## 2 Heliostat Orientation
### Radial Layout  
The methods shown above for azimuth-elevation and tilt-roll models assume that for each heliostat, the y-axis points north and the x-axis points east, so heliostats are initially oriented in a matrix-aligned format (see Figure 5).  
Figure 4: (a) Azimuth-elevation and (b) tilt-roll mechanical models to orient a target-aligned heliostat, showing fixed quantities (vector target \(\overset{\rightarrow}{t}\), vector sun \(\overset{\rightarrow}{s}\), offsets 01, 02) and variable quantities (vector \(\overset{\rightarrow}{c}\), angles \(\alpha\), \(\beta\) around axes a, b). Axis z points to the target and axis x points westerly.  
If heliostats are horizontally aligned with the target and oriented in a radial-aligned format, the new azimuth-elevation model (here named AE/TA) and the new tilt-roll model (here named TR/TA) can be handled by combining the iterative techniques described above for the target-aligned model (the new azimuth-elevation model can be handled by simply changing the primary axis rotation to accommodate the radial angle). If vector \(\overset{\rightarrow}{c}\) points to the heliostat center and vector \(\overset{\rightarrow}{t}\) points to the target, the following steps are taken:  
(1) Build the target referential as follows (vector y points from the target to the heliostat):  
\[\overset{\rightarrow}{\mathbf{y}}=\overset{\rightarrow}{c}\ -\ \overset{ \rightarrow}{t}\ \ \Rightarrow\ \ \mathbf{y}[2]=0\ \ \Rightarrow\ \overset{\rightarrow}{\mathbf{y}}=\overset{ \rightarrow}{\mathbf{y}} \tag{10}\]  
\[\overset{\rightarrow}{\mathbf{z}}=(0,0,1)\ \ \Rightarrow\ \overset{ \rightarrow}{\mathbf{x}}=\overset{\rightarrow}{\mathbf{y}}\times\overset{ \rightarrow}{\mathbf{z}} \tag{11}\]  
(2) Convert the sun vector (plus target, center vectors) to the target referential, using Equation (7) introduced above.  
(3) Repeat steps 1, 2, and 3 of the azimuth-elevation model (AE/TA) or steps 1, 2, and 3 of the tilt-roll model (TR/TA) until convergence is reached for the \(\alpha\) and \(\beta\) angles. In the second case, in the newly aligned referential, the model works exactly like the tilt-roll model in the Earth referential: first, rotate the secondary axis (rotating along the projected heliostat center-target axis), and then rotate the primary axis (rotating perpendicular to the projected heliostat center-target axis).  
(4) Convert the normal and center vectors back to the Earth referential, using Equations (8) and (9) introduced above.  
For all six mechanical models discussed here (AE, TR, TA/AE, TA/TR, AE/TA, and TR/TA), for typical offsets of o1 (30 cm) and o2 (20 cm), fewer than 10 iterations are usually enough to achieve convergence, even for angle tolerances as strict as \(10^{-10}\) rad (the number of iterations increases with offset). Finally, it should be emphasized that the radiation flux distribution collected at the target, resulting from the heliostat mirror reflection, is equal for these six mechanical models when properly applied (changes due to different mirror offsets, o1 and o2, are usually negligible).  
Figure 5: Heliostat fields: (**a**) matrix-aligned; (**b**) radial-aligned. The target (T) points north (N), to collect the sun’s (S) rays reflected by the heliostats.