Modeling And Application Of Subdivision Surfaces

This thesis studies modeling and application of subdivision surfaces. Subdivision method has become an important tool in computer graphics recently. To extend application areas (especially in the area of CAD) more widely, however, many problems have to be solved. The problems include constructing subdivision schemes of higher continuity, fusing subdivision and analytic methods, fairing subdivision surfaces, intersecting subdivision surfaces, developing mathematical tool for analysis of subdivision surfaces and designing new subdivision schemes satisfying all kinds of requirements etc. The thesis investigates some effective modeling approaches for enhancing modeling ability of the subdivision methods. One of the contributions of this thesis is to propose an approach to generate sharp features for subdivision surfaces. Most of the known methods produce the sharp features by modifying weights of geometric rules of primitive subdivision schemes. It is not intuitive to select new weights satisfying conditions. Moreover, continuity analysis must be performed again for introduction of new rules. The approach described in this thesis is based on topology modification of control meshes. The main operation is to update the connectivity relation of edges and vertices associated with sharp features such that the internal edges and vertices become boundary ones. It is easy to guarantee that the surfaces are not smooth along the boundary on account of the property that the boundary curves of the subdivision surfaces are uniquely determined by the boundary polygons of the meshes. The thesis also employs subdivision surfaces to blend parametric surfaces and fill n -sided holes surrounded by parametric surfaces, which is based upon the consideration that while modeling complicated objects we can combine the advantages of both parametric surfaces and subdivision surfaces. An approach (named skirt-removed approach) is presented for generating subdivision surface boundary by removing boundary vertices, edges and faces of control meshes. On basis of the skirt-removed approach a unified framework is built to blend multiple patches of parametric spline surfaces and to fill n -sided hole G 1and G 2 continuously with subdivision surfaces by appropriately constructing primitive control meshes. Generally speaking, the simpler the curvature distribution, the fairer the surface. Planar discrete clothoid splines (PDCSs) and discrete clothoid spline (DCS) surfaces of Schneider and Kobbelt are just constructed by assuming that the curvature of the curves or surfaces is piecewise linearly distributed. The thesis extends the notion of PDCS to three-dimension (3D) space and defines the DCS surfaces on open meshes. Discrete Frenet frame introduced in the thesis is the key ingredient of the work. The curvature of the 3D DCS and the DCS surface on an open mesh obtained by our method is piecewise linear distributed in terms of the least squares. In addition, the DCS surface has 3D DCSs as its boundary. The thesis has also investigated the application of the subdivision method. Firstly, properties of the subdivision method are applied to parametric surface modeling and thus algorithms are designed to interpolate vertices and their normal vectors with quadratic and bicubic B-spline surfaces respectively. Secondly, application on triangle mesh simplification is also considered. In a kind of simplification algorithms based on triangle collapse, new vertices should be computed for substituting the collapsed triangles. Some approaches calculate new vertices by resampling from a quadratic surface fitting the vertices near the collapsed triangles. The fitting operation is finished through solving a linear system which is probably unstable and even hasn't solution due to the ill-conditioned data. Other algorithms simply select the vertices or centers of the collapsed triangles as new vertices. The solution may result in large error because of linear sample. The thesis treats the primitive surfaces as butterfly subdivision surfaces with meshes required decimation as primitive control meshes and makes out new vertices using butterfly scheme. Therefore it overcomes the disadvantages appearing in traditional approaches.