平面Bzier曲线的凸性定理的一个证明

平面Bzier曲线的凸性定理是计算几何中一个重要的定理.最近,苏步青、刘鼎元在[1,2]中给出了凸性定理的证明. 本文的目的是从Bezier曲线的一阶导矢与二阶导矢的几何作图出发,给出凸性定理的另一个证明.     

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.     

Bézier曲线降多阶逼近的一种方法

文献[1,2]讨论了Bezier曲线一次降多阶逼近问题,得到了很好的结果.文献[1]利用广义逆矩阵得到不保端点插值的降多阶逼近曲线的控制顶点的表达式.但却没有得到带端点任意阶插值条件的降多阶逼近曲线的控制顶点的表达式.文献[2]得到了带端点任意阶插值的降多阶逼近曲线的控制顶点的解析表达式.本文首先给出两Bezier曲线间距离的定义;然后根据降阶曲线与原曲线间的距离最小,分别得到了用矩阵表示的不保端点插值和保端点任意阶插值的降多阶逼近曲线的控制顶点的显示表达式.所给数值例子显示,用本文方法得到的降多阶逼近曲线对原曲线有很好的逼近效果.     

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.     

四次C-曲线的性质及其应用

以1,t,t2,t3,…为基底的Bézier曲线和B样条曲线是构造自由曲线、曲面强有力的工具.但是它们不能精确地表示某些圆锥曲线如圆弧、椭圆等,也不能精确地表示正弦曲线.本文利用一组新的基底sint,cost,t2,t,1,构造了两条新的曲线,这两条曲线依赖于参数α>0.当α→0时极限分别是四次Bézier曲线和四次B样条曲线,称之为四次C-曲线:四次C-Bézier曲线和四次C-B样条曲线.它们具有一般Bézier曲线和B样条曲线的性质:如端点插值,凸包,离散等,还可以精确的表示圆弧、椭圆及正弦曲线.作为应用,文章最后给出了四次C-Bézier曲线表示正弦曲线的条件.     

三次Pythagorean Hodograph 曲线在给定总弧长下的$G^2$拼接

Pythagorean-hodograph (PH)曲线因其在弧长和等距线计算方面的优势而被广泛应用于曲线建模中.本文讨论了在总弧长约束下的三次PH曲线$G^2$连续拼接问题.具体地说,给定两个端点和一个拼接点,构造两条三次PH曲线,使其在指定总弧长下插值两个端点,并且在连接点处是$G^2$连续的.这也可以看作是一个曲线延拓问题.根据三次PH曲线的弧长公式和$G^2$连续条件,最终将问题转化为了一个带有约束的极小值问题,同时我们给出了几个具体例子来说明该方法.     

开口曲线上奇异积分算子的一些性质

本文讨论开口曲线上奇异积分算子的稳定性、端点行为和可逆性关系,证明了当曲线的端点为非特异节点时这类奇异积分算子是稳定的, 对该类算子在端点的性态进行了确切描述, 以及通过对奇异积分算子相互关系的分析,给出了算子在其上可逆的函数空间或函数集以及对应的 逆映射. 同时,文章列举了上述性质在奇异积分方程方面的一些应用.     

Dirac方程的特征函数的迹公式

本文研究具有周期有限带位势的Dirac算子, 利用Dirac算子与单值算子的交换性,定义Bloch函数和乘子曲线, 进而获得揭示Bloch函数与位势间内在关系的Dubrovin-Novikov型公式; 通过复球面上的留数计算, 最终分别得到相应于谱带左端点、右端点以及双侧端点的特征函数的迹公式.     

