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 .     

具有重结点的B-样条曲面G1的连续条件

本文得到了关于两个双三次内部重结点B-样条曲面片G1连续的充分必要条件和在公共边界线上控制向量的本征条件.这些条件直接由两个B-样条曲面的控制向量表示.文[10]证明了使用内部单结点的双三次B-样条曲面来构造G1光滑曲面,局部格式不存在.使用本文的这些条件就可以构造出具有局部各式的G1光滑造型.     

具有重结点的B-样条曲面G1的连续条件

本文得到了关于两个双三次内部重结点B-样条曲面片G¹连续的充分必要条件和在公共边界线上控制向量的本征条件.这些条件直接由两个B-样条曲面的控制向量表示.文[10]证明了使用内部单结点的双三次B-样条曲面来构造G¹光滑曲面,局部格式不存在.使用本文的这些条件就可以构造出具有局部各式的G¹光滑造型.     

A weak condition for the convexity of tensor-product Bézier and B-spline surfaces

A sufficient condition for a tensor-product Bézier surface to be convex is presented. The condition does not require that the control surface itself is convex, which is known to be a very restrictive property anyway. The convexity condition is generalised toC ¹ tensor-product B-spline surfaces.     

矩形域与三角域Bezier曲面间的PPG~n变换

1．引言曲线,曲面间的几何连续性问题近几年来引起人们的广泛注意和兴趣［1－10］．在1984年国际CAGD会议上R．E．Barnhill和G．Farm将它列为计算几何尚待解决的重大问题之一．一方面是由于在CAGD中总希望所构造的曲线和曲面有更多的自由度或者有更多的几何控制参数以便使设计人员能自由地控制所要设计的对象的几何形状;另一方面是由于需要研究自由曲线曲面间的基于几何连续的拼接和各种形式的连续过渡曲面来构造更为复杂而灵活的曲线,曲面和几何体山．几何连续性的定义在不同的论文中不尽相同,但其本质是非奇异参数变换下的不变…     

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.     

NUAT T-splines of odd bi-degree and local refinement

This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.     

矩形域与三角域Bézier曲面间的PPGn变换

Geometric continuity is a very important issue in CAGD and has gained more attention. In this paper we present the new concept of polynomial preserving Gn continuity (PPGn) between rectangular and triangular Bézier surfaces. Necessary and sufficient conditions of PPGn are established explicitly. A simple and useful sufficient condition is also presented. The results can be used more effectively in connecting surfaces and composing rectangular and triangular patches.     

Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.     

周期B样条基转换矩阵的表示和计算

周期B样条基以一种简洁的形式表示闭B样条曲线.周期B样条基转换矩阵为闭B样条曲线及相关曲面的不同表示间的转换提供了一个数学模型.本文给出了周期B样条基转换矩阵的存在性条件,给出并证明了周期B样条基转换矩阵的一个简单的递归表示式.在此基础上,本文进一步给出了周期B样条基转换矩阵的计算公式和高效算法.周期B样条基转换矩阵为闭B样条曲线的节点插入、升阶、节点删除和降阶等基本运算提供了一个统一而简单的解决方法,本文给出了一些应用例子.     

一类超曲面构造变量分离法及其应用

以实际工程应用为背景 ,将具有某种物理意义的量 (如时间等 )与空间位置变量分而治之 ,提出一类超曲面构造变量分离的各种算法 ,对它们进行算法分析比较 ,探讨分离次序可变换性条件和变量分离法构造的超曲面的连续性 ,并应用该方法解决飞机表面压强值重建的实际应用问题     

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。     

Developable Bézier-like surfaces with multiple shape parameters and its continuity conditions

To solve the problems of shape adjustment and shape control of developable surfaces, we propose two direct explicit methods for the computer-aided design of developable Bézier-like surfaces with multiple shape parameters. Firstly, with the aim of constructing Bézier-like curves with multiple shape parameters, we present a class of novel Bernstein-like basis functions, which is an extension of classical Bernstein basis functions. Then, according to the important idea of duality between points and planes in 3D projective space, we design the developable Bézier-like surfaces with multiple shape parameters by using control planes with Bernstein-like basis functions. The shape of the developable Bézier-like surfaces can be adjusted by changing their three shape parameters. When the shape parameters take different values, a family of developable Bézier-like surfaces can be constructed and they retain the characteristics of Bézier surfaces. Finally, in order to tackle the problem that most complex developable surfaces in engineering often cannot be constructed by using a single developable surface, we derive the necessary and sufficient conditions for G¹ continuity, Farin−Boehm G² continuity and G² Beta continuity between two adjacent developable Bézier-like surfaces. In addition, some properties and applications of the developable Bézier-like surfaces are discussed. The modeling examples show that the proposed methods are effective and easy to implement, which greatly improve the problem-solving abilities in engineering appearance design by adjusting the position and shape of developable surfaces.     

矩形域与三角域Bezier曲面间的PPG~n变换

Geometric continuity is a very important issue in CAGD and has gained more attention. In this paper we present the new concept of polynomial preserving Gn continuity (PPGn) between rectangular and triangular Bézier surfaces. Necessary and sufficient conditions of PPGn are established explicitly. A simple and useful sufficient condition is also presented. The results can be used more effectively in connecting surfaces and composing rectangular and triangular patches.     

有理B-样条曲线、曲面的包络性质

研究有理Bezier曲线和B-样条曲线、曲面的包络性质,愈来愈广泛,因为它从包络磨光的角度解释了曲线、曲面的一种几何构造特征,形象地说明了模型是由多边形或多面体逐步磨光的结果.     

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．     

Helix splines as an example of affine Tchebycheffian splines

The present paper summarizes the theory of affine Tchebycheffian splines and presents an interesting affine Tchebycheffian free-form scheme, the “helix scheme”. The curve scheme provides exact representations of straight lines, circles and helix curves in an arc length parameterization. The corresponding tensor product surfaces contain helicoidal surfaces, surfaces of revolution and patches on all types of quadrics. We also show an application to the construction of planarC ² motions interpolating a given set of positions. Because the spline curve segments are calculated using a subdivision algorithm, many algorithms, which are of fundamental importance in the B-spline technique, can be applied to helix splines as well. This paper should demonstrate how to create an affine free-form scheme fitting to certain special applications.     

On existence of stable and efficient outcomes in games with public and private objectives

It is well known that game-theoretic models of economies with externalities tend to exemplify the contradiction between efficiency and stability. In formalizations and proofs of this rather loose statement some smoothness conditions are usually required. Here a situation of this kind is studied under continuity assumptions only. The most interesting thing is that there exist such (non-smooth) preferences which guarantee that this contradiction occurs under no circumstances. Assuming some sort of homogeneity of preferences over the set of the players, the necessary and sufficient conditions for such persistent existence of efficient and stable outcomes are derived.     

Another Proof of the Total Positivity of the Discrete Spline Collocation Matrix

We provide a different proof for Morken's result on necessary and sufficient conditions for a minor of the discrete B-spline collocation matrix to be positive and supply intuition for those conditions.     

关于隐式代数曲面的一种几何连续性定义

本文利用函数的幂级数给出了隐式代数曲面的一种几何连续性定义．证明了在此定义下的ｋ阶几何连续即为ｋ阶变尺度（ｒｅｓｃａｌｉｎｇ）连续，并举例说明了其应用的方便性．