利用近似能量极小构造平面C1三次Hermite插值曲线

研究了利用近似能量极小构造平面$C^1$三次Hermite插值曲线的方法.该方法的主要的目是求出$C^1$三次Hermite插值曲线的最佳切矢.通过将应变能、曲率变化能和组合能的近似函数极小化,得到了求解最佳切矢的线性方程组.通过求解发现,近似曲率变化能极小不存在唯一解, 而近似应变能极小和近似组合能极小由于方程系统的系数矩阵为严格对角占优故都存在唯一解.最后, 通过实例表明了本文方法构造平面$C^1$三次Hermite插值曲线的有效性.     

导数振荡与应变能极小的平面分段三次参数Hermite插值

由于分段三次参数Hermite插值的切矢往往被作为变量,故可对其进行优化以使得构造的插值曲线满足特定的要求.为了构造兼具保形性与光顺性的平面分段三次参数Hermite插值曲线,给出了一种通过同时极小化导数振荡和应变能来确定切矢的方法.首先以导数振荡函数和应变能函数为双目标建立了切矢满足的方程系统;然后证明了方程系统存在唯一解,并给出了解的具体表达式;最后给出了误差分析,并通过数值算例表明方法的有效性.结果表明,相对于导数振荡极小化方法和应变能极小化方法,所提出的导数振荡和应变能极小化方法同时兼顾了平面分段三次参数Hermite插值曲线的保形性和光顺性.     

参数保形插值中决定切矢的方法

由分段三次参数多项式曲线拼合成的Ｃ１插值曲线的形状与数据点处的切矢有很大关系．基于对保形插值曲线特点的分析，本文提出了估计数据点处切矢的一种方法：采用使构造的插值曲线的长度尽可能短的思想估计数据点处的切矢，并且通过四组有代表性的数据对本方法和已有的三种方法进行了比较．     

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.     

基于离散点和法矢的曲线重构算法

This paper presents a curve reconstruction algorithm based on discrete data points and normal vectors using B-splines.The proposed algorithm has been improved in three steps:parameterization of the discrete data points with tangent vectors,the B-spline knot vector determination by the selected dominant points based on normal vectors,and the determination of the weight to balancing the two errors of the data points and normal vectors in fitting model.Therefore,we transform the B-spline fitting problem into three sub-problems,and can obtain the B-spline curve adaptively.Compared with the usual fitting method which is based on dominant points selected only by data points,the B-spline curves reconstructed by our approach can retain better geometric shape of the original curves when the given data set contains high strength noises.     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

带有能量约束的类B\'ezier 曲线的构造

本文提出了一类创新的类Bernstein基函数,这是经典Bernstein基函数的推广.我们分析了这类基函数的性质并定义了带有两个形状参数的类B\''ezier曲线.这类基函数和类B\''ezier曲线分别具有类似三次Bernstein基函数和三次B\''ezier曲线的性质.进一步地,我们构造了带有能量约束的类B\''ezier曲线并考虑了带有极小能量约束的$C^1$和$G^1$ Hermite插值问题.最后,给出了一些具有代表性的例子来说明我们方法的应用性和有效性.     

Planar G2 Hermite interpolation with some fair,C-shaped curves

G² Hermite data consists of two points, two unit tangent vectors at those points, and two signed curvatures at those points. The planar G² Hermite interpolation problem is to find a planar curve matching planar G² Hermite data. In this paper, a C-shaped interpolating curve made of one or two spirals is sought. Such a curve is considered fair because it comprises a small number of spirals. The C-shaped curve used here is made by joining a circular arc and a conic in a G² manner. A curve of this type that matches given G² Hermite data can be found by solving a quadratic equation. The new curve is compared to the cubic Bézier curve and to a curve made from a G² join of a pair of quadratics. The new curve covers a much larger range of the G² Hermite data that can be matched by a C-shaped curve of one or two spirals than those curves cover.     

用边界曲线构造C~1 Coons曲面确定扭矢的方法

本文讨论了由四条边界曲线构造Ｃ１Ｃｏｏｎｓ曲面的问题，给出了确定角点扭矢的新方法．该方法沿四边形两对角线方向构造两条四次多项式曲线，每个角点处的扭矢，由一条四次曲线和两条边界曲线确定．跨界切矢由三次埃尔米特插值方法定义．文中还给出了一个用新方法构造曲面的实例．     

以节点与权因子修改为基础的4阶NURBS受限形状控制

改变k阶NURBS曲线的节点,会产生一个单参数NURBS曲线族,该曲线族的包络是用相同控制顶点定义的k-a阶NURBS曲线,这里a是所改变的节点的重数.论文运用这项理论结果,提出了几种建立在修改一个节点与两个连续权因子基础上的4阶NURBS形状控制方法,该方法要受一定的位置与切线方向的约束.     

可调形三次三角Cardinal插值样条曲线

在三次Cardinal插值样条曲线的基础上,引入了三角函数多项式,得到一组带调形参数的三次三角Cardinal样条基函数,以此构造一种可调形的三次三角Cardinal插值样条曲线.该插值样条可以精确表示直线、圆弧、椭圆以及自由曲线,改变调形参数可以调控插值曲线的形状.该插值样条避免了使用有理形式,其表达式较为简洁,计算量也相对较少,从而为多种线段的构造与处理提供了一种通用与简便的方法.     

