Reply to comment on “Dual-quaternion-based iterative algorithm of the three dimensional coordinate transformation”

Abstract Our recent paper (Zeng et al. in Earth Planets Sp 76:20, 2024) presented a Dual Quaternion Algorithm (DQA) of 3D coordinate transformation in weighted total least squares (WTLS). Zeng et al. (Earth Planets Sp 76:20, 2024) demonstrated two numerical experiments involving an actual geodetic datum transformation for small rotation angle case and a simulated big rotation angle case from surface fitting. In order to validate the DQA, the Quaternion Algorithm (QA) from Mercan et al. (2018) was compared in the experiment study. One conclusion of Zeng et al. (Earth Planets Sp 76:20, 2024) is that the DQA is advantageous to the QA partially because of the improvement of estimated precisions of transformation parameters. Comment on our paper (Bektaş in Earth Planets Sp, 2025a) did not denigrate our work, but disagreed the judgement that the DQA method is more precise. It is inferred from Bektaş (Earth Planets Sp, 2025a) that as the same adjustment method (WTLS) and the same data are used, the DQA and QA should give the same results including the estimated transformation parameters and their precisions. We agree with that viewpoint. More numerical tests based on centered coordinates and non-centered coordinates are carried out for the small rotation angle case and it is found that the estimated parameter precisions of DQA are consistent with those of Bektaş (Earth Planets Sp, 2025a) based on centered coordinates. Therefore corrections and clarity are made. Graphical Abstract