Discretization of Laplace-Beltrami Operator

Çakar, Ilgaz
Discrete differential geometry studies the local properties of discrete shapes. Its main purpose is to translate the objects and tools such as curves, surfaces, curvature from smooth category to discrete category so that they can be easily used for computational purposes. One of these tools from smooth category is the Laplace-Beltrami operator whose discrete version is well-known for its applications in geometry processing such as surface smoothing, computing a vector field with prescribed singularities, or mesh parametrization. As the discrete form can be used in computers with more ease, the discretization of the Laplace operator is of utmost importance. In this thesis, after examining two different approaches to discretize Laplacian on triangular meshes, we show that a discrete Laplacian can not preserve all the properties of its smooth counterpart. Finally, we generalize the discretization to general polygonal meshes which allows much more flexibility on applications.


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Citation Formats
I. Çakar, “Discretization of Laplace-Beltrami Operator,” M.S. - Master of Science, Middle East Technical University, 2022.