## 2 Tabular Method for Reducing UGR Calculations, for Every Point on the Room Midline
### Case Study and Illustration of Implementation
#### 2.4.2 Luminaires Equipped with Lenses
As for LED luminaires with diffusers, the tabular method proposed in this research is also investigated for UGR estimation for LED luminaires with lenses. However, previously in Section 2.3, it was anticipated that the proposed method likely does not have enough efficacy for luminaires with non-uniform luminous intensity distribution. An example of a UGR table (luminaire with lens type A's UGR table) is illustrated in Table S23. Similar to studying luminaires with diffusers, by accurately investigating the curves in Figure 10a,b along with UGR tables published by the manufacturers for eight types of LED luminaires with lenses, a relation between them can be understood. By examining eight luminaires A to H and using two sets of the table data for X = 2H and 4H and Y = 2H, 3H, 4H, and 6H (in the red border in Table S23, for example, and illustrated in Tables S24-S31) for 2D interpolation or extrapolation, the similarities between the proposed method and simulation results for the mentioned luminaires are checked (Tables S32-S39). X dimensions (2H and 4H) are selected since the room's X dimension 5H is near these two values and the range of Y from 2H, 3H, 4H, and 6H corresponds to 4 m, 6 m, 8 m, and 12 m, respectively, which properly covers an observer's distance from 10 to 0 m from the opposite-side wall in the room. Tables S32-S39 show data obtained from 2D interpolation or extrapolation of table data and simulation data for X = 10 \(m\) and Y = \(\{y|\ y=n\times 0.5^{m}\,\ n\in\mathbb{Z}\ \&\ 0\leq n\leq 19\ \}\) for each luminaire with lenses. Data used in the UGR table is for X = (2, 4) H and Y= (2, 3, 4, 6) H. The deviation of 2D-interpolated or extrapolated data average from the simulated UGR average, for luminaires A to H, is illustrated in Table 6 as a percentage of 30 units of UGR. As mentioned before, according to Hopkinson criteria, below 10% deviation of full-scale UGR changing (0 to 30) is acceptable. Therefore, the proposed method results are not in the reliable range. Also, from Figures S9-S16, it is obvious that UGR obtained by 2D-interpolated or extrapolated table data does not properly follow the simulated UGR. In addition, for further study and finding any possibility for adapting the proposed tabular method for LED luminaires with lens (with non-uniform luminous intensity distribution), difference between UGR obtained by tabular method and simulation, according to supplementary Tables S32-S39 are illustrated in Table 7 for distances from opposite-side wall equal to 4, 6, and 8 m separately. For mentioned distances from opposite-side wall, corresponding UGRs exist in UGR table and no interpolation or extrapolation in y axis is needed. From Table 7, it can be deduced that even at points that the UGR table directly gives UGR, there are great differences (more than 10% Hopkinson criterion as perceivable UGR change) between the proposed method and simulation results. Consequently, since the proposed tabular method is based on a UGR standard table which is constructed according to the standard UGR calculation method [4], and the result of the proposed method even for points the UGR table directly gives, the UGR value has great difference with results from simulation based on the standard UGR calculation formula. Hence, the UGR calculation method applicability for LED luminaires with lenses could be a unclear as stated in former research (e.g., [4,7]). Therefore, further research and efforts to enhance the usual method are needed to find ways for accurate UGR calculation for luminaires with non-uniform luminous intensity distribution, including LED luminaires equipped with lenses. Consequently, as a UGR-calculation-reducing method in indoor spaces along room midlines for luminaires with lenses, the table data cannot be used. This corresponds exactly to what was anticipated for proposed method efficacy for LED luminaires with lenses in Section 2.3.