A class of algebraic-trigonometric blended splines

This paper presents a new kind of algebraic-trigonometric blended spline curve, called xyB curves, generated over the space {1,t,sint,cost,sin²t,sin³t,cos³t}. The new curves not only inherit most properties of usual cubic B-spline curves in polynomial space, but also enjoy some other advantageous properties for modeling. For given control points, the shape of the new curves can be adjusted by using the parameters x and y. When the control points and the parameters are chosen appropriately, the new curves can represent some conics and transcendental curves. In addition, we present methods of constructing an interpolation xyB-spline curve and an xyB-spline curve which is tangent to the given control polygon. The generation of tensor product surfaces by these new spline curves is straightforward. Many properties of the curves can be easily extended to the surfaces. The new surfaces can exactly represent the rotation surfaces as well as the surfaces with elliptical or circular sections.     

球面上的几何连续插值

In this paper, a new method for geometrically continuous interpolation in spheres is proposed. The method is entirely based on the spherical B′ezier curves defined by the generalized de Casteljau algorithm. Firstly we compute the tangent directions and curvature vectors at the endpoints of a spherical B′ezier curve. Then, based on the above results, we design a piecewise spherical B′ezier curve with G 1 and G 2 continuity. In order to get the optimal piecewise curve according to two different criteria, we also give a constructive method to determine the shape parameters of the curve. According to the method, any given spherical points can be directly interpolated in the sphere. Experimental results also demonstrate that the method performs well both in uniform speed and magnitude of covariant acceleration.     

Curvature variation minimizing cubic Hermite interpolants

In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solution is confirmed. The approach is combined with strain energy minimization in order to obtain G¹ cubic interpolatory spline.     

平面三次Pythagorean-Hodograph Hyperbolic曲线

本文基于Pythagorean-hodograph (PH)曲线和代数双曲线的良好几何特性,构造了Pythagorean-Hodograph Hyperbolic (PH-H)曲线,并给出了PH-H曲线的定义以及相应性质.同时,分别利用Hyperbolic基函数和Algebraic Hyperbolic (AH) B\''ezier基函数,得到了平面三次AH B\''ezier曲线为PH曲线的两个不同的充要条件.此外,三次PH-H曲线也被用于求解具有确定解的$G^1$ Hermite插值问题.文中给出了具体实例来说明我们的方法.     

Stability of critical shapes for the drag minimization problem in Stokes flow

We study the stability of some critical (or equilibrium) shapes in the minimization problem of the energy dissipated by a fluid (i.e. the drag minimization problem) governed by the Stokes equations. We first compute the shape derivative up to the second order, then provide a sufficient condition for the shape Hessian of the energy functional to be coercive at a critical shape. Under this condition, the existence of such a local strict minimum is then proved using a precise upper bound for the variations of the second order shape derivative of the functional with respect to the coercivity and differentiability norms. Finally, for smooth domains, a lower bound of the variations of the drag is obtained in terms of the measure of the symmetric difference of domains.     

带形状参数的代数三角样条曲线曲面的构造（英文）

The construction of trigonometric B-spline curves with shape parameters has become the hotspot in computer aided geometric design.However,the shape parameters of the curves and surfaces are all global parameters and only meet with C~2 continuity in some previous papers.In order to provide more flexible approaches for designers,the algebraic and trigonometric spline(AT-spline) curves and surfaces are constructed as a generalization of the traditional cubic uniform B-spline curves and surfaces.AT-spline curves and surfaces not only inherit the properties of trigonometric B-spline curves,but also exhibit better performance when adjusting its local shapes through two shape parameters.Particularly,the AT-spline rotational surfaces with two local shape parameters are presented.When the shape parameters take special value,it can accurately represent the conic curve and surface.     

与给定多边形相切的可调广义Ball闭曲线

讨论了与给定控制多边形相切的分段三次、五次和六次可调广义Ball曲线的构造方法,所构造的曲线分别是C1,C2和C3连续的,而且对切线多边形是保形的.曲线上的所有广义Ball曲线段的控制点由切线多边形的顶点直接计算产生.给出了在保持公共连接点处相应连续的情况下,内控制点的活动范围.曲线可以在一定范围内做局部修改.计算实例...     

过三次曲线上一点的切线的求法

通过对一道求切线例题的讨论,引申出求过一条三次曲线上一点的切线的求法,进而推出求解这类问题的一个一般性结论.     

Optimization of arches using genetic algorithm

An optimization procedure is presented for the minimum weight and strain energy optimization for arch structures subjected to constraints on stress, displacement and weight responses. Both thickness and shape variables defining the natural line of the arch are considered. The computer program which is developed in this study can be used to optimize thick, thin and variable thickness curved beams/arches. An automated optimization procedure is adopted which integrates finite element analysis, parametric cubic spline geometry definition, automatic mesh generation and genetic algorithm methods. Several examples are presented to illustrate optimal arch structures with smooth shapes and thickness variations. The changes in the relative contributions of the bending, membrane and shear strain energies are monitored during the whole process of optimization.