On Degree Reduction Of Parametric Curves And Surfaces

The degree reduction of parametric curves/surfaces is a key problem for Computer Aided Geometric Designed (CAGD). It has not only important theory signification but also crucial application value. This dissertation presents our researches as follows The development, difficulty and further trend on geometric modeling have been summarized. The importance that uses "Geometric Design and Calculate (GDC)" rather than "Computer Aided Geometric Design" is explained. Because the concept "Geometric Design" can't completely depict CAGD which has not only the strictly mathematic theory but also the important applied background. However, the GDC extends widely research fields. Based on GDC, a brief summarization of the degree reduction of Bezier curves/surfaces are given. Existed algorithms of degree reduction on Bezier curves/surfaces, Rational Bezier curves, as well as Triangle Bezier surfaces are analyzed systematically. Using the geometric properties and algorithms for degree elevation of Bezier curves, a new algorithm is gave for Bezier curves based on the Genetic algorithms, which has the virtues of uncomplicated and high precision and obvious geometric property. By the means of optimization methods, degree reduction of rational Bezier curves is changed to a optimization problem so that both weights and vertices are consider respectively. Using programming method and Genetic algorithms, a new method on rational Bezier curves is presented. Compared to the Existed algorithms, the new method has the follow virtues: Firstly, it is simply to get the result by fitness function, copy process, crossover process, mutation process and selection process.Secondly, the rational Bezier curves can reduced many times and interpolated endpoints. Finally, the reduced curves of Bezier is represented explicitly.