## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Mathematical and Practical Proof of Feasibility of the Method
 \tag{31}\]  
In other words:  
\[\overline{\Delta F_{UWID}(Y_{0})}=\overline{F_{DF}(Y_{0})}\Big{(}\overline{ \Delta F_{T,FW}(Y_{0})}-\overline{\Delta F_{T,WW-1}(Y_{0})}\Big{)}+\overline{ R_{DLO}}\cdot\overline{\Delta F_{T,WW-1}(Y_{0})} \tag{32}\]  
By considering Table 2, it can be deduced that:  
\[\overline{\Delta F_{T,FW}(Y_{0})}-\overline{\Delta F_{T,WW-1}(Y_{0})}\approx \ +\ 0.02 \tag{33}\]  
and  
\[\overline{\Delta F_{T,WW-1}(Y_{0})}=-0.05\ \ \Delta Y_{0}=4 \tag{34}\]  
From Equations (18) and (21) and by investigating luminaines with their luminous intensity distribution in the lower hemisphere around the luminaine, \(F_{\rm DF}\) can be expressed as follows:  
\[ \begin{array}{ll}\big{(}\overline{F_{GL1}}\approx 0.3,\ \overline{F_{GL2}}&\approx 0.1,\ \overline{F_{GL3}}\approx 0.3,\ \overline{F_{GL4}}\approx 0.1,\overline{\Phi_{ Z14}}=\overline{R_{DLO}}\ \big{)}\Rightarrow\overline{F_{DF}}\\ &=0.0003\Phi_{Z1L1}+0.0001\overline{\Phi}_{Z12}+0.0003\Phi_{Z13}+0.1\overline {R}_{DLO}\\ \end{array}  \tag{35}\]  
Usually in luminaines with uniform luminous intensity distribution in bottom hemisphere:  
\[\overline{\Phi_{Z1L}}<\overline{\Phi_{Z12}}<\overline{\Phi_{Z13}}<\overline{ \Phi_{Z14}}\Rightarrow\overline{\Phi_{Z1L}}<\overline{\Phi_{Z12}}<\overline{ \Phi_{Z13}}<\overline{R}_{DLO} \tag{36}\]  
Also, by estimation it can be assumed that Equation (35) for \(F_{\rm DF}\) calculation can be expressed as:  
\[ \begin{array}{ll}\overline{F_{DF}}\approx 0.3\big{(}\overline{\Phi_{Z11}}&=(a) \overline{R_{DLO}}\big{)}+0.1\big{(}\overline{\Phi_{Z12}}=(b)\overline{R_{DLO} }\big{)}+0.3\big{(}\overline{\Phi_{Z13}}=(c)\overline{R_{DLO}}\big{)}\\ &\quad+\big{(}\overline{\Phi_{Z14}}=0.1\overline{R}_{DLO}\big{)}=(0.3a+0.1b+0.3 c+0.1)\overline{R}_{DLO}\\ \end{array}  \tag{37}\]  
From Equation (36) it can be concluded that:  
\[a,b,c<1 \tag{38}\]  
Hence, from Equations (37) and (38):  
\[0.1\overline{R_{DLO}}<\overline{F_{DF}}<0.8\overline{R}_{DLO} \tag{39}\]  
Also, \(R_{DLO}\) and \(F_{DF}\) are numbers smaller than 1. Hence, from Equations (32)-(34), and (39):  
\[\overline{\Delta F_{UWID}(Y_{0})}=-0.021 \tag{40}\]  
According to Equation (12), the indirect component of the illuminance on the walls in terms of its change based on room dimension change would be:  
\[ \begin{array}{ll}\overline{E_{WID}(Y_{0}+\Delta Y)}&=\overline{E_{WID}(Y_ {0})}+\overline{\Delta E_{WID}(Y_{0})}\\ &=1000\Big{(}\overline{F_{UWID}(Y_{0})}+\overline{\Delta F_{UWID}(Y_{0})} \Big{)}\Big{(}\overline{\overline{A}_{W}}(Y_{0})+\overline{\Delta\overline{A }_{W}}(Y_{0})\Big{)}\\ &=1000\Big{(}\overline{F_{UWID}(Y_{0})}-0.021\Big{)}\Big{(}\overline{\overline{A }_{W}}(Y_{0})+\overline{\Delta\overline{A}_{W}}(Y_{0})\Big{)}\\ \end{array}  \tag{41}\]  
\(\frac{N}{A_{W}}(Y_{0})\) for a sample dimension changing range (8 m to 16 m) in both directions of X and Y can be seen in Figure 5.  
From Figure 5 it can be deduced that the average change of \(\frac{N}{A_{W}}\) in the considered dimension change range is 0.125. Consequently, Equation (41) would be:  
\[\overline{E_{WID}(Y_{0}+\Delta Y)}=1000\Big{(}\overline{F_{UWID}(Y_{0})}-0.021 \Big{)}\left(\overline{\frac{N}{A_{W}}(Y_{0})}+0.125\right) \tag{42}\]  
Therefore:  
\[\overline{E_{WID}(Y_{0}+\Delta Y)}=1000\Big{(}\overline{F_{UWID}(Y_{0}) \frac{N}{A_{W}}(Y_{0})}-0.021\overline{\frac{N}{A_{W}}(Y_{0})}+0.125\overline{ F_{UWID}(Y_{0})}-0.0026\Big{)} \tag{43}\]  
For finding the change of the indirect component of the illuminance on the walls according to the change in the room's dimensions:  
\[ \begin{array}{rl}\overline{E_{WID}(Y_{0}+\Delta Y)}&=\overline{E_{WID}(Y_ {0})}+\overline{\Delta E_{WID}(Y_{0})}\\ &=1000\Big{(}\overline{F_{UWID}(Y_{0})\frac{N}{A_{W}}(Y_{0})}-0.021\overline{ \frac{N}{A_{W}}(Y_{0})}\\ &+0.125\overline{F_{UWID}(Y_{0})}-0.0026\Big{)}\end{array}  \tag{44}\]  
The right side of Equation (44) can be rewritten as:  
\[