This dissertation, "Shape-preserving Meshes and Generalized Morse-Smale Complexes" by Feng, Sun, 孙峰, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Discrete representation of a surface, especially the triangle mesh, is ubiquitous in numerical simulation and computer graphics. Compared with isotropic triangle meshes, anisotropic triangle meshes provide more accurate results in numerical simulation by capturing anisotropic features more faithfully. Furthermore, emerging applications in computer graphics and geometric modeling require reliable differential geometry information estimated on these anisotropic meshes. The first part of this thesis proposes a special type of anisotropic meshes, called shape-preserving meshes, provides guaranteed convergence of discrete differential operators on these meshes and devises an algorithm for generating shape-preserving meshes on free-form surfaces based on the mesh optimization framework with centroidal Voronoi tessellation (CVT). To improve the numerical stability in simulation, we discuss how to reduce the number of obtuse triangles in the mesh. The second part of the thesis discusses the non-uniqueness of anisotropic meshes to represent the same anisotropy defined on a domain, shows that of all anisotropic meshes, there exists one instance minimizing the number of obtuse triangles, and proposes a variational approach to suppressing obtuse triangles in anisotropic meshes by introducing a Minkowski metric in the CVT framework. On a complex shape, its topological information is also highly useful to guide the mesh generation. To extract topology properties, the Morse-Smale complex (MSC) is a classical tool and widely used in computer graphics. However, on a manifold with boundary, its MSC is not well defined. The final part of this thesis generalizes the MSC to manifolds with boundaries. Based on this generalized MSC (GMSC), an operator to merge n GMSCs of manifolds partitioning a large manifold is proposed. The merging operator is used in a divide-and-conquer approach on a massive data set, providing the potential to employ the computational power in a parallel manner. DOI: 10.5353/th_b4786963 Subjects: Computer graphics - Mathematical models