## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Mathematical and Practical Proof of Feasibility of the Method
The background luminances for mentioned Xs and Ys obtained by manual calculation and simulation for different types of luminaires are illustrated in Tables 3 and 4, respectively. In the mentioned tables, for each X as the observer's path width, deviation of background luminance for Y = 1 and 2 from background luminance at the start point of path (Y = 12) is also illustrated as a percentage of background luminance at the starting point. Due to what is mentioned in CIE 117:1995 about insensitivity of the UGR to the errors produced in \(L_{b}\) calculation (in which +33% error in \(L_{b}\) leads to only 1-unit error in UGR), by using data in Table 3 for manual calculation and Table 4 for simulation of \(L_{b}\), it can be deduced that starting from a distance of 24 m from the opposite-side wall, for the manual calculation example ending at a distance of at least 4 m and for the simulation ending at a distance of less than 2 m from the opposite-side wall, the UGR calculation error is limited to only 1 unit, which is far from the 3 units of UGR introduced in the Hopkins criterion for perceptible UGR difference. Also, for the previously obtained range of UGR change for \(4\leq X\leq 8\) and \(4\leq Y\leq 8\) for an average value of \(R_{DLO}\) equal to 0.5, the UGR change was obtained as 3.67%, which corresponds to 0.13 units of UGR error. Hence, in addition to \(L_{i}\), \(\omega_{i}\), and \(p_{i}\), which are totally independent of the luminaires and space out of an observer's sight, background luminance also has relatively small changes according to the number of luminaires and change in dimensions of the space both inside and outside the observer's sight. Therefore, in addition to Equations (7)-(9), it was proved that Equation (10) is also true with approximation and it can be assumed that UGR obtained for the main room and corresponding smaller room illustrated in Figure 4a,b are close enough and UGR table can be used for points on the room's midline. More practical samples for evaluating the feasibility of the method will be examined in Section 2.4. The tabular method was originally designed for UGR calculation in the middle of one wall in a room. When using it at a point inside a room on the room's midline other than wall midpoints, it should be shown that for a smaller room size corresponding to the space in front of the observer in a larger room when observer's location is other than the room's margins, by using Equation (1), the obtained UGR for smaller room is close enough to the real UGR for the observer in the larger room. Therefore, the only difference between the parameters of the main and corresponding rooms exists in background luminance. Then, \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**X = 12**} & \multicolumn{2}{c}{**X = 8**} & \multicolumn{2}{c}{**X = 2**} \\ \hline Y & **12** & **2** & **1** & **12** & **2** & **1** & **12** & **2** & **1** \\ \hline \(E_{WID}\) **(simulated)** & 36.3 & 32.9 & 29.3 & 29.7 & 32.3 & 29.0 & 32.0 & 30.1 & 27.1 \\ \(E_{WID}\) **Deviation from X = 12** & 0\% & \(-9\%\) & \(-19\%\) & 0\% & 9\% & \(-2\%\) & 0\% & \(-9\%\) & \(-18\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: An example of simulated background luminance for different room widths while the length also is changing. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**X = 12**} & \multicolumn{2}{c}{**X = 8**} & \multicolumn{2}{c}{**X = 2**} \\ \hline Y & **12** & **2** & **1** & **12** & **2** & **1** & **12** & **2** & **1** \\ \hline \(E_{WID}\) **(calculated manually)** & 25.5 & 28.38 & 12.65 & 31.2 & 21.4 & 13.21 & 28.46 & 26.00 & 19.59 \\ \(E_{WID}\) **Deviation from X = 12** & 0\% & 11\% & \(-50\%\) & 0\% & \(-31\%\) & \(-58\%\) & 0\% & \(-9\%\) & \(-31\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: An example of manually calculated background luminance for different room widths while the length also is changing.