  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.
