B-样条函数的一种快速递推法

 本文提出了常用的等距结点B-样条函数的一种快速递推法及递推计算格式和程序设计。阐明了此法具有计算简便、递推快速和更为直观的特点。它为B-样条函数的数学分析和推广应用提供了方便。此法将易于为一般工程技术人员掌握和运用。

A RAPIDLY RECURSIVE METHOD FOR THE B-SPLINES

A rapidly recursive method is presented for the （Normalized） B-splines of any order on the uniformly spaced kNot sequences.A parametric form of the B-splines is more suitable to matrix methods and possesses definite advantages for its application. Given n points P;（j =1, 2, …, n）, the vectorvalued function for a B-spline curve of order M （degreeM- 1 ）of the j-th segment is given by eq. （ 8 ）. NiM]（ t ） is the i-th B-splinebasis function of order M and AM]] is the basis matrix.Obviously, a directly recursive method for the basis matrix AM]] is of practical importance. According to the analysis of the characteristics of the B-spline basis functions, NiM]' （ t ）can be calculated by algorithmThe continuity condition at the kNot is given by By integrating NiM]' in successive orders, a rapid recursion of the basismatrixAM]] is obtained. The relationships can be expressed as eqs. （14）-（18）. In addition, the equations （21） and （22） can be used to check the numerical calculations in the process of recursion.A concrete application format of the method （see eq. （23）-（27）） including its programming （here omitted） is then presented. Several examples and the results of calculations by FORTRAN program show that this method is more intuitive, rapidly recursive and simple in computation. It is convenient for the mathematical analysis and application of B-splines. It is especially suitable to computer aided geometric design, finite element methods and others for engineers and technicians.