Research On Shape Modification,Subdivision And Extension Of Quartic ?-b(?)zier Curve

In the field of CAD/CAM,parametric curve surface is the main tool that is used to describe the geometry information of the product,and is the basis and core of CAGD.Constructing curve surface with good properties has important value both in theory and practical application.The traditional B(?)zier method has good properties,however,it cannot adjust its shape after giving control vertex.In recent years,B(?)zier curve surface with shape parameters has become a new hot spot in CAGD research.This type of new curve surface inherits the advantage of traditional B(?)zier curve surface,and contains independent shape parameters,and it's easy to regulate its own shape.Based on this background,this paper makes a more in-depth study on the related algorithms of a class of quartic ?-B(?)zier curve with shape parameters,the research contents and achievements include:(1)This paper proposes quartic ?-B(?)zier curve shape modification algorithm based on single point constraint and multi-point constraint optimization.By using the Lagrange multiplier method to optimize the corrections of the two control vertices,so it can realize the shape modification of the quartic ?-B(?)zier curve and make the modified shape of the curve to meet the requirements of the given position vector and tangent vector constraint,and the curve before and after the shape modification also meets certain conformality.Finally,some concrete examples of the shape modification are given,and the results show that the proposed method can modify the shape of quartic ?-B(?)zier curve effectively.(2)Two kinds of segmentation algorithms are given for the segmentation of ?-B(?)zier curves:? The coefficient segmentation algorithm first converts the quartic ?-B(?)zier curve into a traditional quartic B(?)zier curve.Secondly,the control vertices of the subcurve segment after segmentation are obtained for the shape of the curve remains unchanged before and after splitting;? Using the proportional relationship of the first derivative at the end of the curve before and after the segmentation,The tangent segmentation algorithm finds the display expression of the control vertices of the quartic ?-B(?)zier curve after segmentation,At the same time,the segmentation error and numerical example of the curve are given.(3)Three kinds of extension algorithms for the quartic ?-B(?)zier curve are studied.Firstly,the extension curve which extends to a given single point is obtained by using the C2 or G2 continuous condition of the curve,and the optimal extension curve is obtained by optimizing the shape parameter with the minimization of the objective function.Secondly,according to the continuity conditions of C1 and G1 of the curve,two continuous extension curve control points are obtained.Finally,by using the curve of C1 or G1 continuity conditions,the continuation curve which extends to a given curve is obtained,and taking the shape parameter ? as an optimization parameter,the optimal continuation curve was obtained.The numerical examples show that the method is simple and effective,and can realizes fixed-point,fixed-curve continuation of the quartic ?-B(?)zier curve.