有理曲线的多项式逼近

利用曲线摄动的思想给出了用多项式曲线逼近有理曲线的一种新方法．其基本步骤是对有理曲线的控制顶点进行摄动，使之产生一多项式曲线，并使摄动误差在某种范数意义之下达到最小．同时，通过适当控制摄动曲线的顶点，使逼近多项式曲线与有理曲线在两端点保持一定的连续性．这一结果可以与细分（ｓｕｂｄｉｖｉｓｉｏｎ）技术结合给出有理曲线的整体光滑的分片多项式逼近．实例表明，在某些情况下本文中的方法要优于传统的Ｈｅｒｍｉｔｅ插值方法及Ｔ．Ｗ．Ｓｅｄｅｒｂｅｒｇ和Ｍ．Ｋａｋｉｍｏｔｏ（１９９１）提出的杂交曲线逼近算法．     

一类有理Bezier曲线的类de Casteljau算法及de Casteljau—型子分割

Bezier曲线的一个良好性质是de Casteljau算法不仅可以用于升阶，而且可以用于分割。本文主要研究基于有理调配函数的一类有理Bezier曲线的类de Casteljau算法及类de Casteljau－型子分割方法。第一部分从一类有理Bezier曲线的递推关系出发，讨论这一类有理Bezier曲线的类de Casteljau算法。第二部分给出了这一类有理Bezier曲线的de Casteljau－型分割方法。     

三次Bézier曲线的快速生成算法

本文给出了一种三次Ｂéｚｉｅｒ曲线的生成算法，在曲线的逐点生成过程中，只用到加减法，故效率极高．而且，此方法可推广到一般多项式或有理参数曲线     

生成曲线的有理稳定细分方法

本文推广稳定细分方法，得到生成曲线的有理稳定细分方法，并讨论了该方法所产生曲线的几何性质．有理稳定细分方法生成曲线的类型更为丰富，包括了计算机辅助几何设计中常用的有理B-样条曲线．     

一类有理Bezier曲线的类de Casteljau算法及de Casteljau-型子分割

Bezier曲线的一个良好性质是 de Casteljau算法不仅可以用于升阶 ,而且可以用于子分割 .本文主要研究基于有理调配函数的一类有理 Bezier曲线的类 de Casteljau算法及 de Casteljau-型子分割方法 .第一部分从一类有理 Bezier曲线的递推关系出发 ,讨论这一类有理 Bezier曲线的类 de Casteljau算法 .第二部分给出了这一类有理 Bezier曲线的 de Casteljau-型子分割方法 .     

一类有理Bézier曲线的等距线算法及MATLAB实现

构造一类正则有理Bézier曲线,利用改进的有理de casteljau算法求得这类正则有理n次Bézier曲线各点处的切矢,由此得出各点的单位法矢量,应用于原始曲线等距线的计算.该方法几何意义明显,算法简洁,实践效果比较好.同时给出了用Matlab绘制有理Bézier曲线及其等距线的程序,准确快捷,实践效果较好.     

有理圆锥曲线段的参数的几何意义

用代数和几何方法, 得到用有理二次或有理三次Bézier曲线表示的圆锥曲线上的点与其参数域上的点所对应的函数关系; 即给出了有理圆锥曲线段的表达式所描述的映射的逆映射公式．这种公式用圆锥曲线段上此点和控制顶点所决定的三角形面积、角度及有理Bézier曲线的权因子来表示, 或用此点和曲线段首末端点相应的参数角度及有理Bézier曲线的权因子来表示. 这些结果对有理Bézier曲线曲面的最佳参数化和重新参数化等算法实现是极其有益的.     

广义有理样条的几种构造方法

本文利用Ｔｈｉｅｌｅ倒差分方法、Ｐａｄｅ逼近方法、广义Ｑ．Ｄ．算法及ε－算法等构造了几种广义有理样条函数．此外，通过直接法构造了（ｋ－１，ｋ）－型广义有理样条，给出了它的行列式表示和余项表示并证明了广义有理样条算子的存在性、唯一性、齐次性及连续性．     

