Bézier曲线曲面正则性的判别条件

正则性是参数曲线曲面的重要代数性质,是由参数曲线曲面的参数化决定的.在计算机辅助制造过程中,要求所处理的参数曲线曲面是正则的,前提是计算机辅助设计得到的参数曲线曲面是正则曲线曲面.然而,直接按照正则参数曲线曲面的定义,采用解方程或方程组的方法来判断曲线曲面是否正则,其计算相当复杂,实际上也是行不通的.通过将Bézier曲线曲面的导矢曲线(法矢曲面)的参数表示转换为隐式表示,得到了一个判断Bézier曲线曲面正则性的简单而实用的充分条件.     

点云模型法矢调整优化算法

点云中存在奇异情况时,采用最小生成树法进行法矢调整会出现错误,而采用曲面重建方法运算效率又较低,为此提出一种点云模型法矢调整的优化算法.算法分别处理薄壁特征、垂直法向和相邻曲面3种奇异情况.对薄壁特征,算法提取特征点并在该处强制进行法矢取反;对垂直法向,算法通过扩大邻域搜索范围来获得法矢变化趋势;对相邻曲面,算法在K邻域中剔除歧义邻域点,避免在最小生成树中生成错误边.实验结果表明,该算法在点云中存在奇异情况时能够进行正确的法矢调整,并且相较于曲面重建方法具有较高的效率.     

基于能量优化的NURBS曲面几何特征修改

NURBS作为曲面造型技术中的重要方法之一,在计算机辅助几何设计和计算机图形学领域中有着广泛的应用。针对NURBS曲面的几何特征修改,提出了基于能量优化的修改方法。通过对NURBS曲面的控制顶点进行扰动,以曲面的应力能改变为目标函数并使之最小化,实现了NURBS曲面上给定多个点处的位置、一阶偏导矢、二阶偏导矢和法矢等几何特征的修改。数值实例表明该方法用于微调时,可实现对曲面局部形状的多种修改效果,便于交互设计。     

曲线曲面的形态算法及应用

从积分几何中的概念出发,证明了凸体形态和运算的一个重要性质： Ｆ（ Ｓ,ｕ）＝ Ｆ（ Ａ,ｕ） Ｆ（ Ｂ,ｕ）,从而将两物体的形态和归结为法矢相同的点集的形态和,并提出法矢球的概念,将物体表面各点的法矢顺序对应至球,即得到该物体的唯一法矢球表示,通过对法矢球的合并,则得到两物体的形态运算结果,在理论上统一了二维、三维实体的形态运算,并给出二维、三维曲线、曲面的具体形态算法．此外还给出曲线、曲面形态算法的具体应用,如扫成曲面造型、字型合成、非刚体运动的广义内插等．     

超限插值在一类艺术曲面造型中的应用

本文集中讨论了超限插值及其特殊情况,特别地,对于插值交叉线、插值边界曲线及其导矢的情形给出了大量的艺术曲面造型,其计算效率远远高于基于偏微分方程（PDE）的曲面造型方法.因此,用超限插值方法构造艺术曲面造型不失为一种高效的方法.     

曲面造型技术综述

曲面造型技术是计算机辅助设计（CAD）和计算机图形学（CG）中最为活跃、同时也是最为关键的学科分支之一，它随着CAD／CAM技术的发展而不断完善、渐趋成熟。1963年，美国Boeing飞机公司的Ferguson首先提出了将曲线曲面表示为参数的矢函数方法。他最早引入参数三次曲线，构造了组合曲线和由四角点的位置矢量及两个方向的切矢定义的Ferguson双三次曲面片。他所采用的曲线曲面的参数形式从此成为曲线曲面设计中进行形状数学描述的标准形式。1964到1967年，美国麻省理工学院的Coons提出了一个具有一般性的曲面描述方法，给走围成封闭曲线…     

机器人工作空间的界限面及其位置奇异曲面的代数求解方法

本文提出了一个求解运动副转角或移动有限制的任意机械手的工作空间界限面及其位置奇异曲面的代数方法。得出了手部参考点在固定坐标系中的一系列位置奇异曲面方程。并以5R,6R和5R-P机器人为例,使用计算机绘制出其工作空间上述各种曲面的子午截面曲线图。结果与其它方法相比,具有精度高,运算快的特点,不仅具有理论意义,而且还有实用价值。     

