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。     

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．     

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。     

给定极点的有理函数插值序列的收敛性

设Γ∈Ｃ（１，α），α＞０．Ｇ是复平面上以Γ为边界的有界单连通区域．本文考虑了极点位于G外部，以广义Ｆａｂｅｒ-Ｄzｒｂａsｊａｎ有理函数的零点为插值结点的Ｌａｇｒａｎｇｅ插值有理函数序列对Ａ（G）和Ｅ^ｑ（Ｇ）（１＜ｑ＜＋∞）中函数的一致逼近和平均逼近阶的估计．     

广义Bézier曲线的形状修改

根据广义Bézier曲线的性质,提出了通过调整参数α,β和端点的目标导矢D0,Dn及端点目标二阶导矢E0,En的方法,使曲线插值目标点和在端点具有已给切矢或二阶导矢.这种方法对于曲线的交互设计以及过渡曲线的设计具有重要的意义.     

曲线、曲面的广度Offset及其性质

针对广义offset在实际中的广泛应用，给出一种曲线、曲面的广义offset的定义，并讨论它们的性质．     

用边界曲线构造C^1Coons曲面确定扭矢的方法

本文讨论了由四条边界曲线构造C^1Coons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．     

复杂外形设计中的C^1,C^2样条插值曲面

根据几种复杂外形设计的特点，木文构造了三角形域上Ｓ１２样条插值曲面，三角形域上的Ｃ２超限插值曲面，矩形参数域上Ｃ２超限插值曲面和任意四边形域上双三次Ｃ１，Ｃ２样条插值曲面，给出了一类有效的边界条件确定方法．同时，算法皆已应用到人体外形描述和飞机外形设计中．     

对f：Csmooth→Ssmooth的一般二重点公式

本文应用Ｌａｋｓｏｖ和Ｆｕｌｔｏｎ发展的方法，对态射ｆ：Ｃ→Ｓ得到了Ｆｕｌｔｏｎ（１）中二重点公式的一个推广，此处Ｃ，Ｓ分别是光滑的曲线和曲面。在文章的后一部分对得到的公式作了一些评注。     

有理曲线和曲面的分片多项式逼近

本文利用摄动的思想,以摄动有理曲线（曲面）的系数的无穷模作为优化目标,给出了用多项式曲线（曲面）逼近有理曲线（曲面）的一种新方法．同以前的各种方法相比,该方法不仅收敛而且具有更快的收敛速度,并且可以与细分技术相结合,得到有理曲线与曲面的整体光滑、分片多项式的逼近．     

关于曲面的有界性及第二类曲面积分的教学实践

给出了高等数学范畴的曲面有界性定义;总结了对高等数学的教学难点之一,第二类曲面积分的教学实践,使得在解决这一老大难问题时思路清晰,可操作性强,教学效果较好.     

Judging or setting weight steady-state of rational Bezier curves and surfaces

Many works have investigated the problem of reparameterizing rational B~zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.     

旋转曲面面积的曲线积分表示

利用旋转曲面方程,以及曲面积分和曲线积分的计算方法,可将旋转曲面的面积通过第一型曲线积分表示出来并进行计算.     

A novel generalization of Bézier curve and surface

A new formulation for the representation and designing of curves and surfaces is presented. It is a novel generalization of Bézier curves and surfaces. Firstly, a class of polynomial basis functions with n$n$ adjustable shape parameters is present. It is a natural extension to classical Bernstein basis functions. The corresponding Bézier curves and surfaces, the so-called Quasi-Bézier (i.e., Q-Bézier, for short) curves and surfaces, are also constructed and their properties studied. It has been shown that the main advantage compared to the ordinary Bézier curves and surfaces is that after inputting a set of control points and values of newly introduced n$n$ shape parameters, the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters, when the control polygon is maintained. The Q-Bézier curve and surface inherit the most properties of Bézier curve and surface and can be more approximated to the control polygon. It is visible that the properties of end-points on Q-Bézier curve and surface can be locally controlled by these shape parameters. Some examples are given by figures.     

基于几何连续的AT-β-Spline曲线曲面的构造

为了更好地修改给定的样条曲线曲面,构造了满足几何连续的带两类形状参数的代数三角多项式样条曲线曲面,简称为AT-β-Spline.这种代数三角曲线曲面不仅具有普通三角多项式的性质,而且具有全局的和局部的形状可调性.同时还具备较为灵活的连续性.当两类形状参数在给定的范围内任意取值时,这种带两类形状参数的AT-β-Spline曲线满足一阶几何连续性;如果给定两段相邻曲线段中的两类形状参数满足-1≤α≤1,μ_i=λ_(i+1)或μ_i=λ_i=μ_(i+1)=λ_(i+1)时,则带两类形状参数的AT-β-Spline曲线满足C~1∩G~2连续.另外利用奇异混合的思想,构造了满足C~1∩G~2插值AT-β-Spline曲线,解决曲线反求的几何连续性等问题.同时还给出了旋转面的构造,描述了两类形状参数对旋转面的几何外形的影响;当形状参数取特殊值时,这种AT-β-Spline曲线曲面可以精确地表示圆锥曲线曲面.从实验的结果来看,本文构造的AT-β-Spline曲线曲面是实用的有效的.     

On the average curvature of a convex curve in a surface of nonpositive Gaussian curvature

In this paper, the upper bound of the average curvature of a convex curve in a simply connected surface of nonpositive Gaussian curvature is obtained.

     

一类新的细分曲线方法

Subdivision defines a smooth curve or surface as the limit of a sequence of successive refinements based on initial control polygon or grid.Usually the curve refinements is the basis of the corresponding surface rules. In this paper we analyze previous subdivision scheme according to theories about convergence of N.Dyn and M.F Hassan. In terms of binary and ternary subdivision schemes general construction about curve‘s refinements are studied.Two approximating curve subdivision schemes with neighboring four control points are derived,the generating limit curves can both reach the smoothness of C^1 over the initial polygon using the two schemes and the tolerances of them are given according to the method of [7].     

n元椭球面的判定及所围椭球体的体积

给出n元二次曲面是椭球面的充要条件和所对应的n维椭球体的体积计算公式,并且条件和体积的计算中只用到曲面中的系数行列式,使判定和体积计算较为方便.     

A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters

We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points lines, where three interfaces meet, and at the boundary points lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.     

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C^1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.