Research On The Theory And Algorithms Of Q-bézier Curves And Surfaces

As a powerful tool for shape design in CAGD,traditional Bernstein-Bézier curves and surfaces are widely used in mechanical design,industrial design,computer-aided design and manufacturing,computer graphics and other geometric modeling fields.However,the shapes of Bemstein-Bézier curves and surfaces are uniquely determined by their control points,which brings great inconvenience to the shape design of various products.In order to overcome this shortcoming,when the control points are fixed,Q-Bézier curves and surfaces can adjust its local and global shape easily by introducing shape control parameters,so it is very important to study Q-Bézier curve and surface in theory and application.In this paper,the smooth continuity,the conversion between traditional Bézier curves and surfaces,as well as the degree elevation algorithms of Q-Bézier curves and surfaces are mainly studied.The specific work and research results are as follows:(1)First of all,the definition and properties of generalized Bernstein basis functions with parameters are briefly introduced.Then the definitions of Q-Bézier curves and surfaces are given and their important properties are analyzed.At the same time,the influence of shape parameters on the shape of Q-Bézier curve and surface is discussed in detail.Finally,the G1 and G2 geometric continuity conditions of Q-Bézier curves and surfaces are studied respectively,and numerical examples of smooth continuity are given as well.The examples show that Q-Bézier curve and surface can be used as a powerful supplement of complex curves and surfaces design in CAD/CAM system.(2)In order to solve the incompatibility between different curves and surfaces in the same CAD/CAM system,the conversion relationships between Q-Bézier curves and surfaces with traditional Bernstein-Bézier curves and surfaces are studied.First of all,a conversion matrix is derived to transform the nth-degree generalized Bernstein basis function with parameters into the Bernstein basis function of degree n+1.Secondly,based on the conversion matrix,the two explicit conversion formulas of transforming Q-Bézier curves and surfaces into Bernstein-Bézier curves and surfaces are derived.At the same time,combined with the least square theory of the generalized inverse matrix,the approximate algorithms of transforming the traditional Bernstein-Bézier curve and surface into Q-Bézier curve and surface are proposed respectively by solving the overdetermined equations.At last,some representative numerical examples are given to verify the effectiveness and accuracy of the proposed transformation algorithm.(3)As a kind of important algorithm in CAGD field,degree elevation of curves and surfaces can not only realize the exchange of data between different degree curves and surfaces in CAD/CAM system,but also play an important role in the shape control and the construction of complex curves and surfaces.In order to enhance the modeling ability of Q-Bézier method,by using the conversion relationships between Q-Bézier curve and surface and Bézier curve and surface,and combining with the degree elevation formula of traditional Bernstein-Bézier model,the degree elevation algorithms of Q-Bézier curves and surfaces are proposed,respectively.The algorithms proposed in this paper give the constraint equations that the control points and shape parameters of Q-Bézier curve and surface before and after the degree elevation should satisfy,and realize the accurate degree elevation of Q-Bézier curve and surface.Numerical examples show that the algorithms are not only easy to operate,but also verify the curves and surfaces after degree elevation can have different control points and shape parameters.