含有权参数的二次代数双曲B样条曲线

1引 言二次非均匀B样条曲线,由于结构简单,因而非常方便用于曲线曲面造型[1].但当控制多边形和节点向量给定后,曲线的形状是固定的.如果要调整曲线的形状,可以调整相应的控制顶点或节点向量,这意味着再一次计算曲线方程,计算量也随之增大.此外,二次非均匀B样条曲线不能表示除抛物线以外的圆锥曲线.有理形式的二次非均匀B样条曲线虽然可以表示一些圆锥曲线,权因子也具有调整曲线形状的作用,但权因子几何意义不明显,这对使用者来说是不方便的[2].为此,人们引入不同类型的非多项式、非有理形式的样条.     

NURBS曲线的一个插值性质

本介绍了非均匀有理B样条曲线，并给出了非均匀有理B样条曲线的一个插值性质。     

一类非端点插值B样条曲线降阶的方法

降阶算法是B样条曲线和曲面设计的一个基本算法,它广泛应用于组合曲线,蒙皮或扫描曲面等设计中.Piegl与Tiller曾给出B样条曲线的降阶方法.本文给出了解决更一般的非端点插值B样条曲线降阶的方法.新的方法主要是通过对现有的节点插入方法进行分析,给出了一种端点插值递推公式,并利用此公式对Piegl与Tiller降阶方法加以改进,使之能够解决非端点插值均匀及非均匀B样条曲线的降阶问题.     

局部形状可调整的三次有理B样条插值曲线

利用三次非均匀有理B样条,给出了一种构造局部插值曲线的方法,生成的插值曲线是C2连续的.曲线表示式中带有一个局部形状参数,随着一个局部形状参数值的增大,所给曲线将局部地接近插值点构成的控制多边形.基于三次非均匀有理B样条函数的局部单调性和一种保单调性的准则,给出了所给插值曲线的保单调性的条件.     

带参数的Catmull-Clark细分曲面

在均匀B样条曲线的Lane-Riesenfeld细分算法中,每一步细分可看成是对原控制多边形的"切角"操作.文章通过引入一个参数来控制切角的程度,提出加权的Lane-Riesenfeld算法,并从均匀三次B样条曲线出发,得到光滑性为C~1的单参数曲线细分格式.进一步将该算法推广到任意拓扑的四边形网格上,得到除奇异点外处处C~1的细分曲面(称之为带参数的Catmull-Clark(C-C)细分曲面).格式中的参数在一定范围内调整时,可以使细分曲线/曲面不同程度地逼近控制多边形/控制网格,具有较好的灵活性.     

二元三次样条空间S_3~(1,2)(△_(mn)~(2))的样条拟插值

本文首先利用由两组具有局部最小支集的样条所组成的基函数,构造非均匀2型三角剖分上二元三次样条空间S1,23(△(2)mn)的若干样条拟插值算子.这些变差缩减算子由样条函数B1ij支集上5个网格点或中心和样条函数B2ij支集上5个网格点处函数值定义.这些样条拟插值算子具有较好的逼近性,甚至算子Vmn(f)能保持近最优的三次多项式性.然后利用连续模,分析样条拟插值算子Vmn(f)一致逼近于充分光滑的实函数.最后推导误差估计.     

空间圆柱螺旋线的NURBS表示

O．引言用B样条方法或B6zier方法来表示自由曲线、曲面，是在CAD／CAM技术中广泛使用的数学手段．但是由于它们都不能精确地表示除抛物线或抛物面以外的圆锥曲线与初等二次曲面，因此近年来，另一种形式的参数样条-一参数有理多项式方法占据了主导地位．非均匀有理B样条（简称NURBS）已被国际标准组织（ISO）于1991年正式颁布为关于工业产品几何定义的STEP国际标准，将其作为定义产品形状的唯一数学方法．越来越多的CAD系统采用NURBS曲线与曲面来建立图形库，研究各种曲线与曲面的NURBS表示无疑是很有意义的．在描述圆锥曲线…     

两邻接B样条曲面的G1连续条件

本文得到了非均匀重内节点邻接B样条曲面间G1连续的充要条件 ,给出了一类G1连续的充分条件 ;基于对B样条曲线参数连续性的分析 ,本文着重给出了这类充分条件成立的内在约束 ,即对公共边界控制顶点的约束条件 .     

