一类三角Bézier曲线的形状特征

在几何造型的许多应用中,良好的曲线形状应该消除不必要的奇点和拐点,因此 往往需要预知与分析参数曲线的各种形状特征,以避免出现奇异形状的设计风险。为了快速确 定参数曲线的形状特征,利用锥面的齐次性简化了参数曲线的形状条件,得出了一类带 2 个形 状参数的二次三角 Bézier 曲线的尖点条件锥和 2 张重结点边界条件锥;3 张特征锥面及其切平 面将特征空间划分为不同的特征区域。曲线的形状特征完全由特征点在特征空间的分布区域决 定。用垂直于坐标轴的平面切割特征空间,可得到基于包络与拓扑映射方法的所有形状条件分 布图。进而讨论了形状参数变化对各特征区域的影响,相关结果可使设计者明确如何配置控制 顶点或者调节形状参数,使得生成曲线为全局凸或局部凸曲线,或具有所需要的奇点与拐点, 或将当前曲线形状调节为另一种所需的形状。     

有理C-Bézier曲线的形状分析

对有理C-Bézier曲线进行了形状分析,得出曲线上含有奇点、拐点和曲线为局部凸或全局凸的、用控制多边形边向量相对位置表示的充分必要条件,并讨论了权因子变化对曲线形状图的影响.     

有理C-Bezier曲线的形状分析

对有理C—Bezier曲线进行了形状分析，得出曲线上含有奇点、拐点和曲线为局部凸或全局凸的、用控制多边形边向量相对位置表示的充分必要条件，并讨论了权因子变化对曲线形状图的影响．     

三次PH曲线偶的C1 Hermite插值

对计算机图形中一类特殊的多项式曲线——Pythagorean hodograph(PH)曲线的C1Hermite插值问题进行研究.PH曲线具有诸如有精确的有理Offset、弧长函数可由多项式函数表示以及几何解释优美等一系列优良性质.基于复分析方法,避免了实分析讨论中出现的复杂表示及繁琐计算,构造了满足给定C1 Hermite插值条件且以C1拼接连续的三次PH曲线偶.该曲线偶可灵活处理拐点,从而克服了一般三次PH曲线因恒凸而无法处理拐点的缺陷.相应的两条Bézier曲线表示及其控制顶点的计算简单方便.所得4条插值曲线中,通常有1条曲线具有很好的几何形状特征.结果可直接应用于各工业产品设计及加工领域中.     

三次PH曲线偶的C1 Hermite插值

对计算机图形中一类特殊的多项式曲线—— Pythagorean hodograph(PH )曲线的 C1 Herm ite插值问题进行研究 .PH曲线具有诸如有精确的有理 Offset、弧长函数可由多项式函数表示以及几何解释优美等一系列优良性质 .基于复分析方法 ,避免了实分析讨论中出现的复杂表示及繁琐计算 ,构造了满足给定 C1 Hermite插值条件且以C1拼接连续的三次 PH曲线偶 .该曲线偶可灵活处理拐点 ,从而克服了一般三次 PH曲线因恒凸而无法处理拐点的缺陷 .相应的两条 Bézier曲线表示及其控制顶点的计算简单方便 .所得 4条插值曲线中 ,通常有 1条曲线具有很好的几何形状特征 .结果可直接应用于各工业产品设计及加工领域中 .     

参数凸曲线的性质及拐点判别算法

依据参数曲线凸性的原始几何定义,讨论了参数曲线的局部凸和全局凸性,得到了参数曲线局部凸和全局凸的若干性质。给出了参数曲线的拐点定义,讨论了参数曲线的拐点与局部性之间的关系,导出了参数曲线拐点判别的充要条件及算法。     

平面Bezier曲线拐点的两个定理

本文用几何的方法证明了平面 Bezier曲线拐点的一个基本定理 :当 Bezier曲线的控制多边形只有一个可把原多边形分为两个凸的子多边形的拐向点时 ,其 Bezer曲线总会出现拐点或尖点 ,但最多只有一个拐点 ;本文还导出了具有几何直观意义的有关拐点的方程及其有关结论 ;本文的定理和结论具有直观的几何意义 ,能使我们直接从 Bezier曲线的控制多边形的边向量的形状来判断曲线的拐点的情况。     

四次带参Bézier曲线的形状分析

为了明确形状参数对四次带参Bézier曲线形状的影响,利用基于包络理论与拓扑映射的方法对其进行了形状分析,得出了曲线上含有奇点、拐点和曲线为局部凸或全局凸的充分必要条件,这些条件完全由控制多边形边向量的相对位置所表示;并进一步讨论了形状参数对形状分布图的影响及其对曲线形状的调节能力.     

