Generalized shape operators on polyhedral surfaces

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• 作者：Klaus Hildebrandt ; ^a ; ^{klaus.hildebrandt@fu-berlin.de"" rel=""nofollow} ; Konrad Polthier^a
• 关键词：Discrete curvatures ; Generalized shape operators ; Curvatures of polyhedral surfaces ; Approximation of curvatures ; Curvature estimation
• 刊名：Computer Aided Geometric Design
• 出版年：2011
• 出版时间：June 2011
• 年：2011
• 卷：28
• 期：5
• 页码：321-343
• 全文大小：606 K

文摘

This work concerns the approximation of the shape operator of smooth surfaces in 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.