The Curves And Surfaces Research Of The Approximate Algorithm Based On Cubic B-Spline

Curve is one of the mainly studying subjects of the differential geometry. And surface is one of the studying contents in CAGD. Curves and surfaces both are widely used in real life. For example, architectural design, animation design and exterior design of the industrial products. They are all closely related to the curves and surfaces. With the development of the modern science and technology, both manufacturing and industrial production have higher requirements to curves and surfaces in precision. The traditional interpolate spline and the approximate spline algorithms haven’t met the demands in the practical industrial production yet. For this reason, we put forward the approximate algorithm of the curves and surfaces based on cubic B-spline in this paper. This algorithm is based on cubic B-spline. It not only avoids the shortcomings of the traditional interpolate spline and the approximate spline, but also combines the advantages of them. It makes some research in the approximate area of the curves and the surfaces.The study of this paper contains the following points:(1). Study the development and the research status of the free curves and surfaces.(2). Research the quadratic B-spline algorithm. Extend the algorithm based on quadratic B-spline to the cubic B-spline approximate algorithm. Improve this algorithm with the acceleration error method. With this method, the convergence speed is faster and the convergence precision is higher than the algorithm which based on quadratic B-spline.(3). Apply the algorithm to the field of the curve approximate. Prove the convergence theoretical feasibility of the algorithm in curve. By the numerical experiments, we can verify the convergence speed is fast and the iteration time is short.(4). Extend the curve approximate algorithm to the surface approximate algorithm. And verify the advantages of this algorithm by theoretical proof and numerical experiments.(5). Study the smoothness of the surface interface of this algorithm and give the detailed process of theoretical proof. The prove results indicate that cubic B-spline surfaces are second-order derivable. Because second-order differentiability is more security and has more practical values, this algorithm has more practical applications.(6). Make the numerical experiments respectively to the improved algorithm with the acceleration error method and the algorithm with the traditional error method. At the same time, compare their convergence effect. The result demonstrates that our algorithm is much more effective.(7). By the programming tools such as Matlab and C++Builder6.0, we make the approximation numerical experiments to the surface approximation algorithm Which can show the approximation effect more intuitive.