The Extension Of Low-order Bézier Curves And Surfaces

As one of the main methods of shape mathematical description in CAGD,the Bézier method has many excellent properties that are convenient for graph construction because of its unique Bernstein polynomial as the basis function.However,due to the single shape adjustment method and the complex splicing conditions of piecewise curves and piecewise surfaces,the application scope and application value are limited.The purpose of this paper is to construct the low-order Bézier extension curve with shape parameters based on algebraic polynomial space,without increasing the computational complexity of the blending functions and the corresponding curves as much as possible.The extended curves have more flexible shape adjustment ability and relatively simple splicing conditions.At the same time,the calculation formula of shape parameters in the curve is given under three common energy objectives in practical application.With the quintic,sextic and septic Bernstein basis functions as the initial research objects.We introduce shape parameters into the two inner functions in each blending function.Three groups of Bernstein basis functions are divided and redistributed,and a set of blending functions consisting of four functions with a shape parameter and two sets of blending functions consisting of five and six functions with two shape parameters are obtained.Through the linear combination of blending functions and control points,cubic,quartic and quintic Bézier extension curves are defined,and extended them to the surface field.Without increasing the computational complexity,the newly blending functions retain most of the important properties of the traditional Bernstein basis function,so that the newly constructed extension curves not only have the advantages of the traditional Bézier curve,but also can adjust the shape of the relevant curve and surface by changing the value of the shape parameters without changing the position of the control vertex.Compared with the traditional Bézier curve,the approximation degree of the new curves to the control polygon is greatly improved.Most importantly,when the new extension curves are spliced,as long as the general C~2 smooth splicing conditions are satisfied.They can automatically achieve C~2(40)FC~3 continuity at the connection point;in the special case,the G~3 continuity can be achieved automatically when the G~2 smooth connection condition is satisfied.In addition,the formulas for calculating the shape parameters of curves and surfaces with the shortest approximate arc length,the smallest approximate curvature and the smallest change rate of approximate curvature are given.