TY - GEN A1 - Hildebrandt, Klaus A1 - Polthier, Konrad T1 - Generalized Shape Operators on Polyhedral Surfaces N2 - This work concerns the approximation of the shape operator of smooth surfaces in $\mathbb{R}^{3}$ 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. KW - shape operator KW - discrete curvatures KW - generalized curvatures Y1 - 2011 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/867 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-8679 ER -