带形状参数的二次三角多项式曲线

<正>0引言B样条曲线特别是二、三次样条曲线~([1]),因其构造简单使用灵活,广泛应用到工程技术上,在CAGD和CG中占有重要的地位.但其有一定的缺点,如不能表示圆锥曲线等.非均匀有理样条虽然可以表示圆锥曲线,但有求导求积过于复杂,权因子选取不清楚等缺点~([2-4]).三角样条和三角多项式在理论和实际应用中都具有重要意义。文献[4]给出了三角样条,文献[5]构造了C~3连续三角多项式样条曲线.文献[6]构造了均匀三角多项式B样条     

几种有理插值函数的逼近性质

1　引　　言在曲线和曲面设计中,样条插值是有用的和强有力的工具.不少作者已经研究了很多种类型的样条插值[1,2,3,4].近些年来,有理插值样条,特别是三次有理插值样条,以及它们在外型控制中的应用,已有了不少工作[5,6,7].有理插值样条的表达式中有某些参数,正是由于这些参数,有理插值样条在外型控制中充分显示了它的灵活性;但也正是由于这些参数,使它的逼近性质的研究增加了困难.因此,关于有理插值样条的逼近性质的研究很少见诸文献.本文在第二节首先叙述几种典型的有理插值样条,其中包括分母为一次、二次的三次有理插值样条和仅基于函数值…     

NURBS曲线曲面拟合数据点的迭代算法

本文推广了文献[1]的结果,将文献[1]中关于B样条曲线曲面拟合数据点的迭代算法推广至有理形式,给出了无需求解方程组反求控制点及权因子即可得到拟合NURBS曲线曲面的迭代方法．该算法和文献[1]的算法本质上是统一的,而后者恰是前者的一种退化形式．文章还给出了收敛性证明以及一些定性分析．文末的数值实例说明该算法简单实用．     

Curvature-Based Real-Time NURBS Surface Interpolator with Look-Ahead ACC/DEC Control

Real-time parametric surface interpolation is very useful in high-performance machining. A curvature-based NURBS surface interpolator with look-ahead acceleration/deceleration (ACC/DEC) control has been developed. The cutter contact (CC) paths are planned through iso-parametric line method. The CC feedrate profile is optimized using global flexible control strategy and local adaptive ACC/DEC optimization strategy according to the look-ahead algorithm. The real-time surface interpolation algorithm was programmed on Windows XP platform. The stability and efficiency of the proposed interpolation method were verified by a NURBS curve and a NURBS surface. It is shown that the proposed parametric interpolation algorithm can satisfy the high speed and high precision requirements of high-speed CNC systems.     

Shape recovery for sparse‐data tomography

A two‐dimensional sparse‐data tomographic problem is studied. The target is assumed to be a homogeneous object bounded by a smooth curve. A nonuniform rational basis splines (NURBS) curve is used as a computational representation of the boundary. This approach conveniently provides the result in a format readily compatible with computer‐aided design software. However, the linear tomography task becomes a nonlinear inverse problem because of the NURBS‐based parameterization. Therefore, Bayesian inversion with Markov chain Monte Carlo sampling is used for calculating an estimate of the NURBS control points. The reconstruction method is tested with both simulated data and measured X‐ray projection data. The proposed method recovers the shape and the attenuation coefficient significantly better than the baseline algorithm (optimally thresholded total variation regularization), but at the cost of heavier computation.     

