Differential Geometry for Morphological Analysis of the Built Environment: a Reproducible Workflow to Standardise the Interpretation of Point Clouds

Point clouds are now a primary geometric basis for documenting the built environment; however, their analytical value is often constrained when points are treated as mere sets of 3D coordinates, without considering their local behaviour within a neighbourhood. This paper addresses this limitation through a reproducible, robust, and comparable workflow grounded in differential geometry, aimed at deriving slope and curvature descriptors via the numerical estimation of first and second derivatives. The framework sup-ports coherent reading and comparison across datasets: surveys acquired through terrestrial laser scanning and photogrammetry are homogenised and denoised; normal vectors and normal-orientation fields are computed; directional assessments are then performed by extracting profiles along section planes, where controlled smoothing enables stable estimates of derivatives and curvature. Numerical processing combines open-source tools for point cloud handling with scripted routines (Python for automation and MAT-LAB for profile-based calculations). The method is validated on three experimental case studies, differing in scale and domain: (i) the Gleno Dam in Val di Scalve, where slope and curvature maps support the identification of local anomalies and discontinuities; (ii) the bridge over the Carso stream in Nembro, compared with an idealised geometry to assess as-built deviations and behaviour in terms of inflection points and curvature inversions; and (iii) a geotechnical laboratory test on a sandy-slope model conducted in a centrifuge, where pre-/post-test comparisons reveal anisotropic deformation trends and associated curvature variations. The contribution also discusses the influence of key parameters and outlines future developments towards fully open-source implementations and AI-assisted morphological interpretation.