TR requires about 6-7 times fewer yearly rotations than TA/AE. This is mainly due to the front row heliostats (SW, S, SE), which are closer to the target and require significant rotations in winter to properly orient in the TA/AE model. TA models are seldom used due to their enormous installation and maintenance disadvantages, but these results seem to suggest that TA/TR is a very efficient mechanical rotation model, requiring much fewer yearly rotations than TR, the second-best model in this study. The ratio between the total rotations (\(\alpha+\beta\)) needed by TR and TA/TR models amounts to 1.54 in Julich and 1.41 in Protaras, suggesting that TA/TR would be particularly efficient in more adverse light conditions, as observed in Julich.