以三次曲线为特殊积分的二次系统

本文研究以三次曲线为特殊积分的二次系统:的极限环,得出一类二次系统存在极限环的充要条件。     

具三次曲线解的二次系统至多有一个极限环

本文研究具有三次曲线解x^3-x^2-y^2=0的二次系统,证明此类二次系统最多只有一个极限环,进而证明了具有三次的曲线解的二次系统至多有一个极限环。     

平面C-Bézier曲线的奇拐点分析

本文完全地讨论了平面C-曲线和平面C-Bezier曲线的奇拐点和凸性性质:曲线段为且必为下列情形之一:有一各拐点,两个拐点,一个尖点,一个二重结点,处处为凸;并给出了相应的用控制多边形相对位置表示的充分必要条件.     

四次C-曲线的性质及其应用

以1,t,t2,t3,…为基底的Bézier曲线和B样条曲线是构造自由曲线、曲面强有力的工具.但是它们不能精确地表示某些圆锥曲线如圆弧、椭圆等,也不能精确地表示正弦曲线.本文利用一组新的基底sint,cost,t2,t,1,构造了两条新的曲线,这两条曲线依赖于参数α>0.当α→0时极限分别是四次Bézier曲线和四次B样条曲线,称之为四次C-曲线:四次C-Bézier曲线和四次C-B样条曲线.它们具有一般Bézier曲线和B样条曲线的性质:如端点插值,凸包,离散等,还可以精确的表示圆弧、椭圆及正弦曲线.作为应用,文章最后给出了四次C-Bézier曲线表示正弦曲线的条件.     

二次系统存在三次曲线弓形分界线环的充要条件

可证二次系统内含焦点的三次曲线弓形分界线环必由抛物线与直线所围成。定理1 二次系统存在三次曲线弓形分界线环的充要条件是此系统可化为以下形式     

加权二次有理Bézier插值曲线

在本文中,给定一组有序空间数据点列及每个数据点的切矢向量,利用加权二次有理Bézier曲线对数据点作插值曲线,使该曲线具有C1连续性,并且权因子只是对相应顶点曲线附近产生影响,同调整两个相邻的权因子可以调整这两个相邻顶点之间的曲线和它的控制多边形.     

一类二次系统定义的双参数三次代数曲线解族

本文给出一类由二次系统定义的双参数三次代数曲线解族，研究族中曲线解的轨线成为分界线环或其一部份的充要条件及相应系统的全局相图，从而揭示了由代数曲线解确定的二次系统的异宿环（有界或无界）及退化奇点分支出同宿环的某些现象．另外，本文的结果表明文［３］中关于二次系统的三次代数曲线同宿环的结论是不完备的．     

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.     

关于三次函数及其曲线性质的讨论

就三次函数及其曲线的对称性、极值进行了讨论,得出任何三次函数所表示的曲线都存在唯一拐点,并且关于拐点对称;同时讨论了三次函数存在极值的条件,并给出极值的计算公式。     

具有退化三次曲线解的Hamilton二次系统的Poincare分支

具有退化三次曲线解的Hamilton二次系统,经二次微扰后的Poincare分支,是否存在两个极限环?这是一个长期受到困扰的问题.本文证明了在特定条件下,可以分支出两个极限环.     

Rational cubic/quartic Said-Ball conics

In CAGD, the Said-Ball representation for a polynomial curve has two advantages over the Bézier representation, since the degrees of Said-Ball basis are distributed in a step type. One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomial curve runs twice as fast as the de Casteljau algorithm of Bézier curve. Another is that the operations of degree elevation and reduction for a polynomial curve in Said-Ball form are simpler and faster than in Bézier form. However, Said-Ball curve can not exactly represent conics which are usually used in aircraft and machine element design. To further extend the utilization of Said-Ball curve, this paper deduces the representation theory of rational cubic and quartic Said-Ball conics, according to the necessary and sufficient conditions for conic representation in rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball basis. The results include the judging method for whether a rational quartic Said-Ball curve is a conic section and design method for presenting a given conic section in rational quartic Said-Ball form. Many experimental curves are given for confirming that our approaches are correct and effective.     

Application of degree reduction of polynomial Bézier curves to rational case

An algorithmic approach to degree reduction of rational Bézier curves is presented. The algorithms are based on the degree reduction of polynomial Bézier curves. The method is introduced with the following steps: (a) convert the rational Bézier curve to polynomial Bézier curve by using homogenous coordinates, (b) reduce the degree of polynomial Bézier curve, (c) determine weights of degree reduced curve, (d) convert the Bézier curve obtained through step (b) to rational Bézier curve with weights in step (c).     

Geometric meanings of the parameters on rational conic segments

Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces.     

Degree elevation of nurbs curves by weighted blossom

An algorithmic approach to degree elevation of NURBS curves is presented. The new algorithms are based on the weighted blossoming process and its matrix representation. The elevation method is introduced that consists of the following steps: (a) decompose the NURBS curve into piecewise rational Bézier curves, (b) elevate the degree of each rational Bézier piece, and (c) compose the piecewise rational Bézier curves into NURBS curve.     

点集Bézier曲线

提出了点集Bézier曲线的概念,给出了点集Bézier曲线的性质及细分算法.按照点集算术的定义,当点集是长方形闭域或圆盘时,点集Bézier曲线就是区间Bézier曲线或圆盘Bézier曲线,因此,点集Bézier曲线是对区间Bézier曲线和圆盘Bézier曲线的推广.     

两条连续的有理Bézier曲线的逼近合并

目前多项式 Bézier曲线的逼近合并问题已研究得比较深入 ,而有理 Bézier情形主要还是通过两类多项式 h和 H来降阶逼近 ,但是在工业制造中有重要意义的有理 Bézier曲线的合并问题一直缺乏研究 .本文通过控制点的优化扰动将两连续的满足权约束条件的有理 Bézier曲线转化成新的两有理Bézier曲线 ,使它们符合精确合并条件 ;并将合并得到的同阶有理 Bézier曲线看成是原两曲线的有理逼近     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> positivity preserving scattered data interpolation using rational Bernstein-Bézier triangular patch

A local C ¹ positivity preserving scheme is developed using Bernstein-Bézier rational cubic function. The domain is triangulated by Delaunay triangulation method. Simple sufficient conditions are derived on the inner and boundary Bézier ordinates to preserve the shape of positive data. These inner and boundary Bézier ordinates involve weights in their definition. In any triangular patch if the Bézier ordinates do not satisfy the derived conditions of positivity, then these are modified by the weights (free parameters) involved in the construction of Bernstein-Bézier rational cubic function to preserve the shape of positive scattered data.     

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     

Direction monotonicity for a rational Bézier curve

The monotonicity of a rational Bézier curve, usually related to an explicit function,is determined by the used coordinate system. However, the shape of the curve is independent of the coordinate system. To meet the affine invariant property, a kind of generalized monotonicity, called direction monotonicity, is introduced for rational Bézier curves. The direction monotonicity is applied to both planar and space curves and to both Cartesian and affine coordinate systems, and it includes the traditional monotonicity as a subcase. By means of it,proper affine coordinate systems may be chosen to make some rational Bézier curves monotonic.Direction monotonic interpolation may be realized for some of the traditionally nonmonotonic data as well.