基于整数方程的逻辑方程组求解方法研究

为快速有效地求解大量逻辑方程组,根据逻辑运算的特点详细阐述了将逻辑方程转化成等效整数方程的原理和方法,并对得到的整数方程进行化简,提出了整数方程组的一般求解方法,即吴方法和Grobner基理论。接着给出并完善了一种基于快速多项式乘法的消元法,大大降低了求解的复杂度,最后将基于整数方程的逻辑方程组求解方法应用于故障诊断,并举例验证。     

局部形状可调插值曲线曲面及其参数选取方案

以形状可调插值曲线曲面为研究主题的文献多数侧重于分析曲线曲面性质,少有文献介绍可调插值曲线曲面的构造方法,以及调节参数的选取方案。这里以3次Hermite插值曲线为基础,通过在导矢中引入参数来构造形状可调插值曲线,将曲线按照插值数据进行整理,即可得到含参数的插值基函数,进而由之构造张量积插值曲面。为了帮助设计者寻找合适的参数,提供了4种用于确定曲线中形状参数的准则,其中的3种还推广应用于曲面,每种准则都提供了可以直接使用的公式。所给可调插值曲线曲面的构造方法以及参数选取方案具有一般性,数值实例验证了方案的有效性。     

超声逆散射成像方法研究

在研究奇异值分解、最小二乘法的基础上,采用空间域方法研究超声逆散射成像问题。通过脉冲基和点匹配的方法将泛函方程转换为代数方程,运用迭代算法解决方程的非线性问题。利用Picard准则判断方程的不适定程度,并采用均值处理和截断奇异值分解正则化2种方法对方程进行求解。实验结果证明,该方法可以较好地滤除噪声,提高重建图像的质量和可信度,减少迭代过程中的计算量。     

Symbolic parametrization of curves

If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essential to be able to represent these geometric objects in an appropriate way. For some applications an implicit representation by algebraic equations is desirable, whereas for others an explicit or parametric representation is more suitable. Therefore, transformation algorithms from one representation to the other are of utmost importance.We investigate the transformation of an implicit representation of a plane algebraic curve into a parametric representation. Various methods for computing a rational parametrization, if one exists, are described. As a new idea we introduce the concept of working with classes of conjugate (singular or simple) points on curves. All the necessary operations, like determining the multiplicity and the character of the singular points or passing a linear system of curves through these points, can be applied to such classes of conjugate points. Using this idea one can parametrize a curve if one knows only one simple point on it. We do not propose any new method for finding such a simple point. By classical methods a rational point on a rational curve can be computed, if such a point exists. Otherwise, one can express the coordinates of such a point in an algebraic extension of degree 2 over the ground field.     

Observability of real analytic vector fields on compact manifolds

A smooth vector field on a smooth compact manifold is said to be observable if there exists a smooth output function that distinguishes initial states on any time interval. In this paper we prove that a real analytic vector field on a compact manifold is observable if and only if its singularities are isolated. The result remains true for functions. In this case one has to assume that the set of singular points is (p − 1)-dimensional.     

Determining the initial configuration of uninterrupted redundant manipulator trajectories in a manufacturing environment

In a manufacturing environment where a redundant manipulator is programmed to follow a specific trajectory such as welding, painting, and soldering, it is often the case that the manipulator reaches a singular configuration where the program is stopped, the manipulator has to be switched to a new configuration, and the motion continued. This paper presents a method of determining the initial configuration of robotic manipulators to traverse a path trajectory without halting the motion. The method consists of: (1) identifying singular curves/surfaces using the Jacobian rank deficiency method; (2) determining non-crossable curves/surfaces using acceleration-based crossability criteria; (3) finding if the path intersects any non-crossable singular curve/surface. If there is not any intersection, the manipulator does not interrupt with any initial configuration on the trajectory. Otherwise, the intersection is determined, and the non-crossable singular configuration at the intersection point is determined; (4) solving the differential algebraic system of equations (DAE) of index 2 using the Runge–Kutta method; and (5) solving the optimization problem. The method is illustrated through examples.     

Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves

A quaternion rational surface is a surface generated from two rational space curves by quaternion multiplication. The goal of this paper is to demonstrate how to apply syzygies to analyze quaternion rational surfaces. We show that we can easily construct three special syzygies for a quaternion rational surface from a μ-basis for one of the generating rational space curves. The implicit equation of any quaternion rational surface can be computed from these three special syzygies and inversion formulas for the non-singular points on quaternion rational surfaces can be constructed. Quaternion rational ruled surfaces are generated from the quaternion product of a straight line and a rational space curve. We investigate special μ-bases for quaternion rational ruled surfaces and use these special μ-bases to provide implicitization and inversion formulas for quaternion rational ruled surfaces. Finally, we show how to determine if a real rational surface is also a quaternion rational surface.     

Detection of loops and singularities of surface intersections

Two surface patches intersecting each other generally at a set of points (singularities), form open curves or closed loops. While open curves are easily located by following the boundary curves of the two patches, closed loops and singularities pose a robustness challenge since such points or loops can easily be missed by any subdivision or marching-based intersection algorithms, especially when the intersecting patches are flat and ill-positioned. This paper presents a topological method to detect the existence of closed loops or singularities when two flat surface patches intersect each other. The algorithm is based on an oriented distance function defined between two intersecting surfaces. The distance function is evaluated in a vector field to identify the existence of singular points of the distance function since these singular points indicate possible existence of closed intersection loops. The algorithm detects the existence rather than the absence of closed loops and singularities. This algorithm requires general C² parametric surfaces.     

On the computation of switching surfaces in optimal control: aGrobner basis approach

A number of problems in control can be reduced to finding suitable real solutions of algebraic equations. In particular, such a problem arises in the context of switching surfaces in optimal control. A powerful new methodology for doing symbolic manipulations with polynomial data has been developed and tested, namely the use of Grobner bases. We apply the Grobner basis technique to find effective solutions to the classical problem of time-optimal control     

Computing all parametric solutions for blending parametric surfaces

In this paper we prove that, for a given set of parametric primary surfaces and parametric clipping curves, all parametric blending solutions can be expressed as the addition of a particular parametric solution and a generic linear combination of the basis of a free module of rank 3. As a consequence, we present an algorithm that outputs a generic expression for all the parametric solutions for the blending problem. In addition, we also prove that the set of all polynomial parametric solutions (i.e. solutions that have polynomial parametrizations) for a parametric blending problem can also be expressed in terms of the basis of a free module of rank 3, and we prove an algorithmic criterion to decide whether there exist parametric polynomial solutions. As a consequence we also present an algorithm that decides the existence of polynomial solutions, and that outputs (if this type of solution exists) a generic expression for all polynomial parametric solutions for the problem.     

Computing bifurcation points via characteristics gain loci

The authors show how to check the crossing on the imaginary axis by the eigenvalues of the linearized system of differential equations depending on a real parameter μ via feedback system theory. E. Hopf's theorem (1942) refers to a system of ordinary differential equations depending on the real parameter μ in which, when a single pair of complex conjugate eigenvalues of the linearized equations cross the imaginary axis under the parameter vibration, near this critical condition periodic orbits appear. The authors present simple formulas for both static (one eigenvalue zero) and dynamic or Hopf (a single pure imaginary pair) bifurcations, and show some singular conditions (degeneracies) by continuing the bifurcation curves in the steady-state manifold. The bifurcation curves and singular sets of an interesting chemical reactor which possesses multiplicity in the equilibrium solutions and in the Hopf bifurcation points are described     

Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces

Two main results are established in this paper. First, we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three-dimensional motion parameters (within a scale factor for the translations) of a rigid object with curved surfaces. The seven points should not be traversed by two planes with one plane containing the origin, nor by a cone containing the origin. Second, a set of ``essential parameters' are introduced which uniquely determine the motion parameters up to a scale factor for the translations, and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points. The actual motion parameters can subsequently be determined by computing the singular value decomposition (SVD) of a 3×3 matrix containing the essential parameters.     

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C[0,1] consisting of real functions defined and continuous on the interval [0,1]. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra-Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.