Topology-Aware Quality Control of Triangulated Terrain Models Using Local Euler Curvature and Elevation-Filtration Curves

Bekarys Tashmukhanbet 

Triangulated irregular networks (TINs) are widely used to represent topography and bathymetry, but their quali-ty control is usually dominated by geometric criteria such as minimum angle and aspect ratio, which do not test whether connectivity errors have changed the topology of the mesh. We present a topology-aware quali-ty-control framework based on a local Euler-curvature identity and on lower-star Euler characteristic curves (ECCs). For a triangular two-manifold with boundary we classify the affine local functionals of vertex degree, incident-face count and boundary indicator whose total is invariant under mesh refinement and boundary dis-cretization, and reduce them to the canonical normalization κχ(v) = 1 − d(v)/2 + t(v)/3, whose sum equals the Euler characteristic of any finite triangular complex. Across 32 valid meshes built from four triangulation rules and four resolutions on two real Earth-surface benchmark arrays, the largest static residual was 1.74 × 10⁻¹¹. Injecting 1 to 40 missing facets, duplicated facets or orphan vertices returned a residual equal to the number of defects, whereas topology-preserving diagonal remeshing left it at machine precision. A coordinate-only sliver collapsed the minimum plan-view angle from 38.75° to 0.28° without changing the topological residual. ECCs additionally exposed representation uncertainty, with maximum inter-triangulation spreads of 25 and 20 Euler units and raster 4- versus 8-neighbour differences of 80 and 70. The framework separates mesh topology, ele-ment geometry and representation-induced uncertainty in an auditable way.