## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Mathematical and Practical Proof of Feasibility of the Method
\begin{array}{rl}\overline{E_{WID}(Y_{0}+\Delta Y)}&=\overbrace{1000\bigg{(} \overline{F_{UWID}(Y_{0})\frac{N}{A_{W}}(Y_{0})}\bigg{)}}^{\overline{E_{WID}( Y_{0})}}\\ &+\overbrace{1000\bigg{(}-0.021\overline{\frac{N}{A_{W}}(Y_{0})}+0.125 \overline{F_{UWID}(Y_{0})}-0.0026\bigg{)}}^{\overline{E_{WID}(Y_{0})}}\\ \end{array}  \tag{45}\]  
Therefore, average change in \(E_{WID}\) for a \(\pm 4\) m change in observer position would be:  
\[\overline{\Delta E_{WID}(Y_{0})}=1000\bigg{(}-0.021\overline{\frac{N}{A_{W}} (Y_{0})}+0.125\overline{\cdot F_{UWID}(Y_{0})}-0.0026\bigg{)} \tag{46}\]  
From Equation (17), transfer factors \(F_{TFW}\), \(F_{T,WW-1}\), and \(F_{T,CW}\) and distribution factors \(F_{DF}\), \(F_{DW}\), and \(F_{DC}\) are used for obtaining the indirect utilization factor for walls (\(F_{UWID}\)). The average of transfer factors for usual calculations as mentioned in Table 5 in CIE 190:2010 are 0.135, 0.251, and 0.371, for \(F_{TFW}\), \(F_{T,WW-1}\), and \(F_{T,CW}\), respectively. \(F_{DF}\) and \(F_{DW}\) are  
Figure 5: \(\frac{N}{A_{W}}\) as a function of X and Y (room dimensions). smaller than one and \(F_{DC}\) for luminaires with their luminous intensity distribution in lower hemisphere is approximately zero. Therefore:  
\[\overline{F_{UWID}(Y_{0})}=\overline{F_{DF}(Y_{0})}.\overline{F_{T,FW}(Y_{0})}+ \left(\overline{R_{DLO}}-\overline{F_{DF}(Y_{0})}\right).\overline{F_{T,WW-1}(Y _{0})} \tag{47}\]  
or  
\[\overline{F_{UWID}(Y_{0})}=\overline{F_{DF}(Y_{0})}.\left(-0.116\right)+ \overline{R_{DLO}}.0.251 \tag{48}\]  
Since, as shown in Equation (11), \(\overline{E_{WIDTH}(Y_{0})}\) is proportional to background luminance average, from Equation (46) it is known that for finding average change in background luminance, \(\overline{F_{UWID}(Y_{0})}\) and \(\overline{\frac{N}{A_{W}}(Y_{0})}\) are needed. Also, as mentioned before, \(R_{DLO}\) is smaller than one and an average value of 0.5 can be considered for it. Therefore, from Equation (39) and average value of \(R_{DLO}\), in Equation (39) \(\overline{F_{DF}}\) would be equal to 0.225. Hence, \(\overline{F_{UWID}(Y_{0})}\) in Equation (48) would be equal to 0.099. Then, \(\overline{\Delta E_{WID}(Y_{0})}\), based on average values of \(N/A_{W}\) and \(F_{UWID}\) in the studied range, would be:  
\[\overline{\Delta E_{WID}(Y_{0})}=1000(-0.021\times 0.375+0.125\times 0.099-0.0026 )=1.9 \tag{49}\]  
Ultimately, the ratio of background luminance change to its absolute value is:  
\[ \begin{split}\frac{\overline{\Delta L_{b}}}{\overline{L_{b}}}= \frac{\frac{\overline{\Delta E_{WID}(Y_{0})}}{\overline{\frac{E_{WID}(Y_{0} )}{\overline{\pi}}}}}{\overline{\frac{1}{1000}\left(\overline{F_{UWID}(Y_{0} )}\right)}}&=\frac{\overline{\Delta E_{WID}(Y_{0})}}{\overline {E_{WID}(Y_{0})}}\\ &=\frac{1000\left(-0.07\frac{N}{A_{W}}(Y_{0})}+0.25\overline{E_{ UWID}(Y_{0})}-0.0175\right)}{1000\left(\overline{F_{UWID}(Y_{0})}\frac{N}{A_{W}}(Y_{0}) \right)}=\frac{1.9}{51.76}\\ &=3.67\%\end{split}  \tag{50}\]  
Therefore, it is concluded that for an observer position change (or room dimension change) from Y = 16 m to 8 m, background luminance change average is 3.67% for average values of \(R_{DLO}\) = 0.5 and \(F_{DF}\) = 0.225. This value for background luminance change can lead to UGR change as:  
\[ \begin{split}\overline{\Delta UGR}=\overline{UGR_{a}}-& \overline{UGR_{b}}\\ &=8\log\left[\frac{0.25}{\frac{1}{L_{b_{a}}}}\sum\limits_{i=1}^{n }\frac{\overline{\left(\frac{i}{L_{i_{b}}}\right)^{2}\overline{\omega_{i_{a}}} }}{\overline{\left(p_{i_{a}}\right)^{2}}}\right]-8\log\left[\frac{0.25}{L_{b_{ b}}}\sum\limits_{i=1}^{n}\frac{\overline{\left(\frac{i}{L_{i_{b}}}\right)^{2} \overline{\omega_{i_{a}}}}}{\overline{\left(p_{i_{b}}\right)^{2}}}\right]\\ &=8\log\left[\frac{\frac{0.25}{L_{b_{a}}}\sum\limits_{i=1}^{n} \frac{\overline{\left(\frac{i}{L_{i_{b}}}\right)^{2}\overline{\omega_{i_{a}}} }}{\overline{\left(p_{i_{b}}\right)^{2}}}}{\overline{\left(p_{i_{b}}\right)^{ 2}}}\right]\end{split}  \tag{51}\]  
By applying Equations (7)-(9) to Equation (51):  
\[