Research On Degree Reduction Of Generalized Bezier Curves And Surfaces With Parameters

In the field of CAD/CAM,the generalized parametric Bezier curves and surfaces,which inherit the advantages of traditional Bezier curves and surfaces and have good shape adjustability,is of great value in describing product geometry and actual engineering design.In terms of calculation speed and degree reduction,it is better than traditional Bezier curves and surfaces,and is more suitable for the shape optimization design of the curves and surfaces.In the CAGD research field,one of the important branches of geometric approximation is degree reduction.In practical applications,it is necessary to approximate the degree reduction of curves and surfaces to meet various needs in the process of modeling design.This paper mainly researches the approximate algorithms of Q-Bezier curves and SG-Bezier curves and surfaces in this context,.The main contents and results are shown below:(1)The research status of generalized Bezier curves and surfaces with shape parameters are reviewed,and the degree reduction methods of curves and surfaces are summarized;two types of generalized Q-Bezier or SG-Bezier basis functions with multiple shape parameters are introduced.The related properties of the basis functions are discussed,and the definitions and properties of the corresponding Q-Bezier or SG-Bezier curves and SG-Bezier surfaces with multiple shape parameters are given,as well as the influence of the shape parameters on their shapes.(2)Based on L2 norm,this paper studies the degree reduction of Q-Bezier curve by solving the minimum value problem of component function.Secondly,by solving the linear equations without constraints,C0 constraints and C1 constraints,the direct expression of the control points can be obtained,and then the Q-Bezier curve can be reduced.Finally,the influence of shape parameters on the order reduction is considered in the process of order reduction,and the numerical examples and corresponding errors of order reduction under different constraints are given.(3)In this paper,the intelligent algorithm named grey wolf optimizer algorithm is used to realize the degree reduction of SG-Bezier curves and surfaces.Firstly,we transform the degree reduction problem into the optimization problem by minimizing the maximum distance of the curve before and after the degree reduction.Secondly,the continuity conditions of unconstrained,C0 constrained and C1 constrained are derived.The specific steps and simulation pseudo-code of the degree reduction of SG-Bezier curves are given,and the approximate reduction of SG-Bezier surfaces is realized by using the idea of the algorithm.During the order reduction process,the effects of changing the global and local shape parameters on the degree reduction effect are considered respectively.Finally,the experimental results and the errors further verify the effectiveness of this method in the approximate degree reduction of SG-Bezier curves and surfaces.