Grid-based and meshless methods for the computation of the curvatures and related local geometric quantities of a three-dimensional surface
Vu, Phu D.
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Various applications in computer graphics require accurate estimation of local quantities such as the principal curvatures or principal directions. Other local quantities such as the surface energy or chemical potential in crystal growth and surface evolution problems in materials science also need to be computed with high accuracy. Furthermore, these quantities are the intrinsic surface properties that are not affected by the choice of the coordinate system, the position of the viewer relative to the surface and the particular parametrization of the surface. The standard approaches to the computation of local surface quantities are based on finite difference method. These methods employ uniform grids. However, it is not always possible to have data sampled on such strictly structured setup. This thesis has two main contributions. Firstly, we study and successfully develop a generalized finite difference method on non-uniform grids to approximate geometric quantities of a 3D surface. Secondly, we introduce the local meshless method for the approximation in the case the data is sampled without the presence of any grid. We also discuss some enhancements to improve the local meshless method.