## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Mathematical and Practical Proof of Feasibility of the Method
However, \(F_{DC}\) is derived from R\({}_{ULO}\), which is only related to the luminaire's luminous intensity specifications. Then, it is independent of the observer's position in the room. Based on changing parts of \(F_{UWID}\) as the observer's position changes, its range of change will be investigated as follows. \(\Phi_{ZL}\) as an important part of \(F_{UWID}\) is expressed as:  
\[\frac{0.25}{L_{b_{a}}}\cdot\left(\frac{L_{1_{a}}^{2}\omega_{1_{a}}}{p_{1_{a}}^ {2}}+\frac{L_{2_{a}}^{2}\omega_{2_{a}}}{p_{2_{a}}^{2}}+\frac{L_{3_{a}}^{2} \omega_{3_{a}}}{p_{3_{a}}^{2}}+\cdots\right)=\frac{0.25}{L_{b_{b}}}\cdot\left( \frac{L_{1_{b}}^{2}\omega_{1_{b}}}{p_{1_{b}}^{2}}+\frac{L_{2_{b}}^{2}\omega_{ 2_{b}}}{p_{2_{b}}^{2}}+\frac{L_{3_{b}}^{2}\omega_{3_{b}}}{p_{3_{b}}^{2}}+\cdots\right) \tag{6}\]6  
\(\Phi_{ZL1}\), \(\Phi_{ZL2}\), \(\Phi_{ZL3}\), and \(\Phi_{ZL4}\) are only related to each luminaire's luminous flux properties and are not related to the observer's position or the room's dimensions. Then, they are independent of the observer's position in the room. By rewriting \(F_{UWID}\) formula (Equation (17)) considering the change in the observer's position:  
\[ \begin{array}{ll}F_{UWID}(\text{Y}_{0}+\Delta\text{Y})&=F_{DF}(\text{Y}_{0 }+\Delta\text{Y})F_{T,FW}(\text{Y}_{0}+\Delta\text{Y})\\ &+F_{DW}(\text{Y}_{0}+\Delta\text{Y})F_{T,WW-1}(\text{Y}_{0}+\Delta\text{Y}) \\ &+F_{DC}(\text{Y}_{0}+\Delta\text{Y})F_{T,CW}(\text{Y}_{0}+\Delta\text{Y}) \end{array}  \tag{22}\]  
Following that, for finding the changing range for \(F_{UWID}\), the changing range for the right side of Equation (22) will be investigated in detail. For \(F_{DF}\), on the right side of Equation (22), it can be written:  
\[ \begin{split}& F_{DF}(Y_{0}~{}+~{}\Delta Y)=\frac{\Phi_{Z1}(Y_{0}+ \Delta Y)}{\Phi_{0}}\\ &=\frac{(\Phi_{ZZ1}F_{G11}(Y_{0}+\Delta Y)+\Phi_{ZZ3}F_{G12}(Y_{0}+ \Delta Y)+\Phi_{ZZ3}F_{G13}(Y_{0}+\Delta Y)+\Phi_{ZZ4}F_{G14}(Y_{0}+\Delta Y))}{ \Phi_{0}}\end{split}  \tag{23}\]  
As previously mentioned, in Equation (23), \(\Phi_{ZLn}\)s are related to the luminaires' specifications and do not change when the room's specifications change. Geometric factors (\(F_{GLn}\)s) are related to the room's dimensions and ranges of their changing with room dimensions are illustrated in Table 1.  
From Table 1, it can be understood that when a room's Y dimension increases, \(F_{GL1}\) and \(F_{GL2}\) decrease and \(F_{GL3}\) and \(F_{GL4}\) increase. From each row of Table 1 and considering Equation (21), it can be deduced that for luminaires whose \(\Phi_{ZLn}\) values are close to each other, the increasing and decreasing of related geometric factors can cancel each other and CIE zonal flux (\(\Phi_{ZL}\)) can change little when a room's dimensions change. Knowing that, and also since in Equation (18), \(F_{DF}\) is CIE zonal flux (\(\Phi_{ZL}\)) divided by 1000 lm, \(F_{DF}\) change based on changes in room dimensions is not prominent. From Equation (19) and keeping in mind that R\({}_{DLO}\) is not dependent on room dimensions:  
\[F_{DW}=\text{R}_{DLO}-F_{DF}\Rightarrow\Delta_{Y}F_{DW}=\Delta_{Y}\text{R}_{ DLO}-\Delta_{Y}F_{DF}\overset{\Delta_{Y}\text{R}_{DLO}=0}{\Rightarrow}\Delta_{Y}F_{DW}=- \Delta_{Y}F_{DF} \tag{24}\]  
Consequently, \(F_{DW}\) change which is equal to \(F_{DF}\) change but with a negative sign is not also prominent. Since luminaires' R\({}_{ULO}\) value is usually negligible, the \(F_{DC}\) also can be assumed to be a negligible value. Therefore, \(F_{DF}\) and \(F_{DW}\) are the only distribution factors influencing the change of the indirect utilization factor of the walls. Table 2 illustrates a range of transfer factors' change according to the change in a room's dimensions. Validity of using data illustrated in Tables 1 and 2 requires maintaining a trend of change for geometric and transfer factors linearly, both inside and outside of the studied range (X = 2H, 4H, 8H, and 12H, Y = 4H, 6H, and 8H).