广义Bezier曲线与曲面在连接中的应用

通常的贝齐尔曲线、曲面，在其端点或边界只具有ＧＣ＾１阶插值性。本文在保持通常贝齐尔曲线、曲面性质的基础上，定义了一种广义的贝齐尔曲线、曲面，使其在曲线段的端点和曲面片的边界具有高阶光滑插值性，它可方便地光滑连接两条参数型的曲线段和两张以上参数型曲面片，并且连接方式是ＧＣ＾ｒ的，所以广义贝齐尔曲线、曲面在计算机辅助设计应用中更具有独特的意义。     

广义Bézier曲线与曲面在连接中的应用

通常的贝齐尔（Ｂｅｚｉｅｒ）曲线、曲面，在其端点或边界只具有ＧＣ１阶插值性．本文在保持通常贝齐尔曲线、曲面性质的基础上，定义了一种广义的贝齐尔曲线、曲面，使其在曲线段的端点和曲面片的边界具有高阶光滑插值性，它可方便地光滑连接两条参数型的曲线段和两张以上参数型曲面片，并且连接方式是ＧＣｒ（ｒ≥１）的．所以广义贝齐尔曲线、曲面在计算机辅助设计应用中更具有独特的意义．     

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.     

Discussion on relationship between minimal energy and curve shapes

Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics. However, our testing examples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes, the study is carried out by constructing a cubic Hermite curve with satisfactory shape. The cubic Hermite curve interpolates the positions and tangent vectors of two given endpoints. Computer simulation technique has become one of the methods of scientific discovery, the study process is carried out by numerical computation and computer simulation technique. Our result shows that： （1） cubic Hermite curves cannot be constructed by solely minimizing the strain energy; （2） by adoption of a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes, which is a modification of strain energy model. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties： （1） it is easy to compute; （2） it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included.     

The total mixed curvature of open curves in $$E^3$$

The total mixed curvature of a curve in \(E^3\) is defined as the integral of \(\sqrt{\kappa ^2+\tau ^2}\), where \(\kappa \) is the curvature and \(\tau \) is the torsion. The total mixed curvature is the length of the spherical curve defined by the principal normal vector field. We study the infimum of the total mixed curvature in a family of open curves whose endpoints and principal normal vectors at the endpoints are prescribed. In our previous works, we studied similar problems for the total absolute curvature, which is the length of the spherical curve defined by the unit tangent vector, and for the total absolute torsion, which is the length of the spherical curve defined by the binormal vector.     

Curve fitting with a Sobolev gradient method

Consider the problem of constructing a mathematical representation of a curve that satisfies constraints such as interpolation of specified points. This problem arises frequently in the context of both data fitting and Computer Aided Design. We treat the most general problem: the curve may or may not be constrained to lie in a plane; the constraints may involve specified points, tangent vectors, normal vectors, and/or curvature vectors, periodicity, or nonlinear inequalities representing shapepreservation criteria. Rather than the usual piecewise parametric polynomial (B-spline) or rational (NURB) formulation, we represent the curve by a discrete sequence of vertices along with first, second, and third derivative vectors at each vertex, where derivatives are with respect to arc length. This provides third-order geometric continuity and maximizes flexibility with an arbitrarily large number of degrees of freedom. The free parameters are chosen to minimize a fairness measure defined as a weighted sum of curve length, total curvature, and variation of curvature. We thus obtain a very challenging constrained optimization problem for which standard methods are ineffective. A Sobolev gradient method, however, is particularly effective. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of the Mean of a Wiener Sheet

A shifted Wiener sheet is observed above a decreasing curve Γ. By the help of a direct discrete approach and under weaker assumptions than in the paper of Arató [Comput. Math. Appl. 33 (1997), 13–25], an explicit formula is derived for the maximum likelihood estimator of the shift parameter. This estimator is a weighted linear combination of the values at the endpoints of the curve Γ and weighted integrals of the observed process and its normal derivative along the curve Γ. This revised version was published online in June 2006 with corrections to the Cover Date.     

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．     

Some properties of singular integral operators over an open curve Dedicated to the 85th birthday of Prof. Lu Jianke

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated.     

Some properties of singular integral operators over an open curve Dedicated to the 85th birthday of Prof.Lu Jinake

In this paper, the stability properties, the endpoint behavior and the invertible relations of Cauchy-type singular integral operators over an open curve are discussed. If the endpoints of the curve are not special, this type of operators are proved to be stable. At the endpoints, either the singularity or smoothness of the operators are exactly described. And the function sets or spaces on which the operators are invertible as well as the corresponding inverted operators are given. Meanwhile, some applications for the solution of Cauchy-type singular integral equations are illustrated.