The Freedom Of Conic Curves And Surfaces Can Be Accurately Represented Model Design Method

With the development of curve and surface modeling technology,modern industry has higher and higher requirements for geometric design,and more and more applications.The industry scope is not limited to the automobile industry,aviation industry and shipbuilding industry,but also involves the robot design and manufacturing industry,biological engineering,medical diagnosis and other industries.Non-uniform rational B-splines(NURBS),the only mathematical method for the geometric definition of industrial products stipulated by the International Standards Organization(ISO),can accurately represent conic curves except parabola,but there are some defects.The current mainstream classical Bezier method and B-spline method may cause errors because they cannot accurately represent conic curve except parabola,and can flexibly control the shape of curve and surface without changing the control vertex.According to the above problems,this thesis uses the Bezier method and B-spline method while retaining the excellent properties of classical basis function.Three basis functions under different function Spaces are constructed.The following is the work of this thesis:Firstly,A group of quasi-cubic trigonometric Bernstein basis functions with five parameters is constructed on space Bp,q=span{1,3t2-2t3,(1-t)p,tq} and space Pp,q=span{1,sin2t,(1-sint)p(1-esint),(1-cost)q(1-fcost)} which can be obtained by mixing algebraic space with trigonometric space.It is proved that the parametric equation and standard equation of circular arc,elliptic arc and parabolic arc can be obtained by selecting appropriate parameters.Finally,a B-spline form of quasi-cubic triangular Bernstein basis function with five parameters is constructed,and its properties and curve control are analyzed.Secondly,A set of QCT-Bernstein basis function with three parameters is constructed on space Ta,α,β={1,sin2t,(1-sint)2(1-asint)e-αsint,(1-cost)2(1-acost)e-βcost},This basis function is fully positive,and its cut Angle algorithm is given.The conditions of C0 continuous and C1 continuous spliching curves are analyzed.It is proved that the parametric equations and general expressions of elliptic arc,arc and parabolic arc can be obtained by selecting appropriate parameters.At the same time,the QCT-Bernstein basis function with three parameters is extended to the trigonometric domain,and the conditions of C0 continuous and G1 continuous splicing of surface slices with three parameters of QCT-Bernstein basis function are analyzed.Thirdly,the existing basis function is promoted once to obtain the basis function on space Ta,α,β={1,sin2t,(1-sint)(1-a sinπt/2)e-λt,(1-cosπt/2)(1-b cos πt/2),e-μ(1-t)},and mixed with the singular function to obtain the singular mixed basis function with multiple shape parameters.In this thesis,the continuity of the joint of singular mixed Bezier curves and surfaces with multiple shape parameters is studied.The parametric equations and standard equations of ellipsoid arc,arc,parabola arc,ellipsoid and sphere are obtained.The positions of corresponding control vertices are given,which shows that this basis function meets the requirements of geometric design.