Generalized Shape Operators on Polyhedral Surfaces
Please always quote using this URN:urn:nbn:de:0296-matheon-8679
- 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.
Author: | Klaus Hildebrandt, Konrad Polthier |
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URN: | urn:nbn:de:0296-matheon-8679 |
Referee: | John M. Sullivan |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2011/12/11 |
Release Date: | 2011/12/11 |
Tag: | |
Institute: | Freie Universität Berlin |
MSC-Classification: | 53-XX DIFFERENTIAL GEOMETRY (For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx) |
65-XX NUMERICAL ANALYSIS | |
Preprint Number: | 829 |