Uniform convergence of discrete curvatures from nets of curvature lines (Q972611)

The authors study pointwise convergence of curvatures for a special class of polyhedral surfaces. The vertices are sampled from the net of curvature lines on a smooth surface \(M\) and the combinatorics locally reflect that of the underlying smooth net of curvature lines. The curvatures under consideration are obtained from integrated quantities, divided by suitable area terms. The convergence result requires local bounds on maximal edge lengths and shape regularity plus a sampling condition relating these bounds to the maximal norm of the shape operator. The proof relies on the additional structure imposed by the curvature line discretization. In fact, a similar convergence result does not hold for general polyhedral surfaces. The reasons for this failure are lucidly explained.