Efficient computation of the Gauss-Newton direction when fitting NURBS using ODR

We consider a subproblem in parameter estimation using the Gauss-Newton algorithm with regularization for NURBS curve fitting. The NURBS curve is fitted to a set of data points in least-squares sense, where the sum of squared orthogonal distances is minimized. Control-points and weights are estimated. The knot-vector and the degree of the NURBS curve are kept constant. In the Gauss-Newton algorithm, a search direction is obtained from a linear overdetermined system with a Jacobian and a residual vector. Because of the properties of our problem, the Jacobian has a particular sparse structure which is suitable for performing a splitting of variables. We are handling the computational problems and report the obtained accuracy using different methods, and the elapsed real computational time. The splitting of variables is a two times faster method than using plain normal equations.     

A General Hybrid Optimization Strategy for Curve Fitting in the Non-uniform Rational Basis Spline Framework

In this paper, a general methodology to approximate sets of data points through Non-uniform Rational Basis Spline (NURBS) curves is provided. The proposed approach aims at integrating and optimizing the full set of design variables (both integer and continuous) defining the shape of the NURBS curve. To this purpose, a new formulation of the curve fitting problem is required: it is stated in the form of a constrained nonlinear programming problem by introducing a suitable constraint on the curvature of the curve. In addition, the resulting optimization problem is defined over a domain having variable dimension, wherein both the number and the value of the design variables are optimized. To deal with this class of constrained nonlinear programming problems, a global optimization hybrid tool has been employed. The optimization procedure is split in two steps: firstly, an improved genetic algorithm optimizes both the value and the number of design variables by means of a two-level Darwinian strategy allowing the simultaneous evolution of individuals and species; secondly, the optimum solution provided by the genetic algorithm constitutes the initial guess for the subsequent gradient-based optimization, which aims at improving the accuracy of the fitting curve. The effectiveness of the proposed methodology is proven through some mathematical benchmarks as well as a real-world engineering problem.     

Relation between algebraic and geometric view on nurbs tensor product surfaces

NURBS (Non-Uniform Rational B-Splines) belong to special approximation curves and surfaces which are described by control points with weights and B-spline basis functions. They are often used in modern areas of computer graphics as free-form modelling, modelling of processes. In literature, NURBS surfaces are often called tensor product surfaces. In this article we try to explain the relationship between the classic algebraic point of view and the practical geometrical application on NURBS.     

双三次B样条插值曲面

本文研究三次B样条插值曲面。对于给定的拓扑网格点阵Pi,j。导出了其插值三次B样条曲面的控制顶点,每四个顶点Pi,j,Pi l,j,Pi,j 1,Pi 1,j 1由九个三次B样条曲面片构成,整个曲面是C^2连续的,最后,给出了一个数值实例。     

基于层次T网格的分片孔斯曲面重构

本文提出一种基于任意层次T网格的多项式(PHT)样条空间$S(3,3,1,1,T)$的一个新的曲面重构算法.该算法由分片插值于层次T网格上每个小矩形单元对应4个顶点的16个参数的孔斯曲面形式给出.对于一个给定的T网格和相应基点处的几何信息(函数值,两个一阶偏导数和混合导数值),可得到与$S(3,3,1,1,T)$的PHT样条曲面相同的结果,且曲面表达形式更简单,同时,在离散数据点的曲面拟合中,我们给出了自适应的曲面加细算法.数值算例显示,该自适应算法能够有效的拟合离散数据点.     

一种二次保形插值参数曲面

1．引言保形插值是工业设计和制造中经常遇到的问题，有关这方面的研究已有许多文献【‘－u1．设n二Fx；，yi，人，川7一0，1，…；n；j＝0；l，…，。；x；＜x；＋1．l＝0，1，…．n—1；yi＜的十；，J二O，L…，。一卫｝是一给定的数据集，Cadson和［ltsch、Beatson和zejerJ－1985年分别提出的方法只保持被插数据集的轴向单调性；Dodd和Roulier等人于1983和1987年提出的方法只保持被插点集网格线上的轴向凸凹性和单调性；Constantini和FOntanella于1990年提出的方法可保持被插点集在所有于区域的边界及共内部的轴向凸凹性和单调性；…