Study Of Data Reduction Of B-spline Curves And Disk B-spline Curves

B-spline curves and surfaces are basic figure representation tool of most of the Computer Aided Design(CAD) systems.The data storage,error representation and data exchange of them in the CAD systems, are important in the research area of the Computer Aided Geometric Design(CAGD).This paper studies in-depth the data reduction of B-spline curves(includes degree-reduction and knot-removal)and disk/sphere curves and surfaces modeling techsnology, the main results and are summed up as follows:1. This paper summarizes and analyzes the existed degree-reduction algorithms and knot-removal algorithms, point out one of the insufficience of the algorithms is that, they can't simultaneously reduce degrees and remove knots, in the practice which the degree-reduction and knot-removal are both needed, the degree-reduction and knot-removal can only be excuted separately, thus the accumulate errors will be increased. This paper combine the B-spline transition matrix and the least square theory, propose a uniform method to degree-reduction and knot-removal of B-spline curves. This method can reduce multi-degree and remove several knots of the B-spline curves simultaneously by some computation of matrix. It is simple and can be easily realized and can reduce the accumulate errors. The uniform method can be used to singly reduce the degrees or singly remove the knots. When used to degree-reduction, it can reduce multi-degree. When used to knot-removal, it can remove several knots at one time. The approximate curves of the uniform method can interpolate the end-points of the original curves. A lot of examples are provided to testify the feasibility of the method, and compare with some the existed degree-reduction and knot-removal algorithm. This method may be used in the data reduction of B-spline curves and the data exchange of different CAD systems.2. The research of curves and surfaces modeling techenology base on error representation are summarized and analyzed. Base on the disk arithmetic,this paper proposes disk B-spline curves, and discusses their some basic properties and algorithms. When we compare the disk B-spline curves with interval B-splines curves, it also can represent the error range of B-spline curves, besides, it has the advantage of smaller storage information and the error range won't be changed under the rotation transition.3. This paper studies the knot-removal problem of the disk B-spline curves, that is, bounding disk B-spline curves with one-knot-removed disk B-spline curves. Two methods are proposed to solve the problem, both of them can get the analytic solution. The examples are provided , and the approximate error of two methods are compared. The knot-removal algorithm can carry the error information of the original curves and the approximate error information, the result is useful for the subsequent geometric operation(such as to search the intersection points of the curves).4. This paper also studies the degree-reduction problem of the disk B-spline curves,...