Research On The Theory And Applications Of The Generalized Bézier Curves And Surfaces With Shape Parameters

With the rapid development of geometric modeling industry,it is difficult for the traditional Bézier curves and surfaces to satisfy the various needs of industrial design.In the last years,constructing generalized Bézier curves and surfaces with shape parameters is a hot research topic in CAD/CAM,which are extensions of traditional Bézier curves and surfaces.These new curves and surfaces have most properties of the corresponding classical Bézier ones.Furthermore,they hold the property of flexible shape adjustability.Therefore,research on their related theory and algorithms is of great importance about theoretical significance and application value.In this paper,we study the construction and continuity conditions of generalized Bézier curves and surfaces with shape parameters,design develop able surface with shape parameters and construct the surfaces of revolution with shape parameters.Also we present a class of new generalized Bézier curves of degree n with multiple shape parameters and discuss its application to surface modeling in engineering.The main research work is as follows:(1)Firstly,a new geometric model of quartic generalized Bézier surfaces with multiple shape parameters is constructed using a class of quartic generalized Bernstein basis functions.The constructions of some special surfaces degenerating from the quartic generalized Bézier surfaces are discussed.The conditions of G1 and G2 continuity between two adjacent quartic generalized Bézier surfaces are also proposed.Secondly,a new geometric model of quartic generalized C-Bézier surfaces with multiple shape parameters is constructed using the basis functions span{1,t,t2,sint,cost}.and the G1 continuity conditions of quartic generalized C-Bézier surfaces are also proposed.Finally,the unique property of CE-Bézier curves is investigated and the G1 continuity conditions of CE-Bézier surfaces are derived and simplified by choosing the control parameters properly.(2)Based on the analysis of the basis functions and terminal properties,we propose the necessary and sufficient conditions of G1,G2 continuity and C1,C2 continuity between two adjacent ?-Bézier curves,and discuss some properties of the continuity condition for the?-Bézier curves.On the other hand,a new geometric model of the generalized Bézier-like surfaces of order m×n with multiple shape parameters is constructed using a class of generalized Bernstein-like basis functions.Meanwhile,many properties of the generalized Bézier-like surfaces are investigated,and the conditions of G1,G2 continuity between two adj acent generalized Bézier-like surfaces are proposed.(3)A class of novel quasi-Bernstein basis functions is presented to construct quasi-Bézier curves with multiple shape parameters,which is an extension of the classical ones.Following the important idea of duality between points and planes in 3D projective space,the developable quasi-Bézier surfaces with multiple shape parameters are designed by using control planes with quasi-Bernstein basis functions.The necessary and sufficient conditions of G1 continuity,Farin-Boehm G2 continuity and G2 Beta continuity between two adjacent developable quasi-Bézier surfaces are obtained.In addition,some properties of the developable Bézier-like surfaces are discussed.(4)According to the important idea of BTVRI function,the shape-adjustable generalized Bézier rotational surfaces are constructed using a shape-adjustable generalized Bézier curves with multiple shape parameters,and the explicit function expression of the ones is presented.In addition,we discuss some properties of the shape-adjustable generalized Bézier rotational surfaces,applications in rotation surfaces design and the influence rules of the shape parameters on rotational surfaces shape.The modeling examples illustrate that the proposed method provides a valuable way for the design of rotation surfaces.(5)We define a new kind of extension Bernstein basis functions of degree n with explicit expression,thus a kind of parametric curves and surfaces with global and local shape parameters is presented,which are called shape-adjustable generalized Bézier curves and surfaces.The properties and Gl continuity conditions of these new curves are also analyzed.Using shape-adjustable generalized Bézier curves,we construct five different types of engineering surfaces with multiple shape parameters,including general cylinder,bilinear surfaces,ruled surface,swung surfaces and swept surfaces.These new surfaces can be expressed in rectangular shape-adjustable generalized Bézier surface accurately.