C<Superscript>2</Superscript> quartic spline surface interpolation

This paper discusses the problem of constructing C² quartic spline surface interpolation. Decreasing the continuity of the quartic spline to C² offers additional freedom degrees that can be used to adjust the precision and the shape of the interpolation surface. An approach to determining the freedom degrees is given, the continuity equations for constructing C² quartic spline curve are discussed, and a new method for constructing C² quartic spline surface is presented. The advantages of the new method are that the equations that the surface has to satisfy are strictly row diagonally dominant, and the discontinuous points of the surface are at the given data points. The constructed surface has the precision of quartic polynomial. The comparison of the interpolation precision of the new method with cubic and quartic spline methods is included.     

局部可调整C2参数四次插值曲线构造

讨论了局部可调整C2参数四次样条曲线的构造问题.将四次样条曲线降为C2连续可提供自由度用于控制曲线的形状.给出了一个确定自由度的局部化方法.首先用二次样条函数方法局部化地在每个数据点处确定一个切矢量,数据点和切矢量大致决定了四次样条曲线的形状.每段曲线上的自由度由极小化该段样条曲线的变化率确定.对样条曲线上不理想的部分,为其重新定义理想运动矢量,若曲线沿理想运动矢量方向变化可形成理想轨迹,用曲线导矢量和运动矢量的向量叉乘平方的积分定义目标函数,曲线的不理想的部分通过极小化目标函数进行修改.最后,用实例对新方法和其他几种方法构造的曲线形状进行了比较,并给出了对曲线采用向量叉乘技术定义目标函数作局部调整的效果.     

具有局部性质的球面插值样条曲线的构造

高维球面样条曲线拟合技术在计算机动画和惯性导航等领域都受到广泛地关注.实际中常需球面曲线插值给定的数据点,并要求曲线具有一定的连续性和良好的局部性质.此前的方法存在一定的局限性.为此,基于球面Bézier曲线,提出了一种仅利用插值点位置信息便可在任意维空间中构造C 2球面插值样条曲线的新方法.首先,通过映射拟合出了插值...     

高精度三次参数样条曲线的构造

构造参数样条曲线的关键是选取节点，该文讨论了GC^2三次参数样条曲线需满足的连续性方程，提出了构造GC^2三次参数样条曲线的新方法，在讨论了平面有序五点确定一组三次多项式函数曲线，平面有序六点唯一确定一条三次多项式函数曲线的基础上，提出了计算相邻两区间上的节点的算法，构造的插值曲线具有三次多项式函数精,该文还以实例对新方法与其它方法构造的插值曲线的精度进行了比较。     

逼近三次B 样条导矢曲线的四次Hermite 插值样条

给出了形状可调的四次Hermite 插值样条曲线的构造方法。四次样条曲线可提供额 外的自由度用于调整曲线具有合理形状。利用导矢逼近使得四次Hermite 样条曲线具有与三次B 样条曲线相似的形状。通过最小化曲线间的导矢误差给出了确定自由度的方法,提出了四次 Hermite 插值样条曲线的构造方法。该方法增加了自由度控制曲线形状能更好满足保形要求。最 后以实例对构造的四次Hermite 样条曲线和标准三次Hermite 插值样条曲线进行了比较。     

A B-spline collocation method for solving the incompressible Navier-Stokes equations using an ad hoc method: the Boundary Residual method

Collocation methods using piece-wise polynomials, including B-splines, have been developed to find approximate solutions to both ordinary and partial differential equations. Such methods are elegant in their simplicity and efficient in their application. The spline collocation method is typically more efficient than traditional Galerkin finite element methods, which are used to solve the equations of fluid dynamics. The collocation method avoids integration. Exact formulae are available to find derivatives on spline curves and surfaces. The primary objective of the present work is to determine the requirements for the successful application of B-spline collocation to solve the coupled, steady, 2D, incompressible Navier-Stokes and continuity equations for laminar flow. The successful application of B-spline collocation included the development of ad hoc method dubbed the Boundary Residual method to deal with the presence of the pressure terms in the Navier-Stokes equations. Historically, other ad hoc methods have been developed to solve the incompressible Navier-Stokes equations, including the artificial compressibility, pressure correction and penalty methods. Convergence studies show that the ad hoc Boundary Residual method is convergent toward an exact (manufactured) solution for the 2D, steady, incompressible Navier-Stokes and continuity equations. C¹ cubic and quartic B-spline schemes employing orthogonal collocation and C² cubic and C³ quartic B-spline schemes with collocation at the Greville points are investigated. The C³ quartic Greville scheme is shown to be the most efficient scheme for a given accuracy, even though the C¹ quartic orthogonal scheme is the most accurate for a given partition. Two solution approaches are employed, including a globally-convergent zero-finding Newton's method using an LU decomposition direct solver and the variable-metric minimization method using BFGS update.     

