2023/03/30 by Rory Conboye, Conboye, Rory
Engineering · Mathematics · Physics and Astronomy · #3D Shape Modeling and Analysis #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2304.00123
openalex publication_date 2023/03/30 · openalex created_date 2023/04/06 · openalex updated_date 2026/07/28
Discrete forms of the mean and directed curvature are constructed on piecewise flat manifolds, providing local curvature approximations for smooth manifolds embedded in both Euclidean and non-Euclidean spaces. The resulting expressions take the particularly simple form of a weighted scalar sum of hinge angles, the angles between the normals of neighbouring piecewise flat segments, with the weights depending only on the intrinsic piecewise flat geometry and a choice of dual tessellation. The constructions are based on a new piecewise flat analogue of the curvature integral along and tangent to a geodesic segment, with integrals of these analogues then taken over carefully defined regions to give spatial averages of the curvature. Computations for surfaces in both Euclidean and non-Euclidean spaces indicate a clear convergence to the corresponding smooth curvature values as the piecewise flat mesh is refined, with the former comparing favourably with other discrete curvature approaches.