Research On Several Algorithms Of Digital Geometry Processing

It is sometimes argued that after sound, image and video, the next digital mediarevolution will be about 3-dimensional geometry. While this prediction may seem somewhatoptimistic, one has to acknowledge that digital geometry processing(DGP) is becomingmore and more important in computer graphics. Although researchers have made promisingprogresses recently, DGP is still young. Geometry data have some inherent attributes, suchas topology, curvature and non-uniform sampling, etc. These attributes make most toolsused in the area of sound, image and video are not fit to process geometry data. Currentresearch in DGP can not provide a unified, theoretically sound frame to address all theseDGP problems, digital geometry processing still remains an open problem in computergraphics community.In this dissertation, we take the mesh model as our research object. We firstlyreview the development of digital geometry processing, including mesh subdivision, meshsmoothing and the detection of feature lines; then we discuss their mathematics foundationrespectively; Finally, we introduce some in?uential and extensively used algorithms indetail. We also survey the theories and implementations for these methods and point outtheir advantages and disadvantages. In this dissertation, we present several novel algorithmsfor digital geometry processing.We firstly present a locally triangular patch subdivision scheme based on the degreeelevation to eliminate the mismatch between the smoothness of the shading and thenon-smoothness of the geometry while rendering a coarse triangle-based mesh. In thisscheme we firstly modify Nielson's edge-vertex method, thus we can guarantee the convexhull property of the interpolating cubic triangular Be′zier curved patch. So the patchcan produce pleasing shapes; then we approximate the triangular patch using the degreeelevation scheme. In order to improve the visible quality, we use a quadratic interpolationalgorithm on the shading stage. Even if the surface is only C0 continuity, the examples used...