任意三角形网格的基于二元四次箱样条分片C1曲面

提出一种以任意三角剖分为控制网格的二元箱样条曲面算法.二元三方向剖分是方向最少的三角剖分,建立在其上的二元三向四次箱样条在CAGD等领域有着广泛的应用.其规范的箱样条曲面计算仅适用于控制点的价数均为6的网格.从规范的算法出发,提出了一种任意价数控制网格的曲面计算算法,并对算法的连续性等进行了详细的分析.生成的曲面具有保凸性,且是分片C1连续的.该算法可进行3D离散点全局或局部插值,并可应用于3D曲面重构等领域.     

满足数据点切向约束的二次B样条插值曲线

给出一种二次B样条曲线插值方法.利用数据点的参数化和节点向量的自由度,构造在各数据点满足切向约束的二次B样条插值曲线,直观地控制插值曲线达到预期形状.用文中方法构造插值曲线是一个递推过程,不必预先确定数据点参数值和节点向量、不必解线性方程组,而是在插值过程中根据数据点及其切向的约束条件递推地确定数据点的参数值、节点和控制顶点.该文方法允许插值曲线各段的连接点与数据点不一致,以使得二次B样条插值曲线的形状更自然.而且在满足数据点切向约束的条件下,还可利用节点进一步调控插值曲线的形状.另外,用文中方法构造的二次B样条插值曲线对于数据点的改变具有较好的局部性质.文中最后给出一些例子将该文方法与其它一些插值方法进行比较,实验结果表明,该文方法是有效的.     

一类加权有理四次插值样条曲线的形状控制

将插值曲线约束于给定的区域之内是曲线形状控制中的重要问题。利用带导数的和不带导数的分母为线性的有理四次插值样条构造了一类新的加权有理四次插值样条函数,插值函数具有简单的显示表示,这类新的插值样条中含有权系数,因而增加了处理问题的灵活性,给约束控制带来了方便。给出了将该种插值曲线约束于给定的折线、二次曲线之上、之下或之间的充分条件。证明了满足约束条件的加权有理样条的存在性。     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.     

基于局部梯度特征的自适应多结点样条图像插值

为了获得质量更好的插值图像,提出了一种新的C\\+2连续的支撑区间为(-2,2)的三次多结点样条插值核函数.通过增加结点带来的自由度构造了多结点样条插值公式;分析了在适当的边界条件和约束下三次多结点样条插值的逼近阶;将一维多结点样条插值算法推广到二维,建立了用于图像数据的插值公式;如果忽视图像的局部特征,通常双三次多结点样条插值图像的边缘会有模糊的现象,为此,对多结点样条插值应用逆梯度,得到了自适应多结点样条插值算法;实验所得误差图像和实验所得图像的峰值信噪比也证实了用自适应多结点样条插值算法重建的图像具有更高的质量.     

The Cubic Trigonometric Automatic Interpolation Spline

A class of cubic trigonometric interpolation spline curves with two parameters is presented in this paper. The spline curves can automatically interpolate the given data points and become C2 interpolation curves without solving equations system even if the interpolation conditions are fixed. Moreover, shape of the interpolation spline curves can be globally adjusted by the two parameters. By selecting proper values of the two parameters, the optimal interpolation spline curves can be obtained.      

