# GPS-Induced Disparity Correction for Accurate Object Placement in Augmented Reality
## 3 Results
### Results of 3D Object AR Rendering App Implementation and Location Correction
From Equations (11) and (12), we calculate \(P^{\prime}\) and \(P^{\prime}_{e}\) using using these coordinates. Here, \(Z\) is determined using (13). By applying these transformations, we noted the differences between the projected positions with and without GPS errors as \(P^{\prime}-P^{\prime}_{e}=(-6.8521,-0.2200,0)\). This discrepancy is crucial for understanding spatial errors introduced by GPS inaccuracies. To address this issue, we calculated the translation vector \(T_{e}\), which represents the necessary adjustment to correct GPS-induced errors. Utilizing (17), we determined \(T_{e}\) to be \((4.4940,\,-5.6051,\,-0.0120)\). This calculation is predicted on the assumption that the rotation matrix \(R\) is invertible, allowing us to isolate \(T_{e}\) by computing \(R^{-1}\cdot(P^{\prime}-P^{\prime}_{e})\). This experimental validation underscores the practical application of our theoretical model, and demonstrates how GPS inaccuracies can be quantified and subsequently corrected. In particular, the calculation of \(T_{e}\) illustrates the projected position adjustment process for virtual objects in AR, ensuring their accurate alignment with real-world locations, despite the presence of GPS errors. This step is instrumental for object placement precision enhancement within digital twin environments, and contributes to the overall reliability and usability of AR technologies in various applications. An X-axis correction of 4.4940 reflects the distance difference in the east-west direction (longitude), while a Y-axis correction of \(-5.6051\) indicates the distance difference in the north-south direction (latitude). To calculate the mean translation vector using (19), we acquired ten points along the boundary surface and computed their average. This method allowed us to determine  
Figure 5: Positioning accuracy under location errors. The ‘x’ symbol captured by the red camera display the correct positioning for object ‘*’, indicating where the image should ideally appear. In contrast, the ‘+’ symbol captured by the green camera represent the misalignment of objects due to GPS inaccuracies, illustrating the discrepancy in actual versus expected object locations. the average corrections necessary for positioning errors. By averaging the individual translation vectors derived from these multiple observations, the mean translation vector was computed as \(T_{\text{mean}}=(4.7,-5.3)\). This mean translation vector indicates an average correction of 4.7 m in the east-west direction (longitude) and \(-5.3\) m in the north-south direction (latitude). This calculation is critical for refining the positional accuracy of virtual objects in the AR environment and ensuring that they align more precisely with their real-world counterparts. This effectively compensates for the aggregated GPS inaccuracies observed across different data points. Finally, as shown in Figure 6, the final results demonstrate that the objects segmented through building object division are displayed correctly from (a) to (d).