Geometric Modeling With Toric Surface Patches And Semialgebraic Splines

Parametric curves and surface patches play an important role in Computer Aided Geometric Design (CAGD). The existing researches focus on the classical Bezier, B-splines, NURBS (Non-Uniform Rational B-Splines) curves and surface patches. Toric surface patch is a multi-sided generalization of classical Bezier surface patch, which preserves the most geometric properties of Bezier surface patch. In this thesis, we study on the conditions for geometric continuity of toric surface patches, and applications of toric surface patches in data fitting and blending sur-faces of pipes. Multivariate spline is also an important tool in CAGD and geometrical modeling. The main methods for studying on multivariate splines include smoothing cofactor-conformality method, B-net method, homological algebra et al.. In this thesis, we study on the multivariate splines defined on semialgebraic sets with generalized homological approach. The main con-tributons of the thesis are as follows.1. Geometric continuity of parametrical surface patches play a key role in CAGD. We study on the geometric continuity of toric surface patches. Firstly, we derive the properties of the first and second partial derivatives of toric Bernstein basis functions and prove the first and second partial derivatives of toric surface patches are invariant during the toric degeneration along the boundary while the lifting functions satisfy the lifting criterion. Therefore, the sufficient and necessary conditions of G1 and G2 continuity of toric surface patches are given. In addition, the regular control surface can be degenerated into rational Bezier surface patches via setting special lifting function, then we present the practical sufficient conditions of G1 and G2 continuity of toric surface patches by the known results of Bezier surface patches. These sufficient conditions involve the relations of control structure and can be used to construct the toric surface patches with G1 and G2 continuity. Some examples are given to verify our method.2. Plateau problem is a well-known problem in the researches of minimal surface patches. Since toric surface patches is a multi-sided parametrical surface patch, we study Plateau-toric problem with Dirichlet functional and get an approximation of toric surface patches of minimal area. Some examples are given to verify the effectiveness of the presented method.3. Data fitting and constructing blending surface of pipes are extensively used in geometri-cal model design, design of airplane body, manufacturing process and so on. Firstly, fitting data and reconstructing the surfaces with toric surface patches are presented by the interpolation and least square fitting of data. Secondly, we construct the blending surfaces of pipes with toric sur-face patches by geometric conditions. Since toric surface patches are a multi-sided generalization-?- of Beier surface patches, they not only preserve the advantages of Bezier surface patches, but also have the flexibility of domains which can reduce the number of blending surface patches and avoid the joins of adjacent surface patches.4. Homological method is an algebraic method to study the multivariate splines which are an important tool in CAGD. We generalize the homological method of multivariate splines over linear partitions to semialgebraic partitions and focus on two kinds of semialgebraic partitions while algebraic curves of degree n intersect at 1 and n2 points respectively. Then we prove the relations between semialgebraic splines and the splines defined on linear partition with toric degeneration and commutative algebra. Furthermore, the Hilbert polynomial and the dimension spline modules Cr(?) are derived.