Research Of Subdivision Property Of Multi-degree B-splines In CAD

Subdivision method is a fast modeling method for curves and surfaces.It inserts new vertices into the initial given control polygon or mesh and connects them to form a new polygon or mesh by defining rules or schemes.After repeated such steps,smooth curves or surfaces can be obtained.With its simplicity and efficiency,subdivision method has become a powerful modeling method in CAD.B-spline,as a modeling tool commonly used in CAD,has a lot of subdivision methods.From the earlier generalized uniform curve subdivision to the later non-uniform subdivision with variable parameters,the subdivision method has more and more applications.As a generalization of B-spline,variable degree B-spline allows different degrees to be used in different segments,which is a direct generalization of B-spline in variable degree piecewise polynomial space.Compared with B-spline modeling,variable degree B-spline can reduce the number of control vertices,thus reducing the amount of data and computation.This paper presents a subdivision algorithm for variable degree spline curves.In this algorithm,the degree of each segment and the continuity between segments of different degrees can be specified before subdivision,and the degree of each segment can be in[1,4].The continuity between different degree segments can be chosen between C~0 and C~1,and the order of continuity between the same segment is degree minus 1.The algorithm is based on the property of inserting nodes of variable degree spline,which is different from inserting nodes on only one segment at a time.Instead,it adopts the form of global linear interpolation,which has symmetry,and includes Lane-Riesenfeld subdivision of uniform B-spline with degree?4 as a special case.The contents of this paper are as follows.In the second chapter,the definition of variable degree B-spline and its properties related to subdivision are introduced.It is pointed out that variable degree splines with continuity no more than C~1 and variable degree no more than 4 studied in this paper are pointed out.The related properties and marking methods of control polygon are reviewed.In the third chapter,firstly,according to the properties of variable degree spline,which is different from the existing methods of inserting nodes one by one in the non-zero node interval,this paper inserts the midpoint in all non-zero node intervals,and accurately gives the relationship between the basis function before and after subdivision,and then gives the relationship between the control points before and after subdivision.Secondly,based on the properties of interpolation nodes,the process of inserting nodes is decomposed into the form of layer by layer linear interpolation,and the initial control polygon is cut layer by layer,thus the subdivision algorithm of variable degree B-spline with continuity no more than C~1 and variable degree no more than 4 is given.Among them,the open curve subdivision algorithm,by introducing the subscript variable,accurately gives the subscript of the control points generated in each step of subdivision,so as to facilitate the extraction of the next step.The closed curve algorithm clearly shows the process of each step of the algorithm by locating the polygon segments that need to be cut at each step.Then,the properties of the algorithm are discussed,and the symmetry of the algorithm is proved by an example.In the fourth chapter,the plane and space subdivision examples of open curve and closed curve are given respectively,and the results of subdivision 1,2 and 5 times are compared.