周期B样条基转换矩阵的表示和计算

周期B样条基以一种简洁的形式表示闭B样条曲线.周期B样条基转换矩阵为闭B样条曲线及相关曲面的不同表示间的转换提供了一个数学模型.本文给出了周期B样条基转换矩阵的存在性条件,给出并证明了周期B样条基转换矩阵的一个简单的递归表示式.在此基础上,本文进一步给出了周期B样条基转换矩阵的计算公式和高效算法.周期B样条基转换矩阵为闭B样条曲线的节点插入、升阶、节点删除和降阶等基本运算提供了一个统一而简单的解决方法,本文给出了一些应用例子.     

B样条基转换矩阵的解析表示和计算公式

摘要B样条基的转换矩阵具有重要的理论和应用意义。本文研究其最基本的问题：存在性条件、解析表示和计算方法，利用差商展开系数得到了上述问题的有关结果，本文的结果为CAGD中B样条曲线的节点插入、节点删除、升阶、降阶、分割、组合等重要技术提供了一个统一的数学背景和实现方法。     

B样条曲线升阶的新研究

B样条曲线的升阶是CAGD中的一个重要课题。本文根据传统的样条函数理论，提出了一个用高次B样条函数表示低次B样条函数的方法。该方法用于B样条曲线的升阶是快捷、有效的。     

(3,3,1,1)层次B样条与分级T网格上多项式样条的相互转换

1引言 B样条在计算机图形学和几何建模等领域有着广泛的应用[3,8].在应用过程中,通常都需要对得到的模型进行修改以到达更好的效果.对于B样条曲线,利用节点插入算法可以有效地进行局部修改.     

一类双k次B样条曲面的G1连续性条件

本文针对两个k×k次B样条曲面的节点向量为端点插值、内部是单节点的情形 ,给出它们之间的G1光滑拼接条件 ,同时得到它们的公共边界曲线的控制顶点所要满足的本征方程 .其中本征方程是B样条曲面片所独有的现象 .     

递归曲线的矩阵表示和构造方法

本文给出了递归曲线的矩阵表示和构造W曲线以及L曲线的比例因子方法．揭示了Bernstein基函数和等距B样条函数以及不等距重节点B样条函数之间的一种简单的内在关系．     

广义Ball曲线的性质及其应用

本文讨论了任意次数的广义Ball曲线的性质和它们的应用,如一般的升阶公式,Bézier曲线与广义Ball曲线之间的转换,极限定理,对偶基,广义Ball基函数下的Marsden恒等式,降阶赋值算法,单位分解性质等.     

基于B样条的函数型数据表示与配准

针对函数型数据配准问题,首先利用B样条函数来近似表示,并将扭曲函数也限定为于B样条函数空间内.进而将函数型数据配准问题转换为B样条函数升阶后比较控制顶点的问题,可降低计算复杂度.数值实验验证了该方法的有效性.     

Beziel曲线与B样条曲线的升阶公式

Bezier曲线的升阶公式在[1]中给出了简单的递推表达式,而B样条曲线的升阶公式则相对地复杂,本文利用[4]提出的n次多项式的blossom即一个与此多项式一一对应的对称的n—仿射映射,给出了Bezier曲线和B样条曲线直接升r阶的升阶公式。     

广义Ball样条曲线及三角域上曲面的升阶公式和转换算法

Ｔ．Ｎ．Ｔ．Ｇｏｏｄｍａｎ在［９］和［１０］中给出了广义Ｂａｌｌ样条曲线、曲面的奇次升阶公式和有关性质，但未给出偶次广义Ｂａｌｌ样条形式。     

Chebyshev�CSchoenberg Operators

We show that a given space of splines with sections in a given Extended Chebyshev space gives birth to infinitely many positive linear operators of Schoenberg-type. As a consequence of the properties of Chebyshevian B-spline bases such operators are automatically variation-diminishing. Among other results, we show that the set of two-dimensional spaces they reproduce is stable under knot insertion and dimension elevation, and we establish a simple sufficient condition for convergence.     

Quantum B-splines

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q (quantum)-blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden’s identity.     

Knot insertion algorithms for piecewise polynomial spaces determined by connection matrices

We show that many fundamental algorithms and techniques for B-spline curves extend to geometrically continuous splines. The algorithms, which are all related to knot insertion, include recursive evaluation, differentiation, and change of basis. While the algorithms for geometrically continuous splines are not as computationally simple as those for B-spline curves, they share the same general structure. The techniques we investigate include knot insertion, dual functionals, and polar forms; these prove to be useful theoretical tools for studying geometrically continuous splines.     

A unified approach to B-spline recursions and knot insertion, with application to new recursion formulas

We present a unified approach to and a generalization of almost all known recursion schemes concerning B-spline functions. This includes formulas for the computation of a B-spline's values, its derivatives (ordinary and partial), and for a knot insertion method for B-spline curves. Furthermore, our generalization allows us to derive also some new relations for these purposes.     

A unified approach to B-spline recursions and knot insertion,with application to new recursion formulas

We present a unified approach to and a generalization of almost all known recursion schemes concerning B-spline functions. This includes formulas for the computation of a B-spline's values, its derivatives (ordinary and partial), and for a knot insertion method for B-spline curves. Furthermore, our generalization allows us to derive also some new relations for these purposes.     

B-splines with homogenized knots

The use of homogenized knots for manipulating univariate polynomials by blossoming algorithms is extended to piecewise polynomials. A generalization of the B-spline to homogenized knots is studied. The new B-spline retains the triangular blossoming algorithms for evaluation, differentiation and knot insertion. Moreover, the B-spline is locally supported and a Marsden’s identity exists. Spaces of natural splines and certain polynomial spline spaces with more general continuity properties than ordinary splines have bases of B-splines over homogenized knots. Applications to nonpolynomial splines such as trigonometric and hyperbolic splines are made.     

B-splines with homogenized knots

The use of homogenized knots for manipulating univariate polynomials by blossoming algorithms is extended to piecewise polynomials. A generalization of the B-spline to homogenized knots is studied. The new B-spline retains the triangular blossoming algorithms for evaluation, differentiation and knot insertion. Moreover, the B-spline is locally supported and a Marsden’s identity exists. Spaces of natural splines and certain polynomial spline spaces with more general continuity properties than ordinary splines have bases of B-splines over homogenized knots. Applications to nonpolynomial splines such as trigonometric and hyperbolic splines are made.