## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Mathematical and Practical Proof of Feasibility of the Method
Therefore, linear regression was applied to trends of mentioned factors for inside and outside of the studied range where data were available, and R\({}^{2}\)-values of regression for all factors for a common range are \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**X** & **Y** & \(\Delta_{Y}F_{GL1}\) & \(\Delta_{Y}F_{GL2}\) & \(\Delta_{Y}F_{GL3}\) & \(\Delta_{Y}F_{GL4}\) \\ \hline
**2** & **4–8** & \(-\)0.062 & \(-\)0.011 & 0.079 & 0.011 \\ \hline
**4** & **4–8** & \(-\)0.058 & \(-\)0.092 & 0.015 & 0.024 \\ \hline
**8** & **4–8** & \(-\)0.041 & \(-\)0.177 & 0.191 & 0.052 \\ \hline
**12** & **4–8** & \(-\)0.032 & \(-\)0.0209 & 0.199 & 0.071 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric factors changing when room dimensions change for X = 2\(\sim\)12 and Y = 4–8. (The negative numbers show decreasing and the positive numbers show increasing of the geometric factors with increasing Y). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**X** & **Y** & \(\Delta_{Y}F_{TZW}\) & \(\Delta_{Y}F_{Taww-1}\) & \(\Delta_{Y}F_{TCW}\) \\ \hline
**2** & **4–8** & \(-\)0.020 & \(-\)0.044 & \(-\)0.065 \\ \hline
**4** & **4–8** & \(-\)0.027 & \(-\)0.054 & \(-\)0.082 \\ \hline
**8** & **4–8** & \(-\)0.032 & \(-\)0.056 & \(-\)0.092 \\ \hline
**12** & **4–8** & \(-\)0.034 & \(-\)0.057 & \(-\)0.096 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transfer factors changing with the changes in a room’s dimensions for X = 2\(\sim\)12 and Y = 4–8. (All changes are in form of decreasing with increasing Y). illustrated in Tables S3 and S4. Based on data illustrated in Tables S2 and S3, the average of R\({}^{2}\)-value for linear regression of geometric factors is 0.8701 and for transfer factors is 0.9651. The mentioned R\({}^{2}\)-values show acceptable linear correlation between room dimension in the Y direction and geometric and transfer factors. It shows the possibility of generalization of the results obtained in the studied dimension range for outside the range. From Equation (22), the average change of the indirect utilization factor for walls based on the change of distribution and transfer factors can be stated as Equation (25):  
\[ \begin{array}{ll}\overline{F_{UWID}(\mathrm{Y_{0}}+\Delta\mathrm{Y})}&= \overline{F_{UWID}\left(\mathrm{Y_{0}}\right)}+\overline{\Delta F_{UWID} \left(\mathrm{Y_{0}}\right)}\\ &=\left(\overline{F_{DF}(\mathrm{Y_{0}})}+\overline{\Delta F_{DF}(\mathrm{Y_ {0}})}\right)\left(\overline{F_{T,FW}(\mathrm{Y_{0}})}+\overline{\Delta F_{T,FW }(\mathrm{Y_{0}})}\right)\\ &+\left(\overline{F_{DW}(\mathrm{Y_{0}})}+\overline{\Delta F_{DW}(\mathrm{Y_ {0}})}\right)\left(\overline{F_{T,WW-1}(\mathrm{Y_{0}})}+\overline{\Delta F_{ T,WW-1}(\mathrm{Y_{0}})}\right)\\ &+\left(\overline{F_{DC}(\mathrm{Y_{0}})}+\overline{\Delta F_{DC}(\mathrm{Y_ {0}})}\right)\left(\overline{F_{T,CW}(\mathrm{Y_{0}})}+\overline{\Delta F_{T, CW}(\mathrm{Y_{0}})}\right)\end{array}  \tag{25}\]  
From Equations (19), (20), and (25):  
\[ \begin{array}{ll}\overline{F_{UWID}(\mathrm{Y_{0}}+\Delta\mathrm{Y})}&= \left(\overline{F_{DF}(\mathrm{Y_{0}})}+\overline{\Delta F_{DF}(\mathrm{Y_ {0}})}\right)\left(\overline{F_{T,FW}(\mathrm{Y_{0}})}+\overline{\Delta F_{T,FW }(\mathrm{Y_{0}})}\right)+\left(\overline{\mathrm{R}_{DLO}}\right.\\ &-\overline{F_{DF}(\mathrm{Y_{0}})}-\overline{\Delta F_{DF}(\mathrm{Y_{0}})} \right)\left(\overline{F_{T,WW-1}(\mathrm{Y_{0}})}+\overline{\Delta F_{T,WW-1 }(\mathrm{Y_{0}})}\right)\\ &\left.+\left(\overline{R_{ULO}}+\overline{\Delta R_{ULO}}\right)\left( \overline{F_{T,CW}(\mathrm{Y_{0}})}+\overline{\Delta F_{T,CW}(\mathrm{Y_{0}}) }\right)\end{array}  \tag{26}\]  
Since, in many luminaires, luminous intensity in the upper hemisphere is near zero, R\({}_{ULO}\) also equals zero. Therefore, Equation (26) would be:  
\[