由控制多边形内部两点确定的三次有理Bézier曲线

１．引言主要应用于自由曲线设计的有理Ｂｅｚｉｅｒ曲线在ＣＡＧＤ中起了重要作用．有理Ｂｅｚｉｅｒ曲线的几何形状不仅受其控制多边形而且受其权因子的控制,有关这方面的研究正受到越来越多的关注,例如［１－７］．当控制多边形给定时,权因子为有理Ｂｅｚｉｅｒ曲线的形状控制提供了自由度．权因子的性质及其与有理Ｂｅｚｉｅｒ曲线形状的关系较为复杂,目前尚未得到全面研究·文［４,５］给出了当修改有理Ｂｅｚｉｅｒ曲线上的一点时,权因子的计算公式,但该公式不能用于同时修改曲线上两点的情况,从而限制了修改曲线的灵活性．文…     

NURBS曲线曲面拟合数据点的迭代算法

本文推广了文献[1]的结果,将文献[1]中关于B样条曲线曲面拟合数据点的迭代算法推广至有理形式,给出了无需求解方程组反求控制点及权因子即可得到拟合NURBS曲线曲面的迭代方法．该算法和文献[1]的算法本质上是统一的,而后者恰是前者的一种退化形式．文章还给出了收敛性证明以及一些定性分析．文末的数值实例说明该算法简单实用．     

A generalization of rational Bernstein–Bézier curves

This paper is concerned with a generalization of Bernstein–Bézier curves. A one parameter family of rational Bernstein–Bézier curves is introduced based on a de Casteljau type algorithm. A subdivision procedure is discussed, and matrix representation and degree elevation formulas are obtained. We also represent conic sections using rational q-Bernstein–Bézier curves. AMS subject classification (2000)  65D17     

插值细分曲线有理参数点的精确求值

本文提出了求值插值细分曲线上任意有理参数的算法.通过构造与细分格式相关的矩阵,m进制分解给定有理数以及特征分解循环节对应算子乘积,计算得到控制顶点权值,实现对称型静态均匀插值细分曲线的求值.本文给出了四点细分和四点Ternary细分曲线的求值实例.算法可以推广到求值其他非多项式细分格式中.     

Verifying the Implicitization Formulae for Degree N Rational Bézier Curves

This is a continuation of short communication$^{[1]}$. In [1] a verification of the implicitization equation for degree two rational Bézier curves is presented which does not require the use of resultants. This paper presents these verifications in the general cases, i.e., for degree $n$ rational Bézier curves. Thus some interesting interplay between the structure of the $n×n$ implicitization matrix and the de Casteljau algorithm is revealed.     

Some Properties of Reduced Modular Polynomials

In this paper we will discuss some properties of reduced modular polynomials used in SEA algorithm for computing the number of rational points of elliptic curves on finite fields.     

Local behavior of planar analytic vector fields via integrability

We present an algorithm to study the local behavior of singular points of planar analytic vector fields having a first integral which is a quotient of analytic functions. The algorithm is based on the blow-up method. It emphasizes the curves passing through the singular points and avoids the computation of the desingularized systems. Vector fields having a rational first integral are a particular case.     

Tropical descendant Gromov–Witten invariants

We define tropical Psi-classes on\({\mathcal{M}_{0,n}(\mathbb{R}^2, d)}\) and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi- and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin’s lattice path algorithm and counts rational plane tropical curves satisfying certain Psi- and evaluation conditions.     

A unified Pythagorean hodograph approach to the medial axis transform and offset approximation

Algorithms based on Pythagorean hodographs (PH) in the Euclidean plane and in Minkowski space share common goals, the main one being rationality of offsets of planar domains. However, only separate interpolation techniques based on these curves can be found in the literature. It was recently revealed that rational PH curves in the Euclidean plane and in Minkowski space are very closely related. In this paper, we continue the discussion of the interplay between spatial MPH curves and their associated planar PH curves from the point of view of Hermite interpolation. On the basis of this approach we design a new, simple interpolation algorithm. The main advantage of the unifying method presented lies in the fact that it uses, after only some simple additional computations, an arbitrary algorithm for interpolation using planar PH curves also for interpolation using spatial MPH curves. We present the functionality of our method for G¹ Hermite data; however, one could also obtain higher order algorithms.     

A degree bound for families of rational curves on surfaces

We give an upper bound for the degree of rational curves in a family that covers a given birationally ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We introduce an algorithm for constructing examples where the upper bound is tight. As an application of our methods we improve an inequality on lattice polygons.