## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Mathematical and Practical Proof of Feasibility of the Method  
As explained previously, in the proposed method, UGR table numbers or their 2D interpolation or extrapolation are used as the UGR for the observer's position on the room's midline. Therefore, it is assumed that UGR for an observer at any point on a room's midline is equal to UGR for an observer in a room with size equal to the space in front of the first observer when the second observer is placed in the middle of one wall looking toward the opposite-side wall in a smaller room. For proving the validity of idea, it should be proved that UGRs obtained for both observers are adequately close to each other. To start the proof, it is necessary to refer to Equation (1), the well-known CIE 117:1995 formula for UGR calculation [8]. The formula takes into account all but one of the variables which are directly related to the observer's field of view. Therefore, except for one variable, all variables--\(L_{i}\) as the luminance of the luminaire i, \(\omega_{i}\) as the solid angle of the luminaire \(i\), and \(p_{i}\) as the Guth position index of the luminaire \(i\)--are directly related to the luminaires which are in the observer's sight. The only parameter which considers the effect of luminaires out of the observer's sight is \(L_{b}\), the background luminance. Therefore, for two mentioned observers in two different rooms, it is enough to prove that the background luminances are close to each other such that when using them in Equation (1), the resulted UGRs are reasonably close to each other. Consider two mentioned observers. One is in a room in a position which places some of the luminaires in front of them (in their field of view), and some of the luminaires behind them (Figure 4a). The other observer exists in a room containing only the luminaires which are in the first observer's field of view. The nearest wall behind the second observer is at a distance of H/2 from the first row of the luminaires in front of them (Figure 4b). The arrangement of the luminaires is as the standard arrangement explained in CIE 117:1995 [8]. Figure 3: Illustration of the part of the main room (in grey color) which is used as a new room with different dimensions for UGR calculation in the proposed method. For UGR calculation in the two rooms illustrated in Figure 4a,b, Equation (1) can be stated as:  
\[UGR_{main}\ room\ (for\ observer\ position)\] \[=8\log\left[\frac{0.25}{L_{b_{0}}}.\left(\frac{L_{1a}^{2}\omega_{1a}}{P_{1 a}^{2}}+\frac{L_{2a}^{2}\omega_{2a}}{P_{2a}^{2}}+\frac{L_{3a}^{2}\omega_{3a}}{P_{3 a}^{2}}+\cdots\right)\right] \tag{2}\]  
\[UGR_{corresponding}\ smaller\ room(Observer\ located\ in\ the\ middle\ of\ one\ wall)\] \[=8\log\left[\frac{0.25}{L_{b_{0}}}.\left(\frac{L_{1b}^{2}\omega_{1b}}{P_ {1b}^{2}}+\frac{L_{2b}^{2}\omega_{2b}}{P_{2b}^{2}}+\frac{L_{3b}^{2}\omega_{3b}} {P_{3b}^{2}}+\cdots\right)\right] \tag{3}\]  
where the parameters with indices a and b correspond to rooms a and b as illustrated in Figure 4a,b. Figure 4: (**a**) Observer in a position in a room in which some of the luminaires are in front of them (in their field of view), (**b**) observer in a room which contains only the luminaires that are in the first observer’s field of view. It should be proved that:  
\[ \begin{split} UGR_{main\ room}\ (for\ current\ observer^{\prime}s\ position)\\ = UGR_{correspondand\ smaller\ room}(Observer\ is located\ in\ the\ middle\ of\ one\ wall)\end{split}