| Blending is an operation of constructing smoothly connecting surface between surfaces. Implicit algebraic surface blending is an important issue of Computer Aided Geometric Design. The main difficulty in surface modeling is that the degree of resulting blending surface is high. Therefore, how to get low-degree blending surface with a specified order smoothness is an important research problem.In this thesis, we present a new blending method which is based on space partition and algebraic splines. By use of this method, low-degree blending surface can be gotten and there exist free parameters to adjust the blending regions. The thesis is organized as follows:In chapter 3, we present methods to blend arbitrary convex n—hedral (n≥3) angle with G~1 continuity considering that most blending methods can only blend trihedral angle. Two methods are proposed. The first method, which is based on homogeneous spline, can convert 3—dimensional blending problem to a 2—dimensional one. Another method is based on high-dimensional Morgan-Scott partition, which can obtain lowest degree blending surfaces. The condition of simultaneous blending of convex polyhedron is proposed in the second method and the convexity of the blending surface is proved. All these two methods have free parameters to control the blending regions.In chapter 4, considering most existing blending methods can only reach G~1 continuity, a new space partition is presented over which we can construct blending surfaces which blend arbitrary convex n—hedral angle with G~2 continuity. Lowest degree blending surfaces can be obtained by use of this method. Further, We prove that the blending surface of the convex polyhedral angle obtained by this method is convex. For trihedral angle, not only convex angle but also the non-convex one can be G~2 blended. Moreover, we can use this method to blend convex combinations of n quadrics.In chapter 5. we present a method to blend the intersection or union or difference of arbitrary two convex polyhedra. By defining new operations, as long as the blending surface of each polyhedron is obtained, we can construct the blending surfaces of the intersection or union or difference of the two polyhedra. This method is intuitionistic and programmable and there exists free parameter to control the blending regions. The blending surface has lower degree. |
