Discretization and approximation of the shape operator, the Laplace-Beltrami operator, and the Willmore energy of surfaces
Klaus Hildebrandt · Universitätsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver · 2013
This thesis deals with discrete differential geometric properties of polyhedral surfaces in Euclidean 3-space. We develop a description of the curvatures of polyhedral surfaces based on a generalization of the shape operator, and we construct discretizations of the (strong form of the) Laplace-Beltrami operator and of the Willmore energy. The focus of our analysis is on approximation properties of the introduced discretizations. In particular, we show that the generalized shape operators can be used to approximate the classical shape operator of a smooth surface, and we prove the consistency of the discrete Laplace-Beltrami operators and the discrete Willmore energies. In addition, we consider a problem in geometry processing: the fairing of polyhedral surfaces. We develop a scheme for fairing under spatial constraints.