| Geometry processing is one of the major subfields of computer graphics, which deals with the representation, display, and animation of virtual scenes. Geometric models are involved not only in the digitization of three-dimensional (3D) objects, but also in rendering and animation, i.e., the processes of creating digital imagery from such models and deforming shapes over time. The representations of shapes and their relations are crucial to the accuracy and efficiency of geometry processing. Thus improved representations can potentially benefit a wide range of applications, including computer aided design, visualization, simulation, education, training, and electronic entertainment.;Despite the rapid progress in the past few decades, several known challenges remain in geometry processing, including: Storage in runtime volumetric data structure has a space complexity that is cubic in resolution, or worse when the shape to represent changes dynamically; Automatic hexahedral meshing with well-shaped elements, i.e., decomposing a shape into cube-like cells, is highly desirable in finite element approaches in physical simulation, but is still considered as an open problem; Coordinate-system-independent representations of curved surfaces representing the boundary of 3D objects are indispensable in robust 3D model acquisition through scanners and animation, but are often highly redundant, or inefficient in conversion to explicit representations or deformation; Shape correspondence is becoming increasingly important with the rapid growing number of digitized 3D models, but can be hard to establish under large deformation.;We address these seemingly diverse issues with methods based on a uniform framework called discrete differential geometry, which strives to preserve desired structures of smooth shapes in their digitized representations, through carefully designed discretization. In this dissertation, we present novel meshing and correspondence inference techniques, designed for generating high quality representations of both the geometric objects and the mappings among these objects, which benefit subsequent geometry processing, modeling, and scientific computing.;First, we use edge lengths and dihedral angles of polyhedra to represent the first and second fundamental forms, two continuous quantities that measure the lengths and curvatures of curved surfaces, respectively. Mirroring the fundamental theorem of surfaces involving these fundamental forms, a discrete condition for the existence of an immersion of the surface in the discrete representation is also provided, which leads to an efficient method applicable in shape deformation.;Second, in order to express correspondence maps between two shapes, we propose a vector field map representation, extended from a spectral representation called functional map. The generalized representation is capable of handling the constraints for correspondence inference in the important angle-preserving case. The extension also makes it possible to handle function transfer between shapes with different topology.;Third, we present an automatic hexahedral meshing tool, based on a systematic treatment that eradicates common singularities that would lead to degenerate volumetric cells. Such singularities could be abundant in automatically generated edge direction fields guiding the interior and boundary layouts of the hexahedra in the final result.;Finally, we develop a compact data structure for volumetric shapes of arbitrary topology. It offers all commonly used incidence queries among its elements with constant time complexity, and can be adapted to work with dynamically changing topology.;We demonstrate the numerical test results, discuss future work, and point out the potential applications and techniques. |
