Anisotrope Krümmungsflüsse parametrischer Flächen sowie deren Anwendung in der Flächenverarbeitung

The presented dissertation is concerned with anisotropic curvature motion of two-dimensional parametric surfaces as well as their application in surface fairing and surface restauration. Mainly the so called anisotropic mean curvature motion (AMCM) and the anisotropic Willmore-flow are being treated. These flows are generalizations of the classical mean curvature flow and the classical Willmore-flow, respectively. The anisotropies are induced by positive, 1-homogenous and convex functions, which can be regarded as support functions of convex bodies, so called Wulff-shapes. Being a method of fourth order of differentiation in the surface coordinates, the anisotropic Willmore-flow allows the prescription of boundary values for the position of the boundary itself as well as for the surface normals on the boundary of a surface patch under consideration. Hence it is an appropriate method for the reconstruction of partially destroyed surfaces. In this work a numerical scheme for the anisotropic Willmore-flow is presented, which is based on an operator splitting of the fourth order evolution equation into two weak equations of first order of differentiation, which is discretized using linear finite elements in space. In particular the discretization of the AMCM turns out to be one of these equations. Based on the AMCM a method for the fairing of surfaces with crystalline edges is developed. Modifications of the discrete AMCM are also used for surface modeling purposes. Schemes for the artificial aging and for virtual engraving of surfaces are presented. Further on a subdivision scheme based on the isotropic mean curvature motion is introduced. Finally, the isotropic as well as the anisotropic Willmore-flow is employed for the restauration of partially destroyed triangulated surfaces.


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