Research On Polynomial Curves With Parameters

Curve and surface modeling is an important research topic in the theory of function approximation.Bernstein basis function has good properties,such as nonnegative,the unit of decomposition,endpoint properties,symmetry and order property.Bézier curve has good properties,such as the endpoint interpolation,symmetry,order property,variation-diminishing and convexity preserving properties.Therefore,Bézier curves are often used for geometric design,which provides a useful tool for the design of curve and surface modeling.In this paper,using the relation between Bernstein base and power base,the relationship between polynomial function with parameter which are defined in[0,1]interval and Bernstein base is given.It can simplify the construction of the basis function.First,this paper discusses the relation between the Bernstein basis function and the B spline basis function.The relationship between the two or three order uniform B spline basis and the corresponding times of Bernstein base is represented.The relationship between polynomial function with parameter defined in[0,1]interval and the basic functions of Bernstein is given.And the properties of the matrix are studied.Then,the paper uses this conclusion to construct the cubic polynomial basis function with four parameters in[0,1]interval,and discusses its basic properties.It can be regarded as the extension of the secondary Bernstein basis,or the extension of the quadratic uniform B spline basis.Later,by extending the cubic polynomial basis function with four parameters to the whole real axis,the paper obtains a set of segmented basis functions,and its basic properties are discussed.Finally,this paper discusses the curve generated by the segmented basis function and its basic properties.And the Bézier expression of the segmented curve can reach G~1 continuity at the splice point.