Space Partition Of G～k Blending Corners By Piecewise Algebraic Surfaces

The construction of smooth blending surfaces by algebraic surface takes an importantrole in Computer Aided Geometric Design. In algebraic surface blending, low degree couldbe got by piecewise algebraic surface, and the degree is lower while the number of pieces islarger. However, the space partition we defined, when blending the comer, determines thenumber of pieces of the blending algebraic surfaces. Hence, lower degree needs complexspace partition witch leads to complex algebraic conditions. People want to construct apiecewise algebraic surface which has the lowest degree and the least pieces. So, therelationship between the degree and the number of pieces of the blending surface is important.In this thesis, we consider the problem of blending the comers of four planes meeting ata common vertex by G~k continuous piecewise algebraic surfaces of degree k+1. Thegeometric continuity condition for algebraic surface patches meeting at the common vertex isconverted to solving a homogenous linear system of equations over polynomial ring. Andthen we present the relationship between the degree and the number of pieces of the blendingsurface. The result will affect the lowest degree and the least pieces of the piecewise algebraicblending surfaces.The thesis is organized as follows:In chapter 1, the method of blending comers by piecewise algebraic surfaces and thesituation of blending the comers of three coordinate planes with G~k continuous piecewisealgebraic surfaces are introduced.In chapter 2, some preliminary knowledge about Gr(¨|o)bner basis, prime module and thegeometric continuty condition for algebraic surfaces are provided.In chapter 3, the generator basis algorithm of prime module over polynomial ring isbriefly introduced, which is used to solve the system of equations over polynomial ring in thethesis.In chapter 4, the space partition problem of blending the comers of four planes meetingat the common vertex with G~k piecewise algebraic surface patches is proposed. The processis as follows. At first, the geometric continuity condition for algebraic surface patchesmeeting at the common vertex is converted to solving an algebraic system of equations overpolynomial ring. Then the system is solved via the generator basis algorithm of prime moduleover polynomial ring. At last we present the relationship between the degree and the number of pieces of theblending surface.