三次ω-Bézier 曲线的形状分析

基于包络理论与拓扑映射的方法对三次ω-Bézier 曲线进行了形状分析,得出了曲线上含有奇点、拐点和曲线为局部凸或全局凸的充分必要条件.这些条件完全由控制多边形的顶点和频率参数所决定.进一步讨论了频率参数对形状分布图的影响及其对曲线形状的调节能力.     

四次带参Bzier曲线的形状分析

为了明确形状参数对四次带参Bzier曲线形状的影响,利用基于包络理论与拓扑映射的方法对其进行了形状分析,得出了曲线上含有奇点、拐点和曲线为局部凸或全局凸的充分必要条件,这些条件完全由控制多边形边向量的相对位置所表示;并进一步讨论了形状参数对形状分布图的影响及其对曲线形状的调节能力.     

Interpolating G-splines with rational offsets

In (Pottmann, 1995), a geometric characterization of rational PH-curves is presented. Using this result we developed an explicit Bézier representation for interpolating G¹-Hermite PH-splines referring to local coordinate systems. Furthermore a simple geometric criterion for avoiding singularities is proposed.     

三点插值的三次PH曲线构造方法

为推广三次PH曲线的实际应用,研究在给定3个平面型值点条件下的三次PH曲线构造方法.三次PH曲线具有鲜明的几何性质和代数特征,采用平面参数曲线的复数表示方法,三次PH曲线的充分必要条件被表述为复代数系统.通过对给定型值点进行参数化,将复代数系统转化为一元二次复方程,求解方程即得三次PH曲线的控制顶点,从而得到2条构造曲线.应用该方法对模拟给定的若干平面型值点数据进行实验,比较了均匀参数化、弦长参数化、弧长参数化方法的不同效果,并计算弧长、弯曲能量、绝对旋转数来选取最优构造曲线.实验结果表明,该方法有效且易于计算,可应用于三次PH样条构造.     

Classification of kinematic singularities in planar robot manipulators

In [13, 14] we have proclaimed a singularity theory based programme of investigations of kinematic singularities in robot manipulators. The main achievement of the programme consists in providing local candidate models of kinematic singularities. However, due to the specific form of the manipulator kinematics, fitting the candidate models into the prescribed robot kinematics is a fairly open problem. The problem is easily solvable only around non-singular configurations of manipulators, where locally the kinematics can be modelled by linear injections or projections. In this paper we are concerned with planar manipulator kinematics, and prove that, under a mild geometric condition, such kinematics can be transformed around singular configurations to simple quadratic models of the Morse type. The models provide a complete local classification of generic planar kinematics of robot manipulators.     

Approximate convolution with pairs of cubic Bézier LN curves

In this paper we present an approximation method for the convolution of two planar curves using pairs of compatible cubic Bézier curves with linear normals (LN). We characterize the necessary and sufficient conditions for two compatible cubic Bézier LN curves with the same linear normal map to exist. Using this characterization, we obtain the cubic spline approximation of the convolution curve. As illustration, we apply our method to the approximation of a font where the letters are constructed as the Minkowski sum of two planar curves. We also present numerical results using our approximation method for offset curves and compare our method to previous results.     

Detecting and quantifying envelope singularities in the plane

The mathematical envelopes of families of both rigid and non-rigid moving shapes play a fundamental role in a variety of problems from very diverse application domains, from engineering design and manufacturing to computer graphics and computer assisted surgery. Geometric singularities in these envelopes are known to induce malfunctions or unintended system behavior, and the corresponding theoretical and computational difficulties induced by these singularities are not only massive, but also well documented. We describe a new approach to detect and quantify the envelope singularities induced by 2-dimensional shapes of arbitrary complexity moving according to general non-periodic and non-singular planar affine motions. Our approach, which does not require any envelope computations, is reframing the problem in terms of “fold points” and “fold regions” in the neighborhood of geometric singularities, and we show that the existence of these fold points is a necessary condition for the existence of singularities. We establish a mathematically well defined duality between the 2-dimensional Euclidean space in which the motion takes place and a 2+1 spacetime domain. Based on this duality, we recast the problem of detecting and quantifying geometric singularities into inherently parallel tests against the original geometric representation in the 2-dimensional Euclidean space. We conclude by discussing the significance of our results, and the extension of our approach to 3-dimensional moving shapes.     

