## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Mathematical and Practical Proof of Feasibility of the Method
 \tag{4}\]  
Therefore, from Equations (2)-(4):  
\[8\log\left[\frac{0.25}{L_{b_{a}}}\cdot\left(\frac{L_{1_{a}}^{2}\omega_{1_{a} }}{p_{1_{a}}^{2}}+\frac{L_{2_{a}}^{2}\omega_{2_{a}}}{p_{2_{a}}^{2}}+\frac{L_{3 _{a}}^{2}\omega_{3_{a}}}{p_{3_{a}}^{2}}+\cdots\right)\right]=8\log\left[\frac {0.25}{L_{b_{b}}}\cdot\left(\frac{L_{1_{b}}^{2}\omega_{1_{b}}}{p_{1_{b}}^{2}}+ \frac{L_{2_{b}}^{2}\omega_{2_{b}}}{p_{2_{b}}^{2}}+\frac{L_{3_{b}}^{2}\omega_{3 _{b}}}{p_{3_{b}}^{2}}+\cdots\right)\right] \tag{5}\]  
Or:  
\[\frac{0.25}{L_{b_{a}}}\cdot\left(\frac{L_{1_{a}}^{2}\omega_{1_{a}}}{p_{1_{a}}^ {2}}+\frac{L_{2_{a}}^{2}\omega_{2_{a}}}{p_{2_{a}}^{2}}+\frac{L_{3_{a}}^{2} \omega_{3_{a}}}{p_{3_{a}}^{2}}+\cdots\right)=\frac{0.25}{L_{b_{b}}}\cdot\left( \frac{L_{1_{b}}^{2}\omega_{1_{b}}}{p_{1_{b}}^{2}}+\frac{L_{2_{b}}^{2}\omega_{ 2_{b}}}{p_{2_{b}}^{2}}+\frac{L_{3_{b}}^{2}\omega_{3_{b}}}{p_{3_{b}}^{2}}+\cdots\right) \tag{6}\]  
As a result, by proving:  
\[L_{i_{a}}=L_{i_{b}} \tag{7}\]  
\[w_{i_{a}}=w_{i_{b}} \tag{8}\]  
\[p_{i_{a}}=p_{i_{b}} \tag{9}\]  
\[L_{b_{a}}=L_{b_{b}} \tag{10}\]  
the equality in Equation (6) will be proved. As can be deduced from Figure 4a,b, since there are same luminaires in both observers' fields of view in the mentioned figures, Equations (7)-(9) are true. However, \(L_{b}\) in Equation (1) is the only variable which is not related only to the luminaires in the observer's field of view. Therefore, the aim is to prove that the values of \(L_{b}\) for both of the rooms and observers' positions are reasonably close to each other. For this aim, the range of changes in background luminance will be obtained based on observers' position changes; we also examine whether background luminance change can be ignored. In this research, the CIE 190:2010 [31] method for \(L_{b}\) calculation will be used:  
\[L_{b}=\frac{E_{WID}}{\pi} \tag{11}\]  
where, \(E_{WID}\) is an indirect component of the illuminance on the wall produced by luminaires. In turn, \(E_{WID}\) is:  
\[E_{WID}=\frac{F_{\text{{{LIWID}}}}\cdot N\cdot\Phi_{0}}{A_{W}} \tag{12}\]  
where \(F_{\text{{{LIWID}}}}\) is the indirect utilization factor for walls, \(N\) is the number of luminaires, \(A_{W}\) is the total area of walls (m\({}^{2}\)) (\(W\) stands for wall) between the reference plane and the luminaires' installation plane, and \(\Phi_{0}\) equals 1000 lm (reference luminous flux) when the luminaire's luminous intensity distribution is given in cd per 1000 lm. It should be noted that in case of giving luminous intensity distribution based on luminaire luminous flux, the luminous intensity distribution numbers should be divided by \(\Phi_{\text{{luminaire}}}/1000\) to change the luminous intensity unit to cd per 1000 lm, which makes the use of Equation (12) possible. As the observer moves toward the opposite-side wall in the main (first) room, the size of the smaller (second) room corresponding to the observer's position in the first room (In the second room, the observer's distance from the opposite-side wall is the same as the first observer, but the nearest wall behind the second observer is exactly H/2 distance from the first row of the luminaires in front of them) changes as illustrated in Figure 4a,b, and the length of the second room in the first observer's path decreases. Meanwhile, the number of luminaires in both observers' fields of view decreases.