三次Hermite参数曲线与曲面的扩展

在给定插值条件时,标准三次Hermite参数曲线与曲面的形状无法调整。为克服标准三次Hermite参数曲线与曲面的不足,首先通过提高基函数次数的方法给出了一种带形状参数的四次Hermite基函数,然后生成了相应的带形状参数的四次Hermite参数曲线与曲面。所生成的曲线与曲面是标准三次Hermite参数曲线与曲面的扩展,不仅与标准三次Hermite曲线与曲面具有完全相同的性质,而且当插值条件给定时,其形状可通过修改形状参数的取值进行局部或整体调节,为插值曲线与曲面的构造提供了一种新方法。     

基于四次B样条的曲线逼近算法

考虑到插值算法增减节点困难,传统逼近算法精度不够等缺点,有文献提出一种基于三次B样条的曲线逼近算法。该算法通过迭代逼近,提高了计算速度与精度。在系统研究此算法的基础上,将该算法推广到四次B样条,使其具有三阶可导性,并给出该算法收敛性的理论证明。最后用该算法对常用函数进行逼近效果实验。结果表明,所提出的四次B样条的曲线逼近算法收敛速度更快,且能够满足更高精度的实际工业生产需要。     

一种B样条表面插值控制点的缩减方法

研究了从给定节点向量中选择节点进行B样条曲线插值的方法，并将此方法应用到行数据点不相同的B样条曲面插值，得到了一个通过对行节点矢量调整传递的曲面插值方法，理论分析和实验表明该方法可大量减少曲面控制点的数目．     

基于改进Catmull-Rom样条的图像缩放算法

Catmull-Rom样条函数具有C1连续性质与快速插值能力,其每一个节点的导数值都是由其相邻的2个节点确定。文中通过相邻4个节点确定样条上各节点的导数值,由此重新构造了Catmull-Rom样条以进一步减少插值误差。实验结果表明,将这种改进的Catmull-Rom样条用于图像的缩放处理,插值误差明显减少,插值效率高,适用于对图像缩放的质量与速度都有较高要求的场合。     

C~2-(C~3-) Continuous Interpolation Spline Curve and Surface

Free-formed or sculptured surfaces in engineering products are frequently constructed from a set of measured 3D data points.C^2-(C^3-)continuity approach is important in this field.This paper presents a method of rectangular interpolation of given 3D data array which is regularly arranged.The interpolation surface which is constructed by tensor product has dsirable properties(second-order or third-order continuity,locality)and is implemented and adjusted easily,Higher order continuity methods are also briefly discussed.     

任意三角形网格的基于二元四次箱样条分片C1曲面

提出一种以任意三角剖分为控制网格的二元箱样条曲面算法.二元三方向剖分是方向最少的三角剖分,建立在其上的二元三向四次箱样条在CAGD等领域有着广泛的应用.其规范的箱样条曲面计算仅适用于控制点的价数均为6的网格.从规范的算法出发,提出了一种任意价数控制网格的曲面计算算法,并对算法的连续性等进行了详细的分析.生成的曲面具有保凸性,且是分片C¹连续的.该算法可进行3D离散点全局或局部插值,并可应用于3D曲面重构等领域.     

C3连续的单位四元数插值样条曲线

目的 构造一类C³连续的单位四元数插值样条曲线,证明它的插值性和连续性,并把它应用于刚体关键帧动画设计中。方法 利用R³空间中插值样条曲线的5次多项式调配函数的累和形式构造了S³空间中单位四元数插值样条曲线,它不仅能精确通过一系列给定的方向,而且能生成C³连续的朝向曲线。结果 与Nielson的单位四元数均匀B样条插值曲线的迭代构造方法相比,所提方法避免了为获取四元数B样条曲线控制顶点对非线性方程组迭代求解的过程,提高了运算效率;与单位四元数代数三角混合插值样条曲线的构造方法（Su方法）相比,所提方法只用到多项式基,运算速度更快。本例中创建关键帧动画所需的时间与Nielson方法和Su方法相比平均下降了73%和33%。而且,相比前两种方法,所提方法产生的四元数曲线连续性更高,由C²连续提高到C³连续,这意味着动画中刚体的朝向变化更加自然。结论 仿真结果表明,本文方法对刚体关键帧动画设计是有效的,对实时性和流畅性要求高的动画设计场合尤为适用。     

基于参数样条插值的同坡曲面构造方法

传统的同坡曲面构造方法都是在导线方程为已知的前提下进行的。然而在实际工程中,导线方程往往是很难得到的,只能通过测量得知导线通过一列数据点。针对这一问题,给出了一种实际工程中同坡曲面的构造方法,该法首先根据测量数据点,利用三次参数样条曲线插值方法构造出同坡曲面的导线方程,然后再从同坡曲面的形成原理入手建立其参数方程,最后通过实例表明该方法是可行有效的。