G～k Blending Of Corners With Piecewise Algebraic Surfaces

The method of constructing a smooth algebraic surface to blend the given surfaces widely appears in Computer Aided Geometric Design. Here we concentrate an important problem: blending corner. In this thesis, we present an approach to blend the corners of three coordinate planes with G~k continuous piecewise algebraic surfaces. The geometric continuity condition for algebraic surface patch complexes meeting at a common vertex is converted to solve an algebraic system of equations over polynomial ring. G~k continuous piecewise algebraic blending surface patches are obtained by solving the system via the generator basis algorithm of prime module over polynomial ring. Compared with the known methods, the blending surface of this paper has relatively lower degree, and the defining region and the shape of the blending surface can be easily controlled by a few parameters. The result can be combined with potential method to blend any three algebraic surfaces meeting at a common point.The thesis is organized as follows. In chapter 1, some methods of constructing blending surfaces are reviewed and introduced. In chapter 2, some preliminary knowledge about Grobner basis and syzygy module and the geometric condition for algebraic surfaces are provided. In chapter 3, the generator basis algorithm of prime module over polynomial ring is briefly introduced, which is used to solve the system of equations over polynomial ring in the thesis. In chapter 4, the problem of blending the corner of any three algebraic surfaces is converted to blend the corner of three coordinate planes in parameter space, Then an approach to blend the convex and non-convex corner with Gk piecewise algebraic surface patches is proposed. The process is as follows. At first, the geometric continuity condition for algebraic surface patch complexes meeting at a common vertex is converted to solve an algebraic system of equations over polynomial ring. Then Gk continuous piecewisc algebraic blending surface patches are obtained by solving the system via the generator basis algorithm of prime module over polynomial ring, and at the same time the conditions which the space partition should satisfy are given. Numerical examples are provided to illustrate the blending method in the end of the thesis.