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．     

基于刘徽割圆术的等距线逼近算法

给出了一种基于刘徽割圆术的平面NURBS曲线的等距线的逼近算法。利用正多边形代替圆所扫掠出的区域边界来近似等距曲线，所得到的逼近曲线是与基曲线同次的NURBS曲线，并且可以达到任意的精度。     

最小二乘法在光谱定量分析中的应用

光谱定量分析是测定矿和中某元素含量的一种方法。本文介绍在计算机中利用最小二乘法，根据实验数据，用最佳逼近多项式来拟合这些数据，从而拟合得到元素含量和黑度之间关系的一条工作曲线，用这条工作曲线可以很方便地根据待测样品的黑度求出对应元素的百分含量。     

Bézier曲线降多阶逼近的一种方法

文献[1,2]讨论了Bezier曲线一次降多阶逼近问题,得到了很好的结果.文献[1]利用广义逆矩阵得到不保端点插值的降多阶逼近曲线的控制顶点的表达式.但却没有得到带端点任意阶插值条件的降多阶逼近曲线的控制顶点的表达式.文献[2]得到了带端点任意阶插值的降多阶逼近曲线的控制顶点的解析表达式.本文首先给出两Bezier曲线间距离的定义;然后根据降阶曲线与原曲线间的距离最小,分别得到了用矩阵表示的不保端点插值和保端点任意阶插值的降多阶逼近曲线的控制顶点的显示表达式.所给数值例子显示,用本文方法得到的降多阶逼近曲线对原曲线有很好的逼近效果.     

三次拟Bézier曲线的光顺逼近

利用 Friedman的 URN模型构造出带有参数的调配函数 ,用其生成三次拟Bézier曲线 .通过对这种新曲线进行分析 ,利用最小二乘法和非线性泛函的极小值优化计算 ,来对平面数据点进行光顺逼近 ,得到最优的光顺逼近曲线 .     

一类有理Bézier曲线及其求积求导的多项式逼近

用多项式曲线来逼近有理曲线在计算机辅助几何设计(CAGD)系统中可简化求积求导等繁琐的计算.然而,按现有的方法能检验一条已知的有理曲线是否具有收敛的多项式逼近曲线却不易选择适当的权因子来产生能用多项式曲线来加以逼近的有理曲线,即不易做到事先设计;同时,要减少求积、求导的逼近误差只能依靠提高多项式曲线的次数.文中给出一类有理Bézier曲线及其多项式逼近算法较好地克服了这两种缺陷,具有推广应用的价值.     

两个回归函数相等的Bootstrap检验

两个回归参数相等性检验一直是统计界感兴趣的问题之一．在这篇文章中，四个检验统计量被用于度量两曲线的差异，在原假设下统计量的分布采用向量数据的重复抽样来逼近，并给出了—些模拟结果．     

参数曲线近似弧长参数化的三角函数方法

本描述了一种局部的近似弧长参数化插值方法，用三角函数对曲线的弧长函数进行分段逼近，段与段之间是相互独立的，且插值曲线在插值点处的弧长与原参数曲线的真实弧长相等。     

关于B样条V·D性质的另一证明

样条函数的变差缩减方法(简称V·D逼近)是利用B样条构造曲线的一种十分有效的方法。这种方法具有模拟被逼近曲线几何形态的特点,且计算简单,特别适用于自由形式的曲线和曲面的设计,古典的Bernstein多项式逼近是V·D逼近的特例,而V·D逼近的理论基础是B样条所具有的V·D性质。本文采用与以往证明不同的途径,对B样条的V·D性质给出了一种纯代数的证明。该证明简单、自然。     

三次均匀B样条曲线的一种新扩展

构造了一组带形状参数的三次B样条曲线,该曲线与经典三次B样条曲线具有相同的基本性质,且可在不改变控制顶点的情况下,通过改变形状参数的取值实现对曲线形状的调整;选取适当的控制顶点,并对形状参数选取适当的取值,构造的三次λ-B样条曲线可以很好的逼近圆和椭圆;提供了插值于已知数据点的λ-B样条曲线的构造方法;最后,通过图例体现了新方法的有效性.     

