Uncertainty Propagation in Curvature-Based Surface Form Metrology: A Monte Carlo and Differential Geometry Approach

Curvature-based descriptors are increasingly used in surface metrology for the characterization of complex geometries. However, their sensitivity to measurement uncertainty remains insufficiently understood, particularly in comparison with conventional deviation-based metrics. This study investigates the propagation of coordinate measurement noise into curvature estimation using a numerical framework combining differential geometry, local quadratic surface fitting, and Monte Carlo simulation. A set of nominal surfaces, including spherical, cylindrical, and free-form geometries, was analyzed under controlled stochastic perturbations. The results show that curvature uncertainty increases nonlinearly with coordinate noise and is significantly more sensitive to measurement errors than point-wise deviations. Even small perturbations in measured coordinates lead to amplified variability in curvature due to its dependence on second-order derivatives. The analysis further reveals the presence of systematic bias in curvature estimation and demonstrates that the resulting distributions deviate from normality, despite Gaussian input noise. This finding highlights the limitations of classical uncertainty evaluation approaches based on linear propagation and normality assumptions. In addition, the study shows that increasing sampling density does not necessarily improve estimation reliability, while the size of the local fitting window plays a key role in stabilizing curvature estimation, acting as an implicit regularization parameter. The comparison with conventional form deviation metrics confirms that curvature-based analysis provides complementary information about local geometric stability, which is not captured by global measures. The proposed simulation-based approach offers a practical framework for evaluating uncertainty in nonlinear geometric measurements and supports the integration of curvature-based descriptors into advanced metrological applications. The proposed framework can support uncertainty-aware evaluation of free-form surfaces in practical measurement tasks, including coordinate measurement of turbine blades and aerodynamic components in the aerospace industry, as well as optical scanning and verification of patient-specific biomedical implants, where accurate curvature characterization is essential for quality assessment.