<p>The subject of the study is a geometric model of rigid-body motion within a virtual three-dimensional metrology laboratory, together with the related problem of evaluating measurement uncertainty. The relevance of the topic stems from the shift of measurement processes into digital environments, while existing approaches based on Euler-angle parametrisation remain prone to singularities and disregard error sources specific to virtual environments. The purpose of the work is to construct an integral geometric model on the special Euclidean Lie group and to extend the uncertainty budget with a separate component reflecting the </p><p> effects of virtualisation. The methodology consists of a step-by-step derivation of the governing formulas through Lie algebra and the matrix exponential, followed by verification of the computational core on certified statistical reference datasets of the United States National Institute of Standards and Technology, covering </p><p> 176 real measurements. As a result, all certified quantities were reproduced with high accuracy, the Kalman filter reduced estimation uncertainty severalfold, and the error component specific to the virtual environment was quantified for the first time on a real calibration record. The obtained results were implemented in software that has passed state registration and complies with the international quality management standard for testing and calibration laboratories. The findings can be applied in the digitalisation of national metrology infrastructure, in educational practice, and in industrial calibration processes. It is concluded that the proposed model provides stable, singularity-free computation and increases the reliability of virtual measurement results.</p>