Fundamental Technology Research On Subdivision Surface Modeling Based On Quad Meshes

Subdivision provides powerful tools for designing and manipulating surfaces ofarbitrary topological type. Among all kinds of subdivision surfaces, Catmull-Clarksubdivision surfaces and non-uniform Catmull-Clark subdivision surfaces arerespectively generalizations of uniform B-spline surfaces and non-uniform B-splinesurfaces. These subdivision surfaces are easy to be blended with NURBS surfaces.The thesis researches Catmull-Clark subdivision surfaces and non-uniformCatmull-Clark subdivision surfaces modeling techniques from different points ofview in order to further improve modeling abilities of subdivision surfaces andconstruct more relations between subdivision surfaces and NURBS surfaces. Its maincontents and contributions are as follows: 1. By improving the algorithm to construct n-sided patches given by Cotrina J. etal, a n-sided patch is constructed with uniformly cubic B-Spline boundary, whosecontrol mesh is the mesh formed by 5-ring of an extraordinary point in a C-Csubdivision mesh. The patch and the C-C subdivision surface can compose a surfacewith G2 continuity. On this basis, how to form and fill n-sided holes in C-Csubdivision surfaces is discussed. A method is given to solve the problem onmodeling low-degree G2 continuous free-form surfaces based on arbitrary topologymeshes. The blend is realized between surfaces constructed by manifolds and C-Csubdivision surfaces. 2. The subdivision method is presently the simplest method to blend parameterpatches. However, it is still a problem how to make programs automatically calculatevertices of the initial subdivision mesh of a blending surface. The paper analysesproperties of the mesh and classifies mesh curves on it. Subsequently, an optimizationmodel is presented by fairing of mesh. Aiming at characteristics of the model, keyvertices of the mesh are firstly calculated and the model are converted to a linearsystem. The approach can automatly determine vertices of the initial mesh. 3. To construct n-side patches is the base of blending patches and filling n-sideholes. Using the non-uniform Catmull-Clark subdivision scheme, two approaches arepresented to construct n-side patches — the skirt removed approach and theinterpolating corner vertices approach. The patches obtained by either of the twoapproaches have non-uniform B-spline boundaries. Based on the skirt removedapproach under the non-uniform Catmull-Clark subdivision scheme, an efficientmethod is given for points and normals interpolation of non-uniform B-splinesurfaces. 4. Minimum distance computation is vital in many application fields. Applyingthe sufficient and necessary condition of biquadratic Bézier patch being non-negative,a method is presented to see whether two points are the closest points between twobiquadratic NURBS surfaces when the two points are respectively on the two surfaces.Based on the method, the surface split algorithm is improved while the surfaces arebiquadratic NURBS surfaces. At the same time, the surface split algorithm is alsoused to compute minimum distance between non-uniform Catmull-Clark subdivisionsurfaces. 5. Owing to the adaptability to arbitrary, subdivision surfaces are suitable forconstructing models of objects with complicated shape in reverse engineering. Analgorithm is put forward to fit Catmull-Clark subdivision surfaces from an irregularand dense triangular mesh of arbitrary topology type. The shrink-wrapping algorithmunder triangular meshes is used in quadrangle meshes. As for subdivisionconnectivity meshes, a new mesh progression transmission method is given byemploying the Catmull-Clark push-back subdivision scheme. Finally, An iterationmethod is presented to calculate initial control mesh vertices while Catmull-Clarksubdivision surfaces are required to interpolate given points. The convergence of theiteration method is analysed.