Piecewise flat approximations of local extrinsic curvature for non-Euclidean embeddings

Abstract: 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.