## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Mathematical and Practical Proof of Feasibility of the Method
\begin{split}\overline{\Delta UGR}&=8\log\left[ \frac{\overline{L_{b_{b}}}}{\overline{L_{b_{a}}}}\right]=8\log\left[\frac{ \overline{\frac{E_{WIDTH}(Y_{0})+\overline{\Delta E_{WID}(Y_{0})}}{\overline {\pi}}}}{\overline{\frac{E_{WIDTH}(Y_{0})}{\overline{\pi}}}}\right]\\ &=8\log\left[\frac{\overline{E_{WIDTH}(Y_{0})+\overline{\Delta E _{WID}(Y_{0})}}}{\overline{E_{WIDTH}(Y_{0})}}\right]=8\log\left(\frac{51.76-1 8}{51.76}\right)=0.13\end{split}  \tag{52}\]  
Therefore, on average, while an observer moves from a distance of 16 m from the opposite-side wall toward it, at distance of 8 m, only an approximate change in UGR of 0.13 units occurs for them. This change is far from the 3 UGR units of change which is perceptible according to the [PERSON] criterion. Remembering that previously it was shown that geometric and transfer factors have linear correlation with dimension change, the obtained result can be generalized and extrapolated for dimensions greater or smaller than values studied in this proof. For evaluating the difference between the background luminance in the main room and the corresponding smaller room in another manner, the effect of a change in observer position on the background luminance for rooms with widths of 24 m, 16 m, and 4 m, and lengths of 24 m, 4 m, and 2 m (\(\{\)X, Y \(\mid\) (X = 12, 8, 2) & (Y = 12, 2, 1) \(\}\)) is investigated both by manual calculation and simulation. The simulation has been performed by DIALux software. The mentioned maximum and minimum dimensions for X and Y are approximately the minimum and maximum values which are usually used in issued tables for UGR calculation. X changing is related to the width of the observer's path of movement. Small and large values of X correspond to narrow and wide pathways, respectively. This can give a view of the effect of the distance between side walls on the indirect light reflectance of the wall surfaces. Y value is the distance between the observer and the opposite-side wall. Therefore, this comprehensive study on usual dimensions in UGR tables can provide a good view of the range of difference between background luminance in the main room and the smaller rooms corresponding to the observer's position. For mentioned Xs and Ys, distribution factors (\(F_{DF}\), \(F_{DW}\), and \(F_{DC}\)), and consequently the indirect utilization factor for walls from Equations (17)-(21), can be obtained. The obtained distribution factors are illustrated in Supplementary Table S4 for the luminaire used for comparing background luminance in main and corresponding smaller rooms. The next step in obtaining background luminance based on Equations (11) and (12) is to find \(N/A_{W}\). According to Equations (13) and (14) for the main room:  
\[\frac{N}{A_{W}}=\frac{\mathrm{XY_{L}}}{8(\mathrm{X+Y_{L}})} \tag{53}\]  
For X and Y = 1, 2, 12, based on Equations (15) and (16) for the corresponding smaller room:  
\[\frac{N}{A_{W}}=\frac{\mathrm{X(Y_{0}+\Delta Y)}}{8(\mathrm{X+(Y_{0}+\Delta Y )})} \tag{54}\]  
Therefore, for X = 12, and Y0 + \(\Delta Y\) = 1, 2, 12:  
\[\frac{N}{A_{W}}(1)=\frac{12(1)}{8(12+(1))}\approx 0.115 \tag{55}\]  
\[\frac{N}{A_{W}}(2)=\frac{12(2)}{8(12+(2))}\approx 0.214 \tag{56}\]  
\[\frac{N}{A_{W}}(12)=\frac{12(12)}{8(12+(12))}=0.75 \tag{57}\]  
And, for X = 8, and Y0 + \(\Delta Y\) = 1, 2, 12:  
\[\frac{N}{A_{W}}(1)=\frac{8(1)}{8(8+(1))}\approx 0.111 \tag{58}\]  
\[\frac{N}{A_{W}}(2)=\frac{8(2)}{8(8+(2))}\approx 0.200 \tag{59}\]  
\[\frac{N}{A_{W}}(12)=\frac{8(12)}{8(8+(12))}=0.600 \tag{60}\]  
Also, for X = 2, and Y0 + \(\Delta Y\) = 1, 2, 12:  
\[\frac{N}{A_{W}}(1)=\frac{2(1)}{8(2+(1))}\approx 0.083 \tag{61}\]  
\[\frac{N}{A_{W}}(2)=\frac{2(2)}{8(2+(2))}\approx 0.125 \tag{62}\]\[\frac{N}{A_{W}}=\frac{\mathrm{X(Y_{0}+\Delta Y)}}{8(\mathrm{X+(Y_{0}+\Delta Y )})} \tag{54}\]0  
Then, having distribution and transfer factors, \(N/A_{W}\), and \(\Phi_{0}=1000\) lm, values for \(E_{WID}\) and in turn, \(L_{b}\) (background luminance) for \(\{X,Y\mid(X=12,\ 8,\ 2)\ \&\ (Y=12,\ 2,\ 1)\}\) can be obtained.