Key Technology Research On Approximation Subdivision Surfaces Based On Quad Meshes

Subdivision is a method for surfaces modeling broadly concerned these years. It has greater topological representation capability than NURBS and has been extensively studied and applied in the realm of Computer Graphics and Computer Animation, but seldom researched in the field of CAD/CAM. A main reason is that basic modeling and geometric processing algorithoms for subdivision surfaces are not perfect and many issues remain to be resolved. The main objective of this dissertation is to further investigate some CAD modeling related key technologies about subdivision surfaces and accelerate their application in engineering. The main research contents and achievements are as follows:Two approximate subdivision surfaces: Catmull-Clark surfaces and Doo-Sabin surfaces are introduced. Parameterization representation method, which is presented in this paper, together with the conventional method based on recursive refinement, makes subdivision surfaces union of discrete polygonal meshes and continuous parametric surfaces, which facilitates theory deducing and practical application.By generalizing knot-insertion algorithm of quadratic quasi-uniform B-spline, special subdivision rules for Doo-Sabin scheme in features, such as creases, corners and darts are presented and continuity nearby is analyzed. Both complicated surfaces of arbitrary topology with sharp features and quadratic regular surfaces frequently used in CAD, such as sphere, cylinder, cone and annulus can be represented by subdivision surfaces, which make them more helpful in engineering.A method of fillet operation for Doo-Sabin subdivision subdivision surface is put forward. By topological splitting, reassigning sharp features and resetting weights, a surface with fillet is constructed with revised Doo-Sabin subdivision scheme. Compared with other methods presented, ours can produce precise iso-radius blending for some surfaces and no local flat area can be seen in the transition surface.Algorithms of interactive shape modification for subdivision surfaces are discussed. Constraints of points, normal vectors and local isoparametric curves on subdivision surfaces, which can be converted into those on control vertices, are specified via setting up local coordinate systems in real-time operation. A global linear system is obtained and the shape of sudivision surfaces can be modified with the various geometric constraints. Two methods based on least-square and energy optimization are presented. The former is suited for local, precise modification, while the latter is a global modification with good fairness.Based on the two representations for subdivision surfaces mentioned above, algorithms of precise intersecting and trimming for subdivision surfaces are put forward. The control meshes are subdivided with revised skirt-removed method and the intersecting facets are detected among them. Initial intersections with parameters are got based on the Divide-Conquer idea and precise ones can be obtained with iteration method. According to the trimmed region required, directions of all intersecting lines are set. Then the facets of control mesh of the surface to be trimmed are classified into three types: reserved facets, trimmed facets and discarded facets. The trimming domains of the trimmed facets are set automatically and the surface can be precisely trimmed.A algorithm for subdivision surfaces reconstruction from complex triangle mesh of arbitrary topology, which may have sharp features, is presented with Squared Distance Minimization (SDM) method. Sharp features are recognized and original triangle meshes are quadrangulated interactively. Then the input data are parameterized and partitioned on each segmented region. The initial control mesh of the approximation surface is built with vertexes of each partition. Local coordinate systems are set up and parameters of each fitted data are optimized. Data to be fitted are downsampled and linear system based on the framework of SDM is constructed. Positions of control vertexes and parameters of the fitted data are recalculated in turn until the given error bound is satisfied. Compared with other existing methods such as Least Square Fitting (LSQ), ours is much faster, which is second order approximation.Experimental results demonstrate the proposed methods.