Some remarks on Bézier curves

Parametric polynomial curves in Bézier-Bernstein representation are considered as prohections of rational norm curves of degree n in n-space; from this point of view the singularities of a planar Bézier cubic are determined and expressed by its affine invariants. Secondly, for an arbitrary pair of adjacent parametric curves in homogeneous coordinates, the general conditions for geometric continuity of any order k, G^k, are established.This result generalizes the corresponding conditions in the non-homogeneous (affine) case, recently obtained by [Goodman '84]. Some applications are given for Bézier curves. In particular, for γ-splines [Boehm '85], the existence of a global rational parameter that makes it to a C² parametric curve is shown. Furthermore, for two adjacent rational Bézier curves the complete conditions for G³ are stated using the projective properties of the control points only.     

The invariant representations of a quadric cone and a twisted cubic

Up to now, the shortest invariant representation of a quadric has 138 summands and there has been no invariant representation of a twisted cubic in 3D projective space, which limit to some extent the applications of invariants in 3D space. We give a very short invariant representation of a quadric cone, a special quadric, which has only two summands similar to the invariant representation of a planar conic, and give a short invariant representation of a twisted cubic. Then, a completely linear algorithm for generating the parametric equations of a twisted cubic is provided also. Finally, we exemplify some applications of our proposed invariant representations in the fields of computer vision and automated geometric theorem proving.     

Collocation and galerkin finite element methods for viscoelastic fluid flow—I. : Description of method and problems with fixed geometry

Collocation and Galerkin finite element methods are developed for viscoelastic fluid flow in a fixed geometry. The collocation methods use Hermite cubic polynomials with a global coordinate transformation to permit irregular geometry. The Galerkin method uses isoparametric elements (transformed element by element) with bilinear polynomials for pressure and quadratic polynomials for velocity. Both methods are applied to two-dimensional flow in planar geometry and the Galerkin method is applied to axisymmetric cylindrical geometries as well. The fluid model is a nonlinear Maxwell model but is limited to small elastic components.

The two methods are applied to several test problems. Entry-length problems test the ability to model pressure singularities are velocity discontinuities. Stick-slip problems test the ability to model pressure singularities and stress discontinuities. Both test problems have analytic or accurate numerical solutions for Newtonian fluids so that the accuracy of the two methods is compared.     

基于脊波变换的体数据压缩编码方法

体数据的数据量大、数据间的相关性强、拥有大量的线或面结构,因此需要研究有效的压缩编码方法。脊波变换作为一种新的时频分析工具,在处理线或面的奇异性时有它适用的一面。在介绍脊波变换理论的基础上,将脊波变换的思想应用到体数据的压缩编码中。文中两种压缩策略的主要思想分别为：策略1先将体数据划分成切片组,再对每一张切片做二维脊波变换,然后进行量化和熵编码;策略2直接对体数据做类似于三维脊波变换的变换,然后进行量化和熵编码。比较而言,策略1实现简单,策略2能获得更高的压缩比。两种策略都具有较强的鲁棒性,且能实现嵌入式编码。该方法已应用到实际工业CT体数据的压缩编码中,还可用于其它类型体数据的压缩编码中。     

A basis for the implicit representation of planar rational cubic Bézier curves

We present an approach to finding the implicit equation of a planar rational parametric cubic curve, by defining a new basis for the representation. The basis, which contains only four cubic bivariate polynomials, is defined in terms of the Bézier control points of the curve. An explicit formula for the coefficients of the implicit curve is given. Moreover, these coefficients lead to simple expressions which describe aspects of the geometric behaviour of the curve. In particular, we present an explicit barycentric formula for the position of the double point, in terms of the Bézier control points of the curve. We also give conditions for when an unwanted singularity occurs in the region of interest. Special cases in which the method fails, such as when three of the control points are collinear, or when two points coincide, will be discussed separately.