Generalized shape operators on polyhedral surfaces

This work concerns the approximation of the shape operator of smooth surfaces in R3R3 from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator, including mean and Gaussian curvature, principal curvatures, and principal curvature directions. The estimates are confirmed by numerical experiments.

► We rigorously define generalized shape operators on smooth and polyhedral surfaces. ► We prove approximation estimates for the gen. shape operators in the operator norm. ► We prove estimates for the pointwise approximation of the (classic) shape operator. ► The estimates are derived in a general setting and explicit bounds are provided. ► We show experimental results that confirm our estimates.