Rational points and Galois points for a plane curve over a finite field

We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the Hermitian, Klein quartic or Ballico–Hefez curve. The author proposes a problem: Does the converse hold true? If the curve of genus zero or one has a rational point, we have an affirmative answer.     

On a class of locally convex closed curves

Summary For a certain type of locally convex plane curves some relations are shown between the numbers of double points and double tangents and the order and class of the curve. It is proved that the curve has an equal number of double points and double tangents and at least v-1 where 2v denotes the order (and class) of the curve. At last it is shown how a curve with more that v-1 double points (tangents) may be deformed into a curve with the minimum number. To Enrico Bompiani on his scientific Jubilee     

Determining knots by optimizing the bending and stretching energies

For a given set of data points in the plane, a new method is presented for computing a parameter value (knot) for each data point. Associated with each data point, a quadratic polynomial curve passing through three adjacent consecutive data points is constructed. The curve has one degree of freedom which can be used to optimize the shape of the curve. To obtain a better shape of the curve, the degree of freedom is determined by optimizing the bending and stretching energies of the curve so that variation of the curve is as small as possible. Between each pair of adjacent data points, two local knot intervals are constructed, and the final knot interval corresponding to these two points is determined by a combination of the two local knot intervals. Experiments show that the curves constructed using the knots by the new method generally have better interpolation precision than the ones constructed using the knots by the existing local methods.     

SINGULARITIES OF SPHERICAL FOUR-BAR MOTION

The |ink curve of a sphericel four-bar mechanism can be separated into two branches.The condition that the double points of the curve must satisfy is given. The double points maybe the double points of either branch, or the intersection points of the two branches.     

Codes from curves with total inflection points

The concept of pure gaps of a Weierstrass semigroup at several points of an algebraic curve has been used lately to obtain codes that have a lower bound for the minimum distance which is greater than the Goppa bound. In this work, we show that the existence of total inflection points on a smooth plane curve determines the existence of pure gaps in certain Weierstrass semigroups. We then apply our results to the Hermitian curve and construct codes supported on several points that compare better to one-point codes from that same curve.      

An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known. This is due to the avoiding, through a sufficiently good change of coordinates, of real root computations on polynomials with coefficients in a simple real algebraic extension of to deal with the critical points of the considered curve. In fact, one of the main features of this algorithm is that its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve.     

On parametrization of interpolating curves

In parametric curve interpolation there is given a sequence of data points and corresponding parameter values (nodes), and we want to find a parametric curve that passes through data points at the associated parameter values. We consider those interpolating curves that are described by the combination of control points and blending functions. We study paths of control points and points of the interpolating curve obtained by the alteration of one node. We show geometric properties of quadratic Bézier interpolating curves with uniform and centripetal parameterizations. Finally, we propose geometric methods for the interactive modification and specification of nodes for interpolating Bézier curves.     

基于轮廓关键点的B样条曲线拟合算法

针对逆向工程中的点云切片轮廓数据点列,提出一种基于轮廓关键点的B样条曲线拟合算法.在确保扫描线点列形状保真度的前提下,首先对其进行等距重采样等预处理,并遴选出曲线轮廓关键点,生成初始插值曲线;再利用邻域点比较法求出初始曲线与各采样点间的偏差值,在超过拟合允差处增加新的关键点,并生成新的插值曲线,重复该步骤至拟合曲线满足预定精度要求.实验表明,在对稠密的二维断面数据点进行B样条逼近时,该算法能有效压缩控制顶点数目,并具有较高的计算效率.同时,由于所得控制顶点的分布能准确反映曲线的曲率变化,该方法还可作为误差约束的曲线逼近中的迭代步骤之一.     

三次Pythagorean Hodograph 曲线在给定总弧长下的$G^2$拼接

Pythagorean-hodograph (PH)曲线因其在弧长和等距线计算方面的优势而被广泛应用于曲线建模中.本文讨论了在总弧长约束下的三次PH曲线$G^2$连续拼接问题.具体地说,给定两个端点和一个拼接点,构造两条三次PH曲线,使其在指定总弧长下插值两个端点,并且在连接点处是$G^2$连续的.这也可以看作是一个曲线延拓问题.根据三次PH曲线的弧长公式和$G^2$连续条件,最终将问题转化为了一个带有约束的极小值问题,同时我们给出了几个具体例子来说明该方法.