P-nary Subdivision Curve Modeling And Its Applications

This thesis studies modeling and application of subdivision curves. Subdivision method has become an important tool in computer aided geometric design and computer graphics recently. Subdivision allows to generate smooth curves and surfaces by applying simple refinement rules to the given control polygon and control mesh .The thesis investigates some effictive modeling approaches for enhancing the ability of modeling smooth subdivision curves of the subdivision methods. The involved problems include binary subdivision curve modeling, ternary subdivision curve modeling and p-nary subdivision curve modeling.Afer briefly reviewing the classification and history of curve and surface modeling, this paper discusses the basic constructing ideas, the history, the classification, the characters and the advantages of subdivision schemes. A survey of the classical subdivision curves and surfaces is presented.On binary subdivision curves modeling the thesis first present a kind of four-point binary subdivision scheme with two parameters, which has a better smoothness property than classical four-point subdivision scheme, and another kind of three-point binary subdivision scheme. The thesis proves the sufficent conditions of the uniform convergence properties and Ck continuity properties of these two binary subdivision schemes based on the results of the convergence properties and the continuity properties of binary subdivision schemes presented by Dyn. One can modify the shape of the subdivision curves and get smooth curves by choosing these two parameters appropriately. Using the four-point binary subdivision scheme with two parameters presented in the thesis one can not only model smooth interpolating subdivision curves but also can model approximating subdivision curves with high smoothness. It is proved that the four-point binary subdivision scheme presented in this thesis can generate C4 functions, and the three-point binary subdivision scheme can produces C3 functions.On ternary subdivision curves modeling the thesis present a kind of three-point ternary interpolating subdivision scheme with one parameter in chapter 3, and later present a kind of four-point ternary interpolating subdivision scheme with three parameters in chapter 4. The thesis proves the sufficent conditions of the uniform convergence properties and C1 continuity properties of the three-point ternary interpolating subdivision scheme with one parameter. The thesis also proves thesufficent conditions of the uniform convergence properties, C1 and C2 continuity properties of the four-point ternary interpolating subdivision scheme with three parameters based on the results of the uniform convergence properties and Ck continuity properties of p-nary subdivision schemes presented in chapter 4.The thesis generalizes the binary subdivision schemes in chapter 2 and the three-point ternary interpolating subdivision scheme in chapter 3 to p-narysubdivision schemes. Binary subdivision schemes is a special case of p-nary subdivision schemes when p equals 2, while ternary subdivision schemes is a special case of p-nary subdivision schemes when p equals 3. The thesis also proposes and proves the sufficent conditions of the uniform convergence properties and Ck continuity properties of p-nary subdivision schemes. At the end of the thesis a kind of four-point ternary interpolating subdivision scheme with three parameters is presented to describe the application of the p-nary subdivision theory presented in the thesis.