一类C~2连续且保凸的插值三次参数样条曲线的构造

本文给出在平面上插值点列为凸的时,构造一类 C~2连续且保凸的插值三次参数样条曲线的方法.这里通过选择插值节点 P_i 处插值曲线 p(t)的切矢方向和长度来代替以往常用的参变量,从而得到一类新的方法.     

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．     

一种保凸的Spline插值方法

本文构造笛氏坐标系中一条光滑曲线,使它通过两个有特定切向的给定点,给出了此种插值函数保凸的一个充要条件.文中考虑逐段三次多项式的保凸插值函数,其收敛性的阶与三次样条多项式一样.最后叙述插值函数形状控制的方法.     

保凸分段三次Bézier插值样条曲线

本文讨论分段三次 Bézier曲线的保凸插值 ,对给定的凸数据点列在相邻两型值点之间构造两个三次 Bézier曲线子段 ,两段之间 G2连续的 ,所构造的曲线插值所有型值点且是 G1的和保凸的     

平面点列的自动光顺算法

本文考虑平面点列的光顺问题并将该问题化成最小能量曲线的构成问题，即在原点列和相应允许误差构成的带状区域内构造一条最小能量曲线并给出一种自动算法．整个光顺过程分成两步，第一步利用凸分析原理在原点列的允许变动范围内除去多余拐点；第二步在保凸的前提下构造插值点列的最小能量曲线并通过对最小能量曲线进行修正而达到对原型值点列进行光顺的目的．光顺结果不仅可以得到一光顺点列，同时还得到了一条插值点列的光顺曲线．该方法可以对分布不均匀甚至有较大转角的点列进行光顺，与已有的方法比起来具有光顺能力强光顺范围广的特点．     

曲面上离散点集的光滑插值

本文主要解决了如下问题：给定Ｒ３凸曲面上的任意个离散点｛Ｐｉ｝Ｎｉ＝１及其对应的函数值｛ｆｉ｝Ｎｉ＝１，要求构造曲面上的插值函数ｆ（ｘ），使得ｆ（Ｐｉ）＝ｆｉ，（ｉ＝１，２，…，Ｎ）．本文方法推广了球面上离散点集的Ｍｕｌｔｉｑｕａｄｒｉｃ插值方法，并且对分区域插值方法也给予了讨论．     

C^3连续的保形插值三角样本曲线

本给出了构造保形插值曲线的三角样条方法，即在每两个型值点之间构造两段三次参数三角样条曲线。所构造的插值曲线是局部的，保形的和C^3连续的而且曲线的形状可由参数调节。     

凸函数样条及其应用

本文主要结果是利用ｍ条凸曲线的乘积(?)f_i（ｘ，ｙ）或ｍ张凸曲面的乘积(?)f_i（ｘ，ｙ,z）代替文献［２］中ｍ条直线的乘积(?)l_i和ｍ张平面的乘积(?)m，构造出一族具有高阶光滑接触边界条件的凸曲线和凸曲面，获得比文［２］更一般的结果．     

多元分次Lagrange插值

对多元多项式分次插值适定结点组的构造理论进行了深入的研究与探讨.在沿无重复分量代数曲线进行Lagrange插值的基础上,给出了沿无重复分量分次代数曲线进行分次Lagrane插值的方法,并利用这一结果进一步给出了在R~2上构造分次Lagrange插值适定结点组的基本方法.另外,利用弱Gr(o|¨)bner基这一新的数学概念,以及构造平面代数曲线上插值适定结点组的理论,进一步给出了构造平面分次代数曲线上分次插值适定结点组的方法,从而基本上弄清了多元分次Lagrange插值适定结点组的几何结构和基本特征.     

一类奇异积分方程组的样条间接近似解法

本文利用三次复插值样条函数给了定义于复平面上光滑封闭曲线上的一类奇异积分方程组（１）的一种间接近似解法，讨论了误差估计和一致收敛性。     

平面三次Pythagorean-Hodograph Hyperbolic曲线

本文基于Pythagorean-hodograph (PH)曲线和代数双曲线的良好几何特性,构造了Pythagorean-Hodograph Hyperbolic (PH-H)曲线,并给出了PH-H曲线的定义以及相应性质.同时,分别利用Hyperbolic基函数和Algebraic Hyperbolic (AH) B\''ezier基函数,得到了平面三次AH B\''ezier曲线为PH曲线的两个不同的充要条件.此外,三次PH-H曲线也被用于求解具有确定解的$G^1$ Hermite插值问题.文中给出了具体实例来说明我们的方法.     

关于平面凸多边形的一个Bonnesen型不等式

本文探索了关于平面凸多边形的Bonnesen型不等式.利用分析方法,先构造一个解析函数的不等式,进而得到了一个关于平面凸多边形的Bonnesen型不等式.     

Identification of Planar Sextic Pythagorean-Hodograph Curves

Pythagorean-hodograph (PH) curves offer computational advantages in Computer Aided Geometric Design, Computer Aided Design, Computer Graphics, Computer Numerical Control machining and similar applications. In this paper, three methods are utilized to construct the identifications of planar regular sextic PH curves. The first exhibits purely the control polygon legs'' constraints in the complex form. Such reconstruction of a PH sextic can be elaborated by $C^1$ Hermite data and another one condition. The second uses polar representation in two cases. One of them can produce a family of convex sextic PH curves related with a quintic PH curve, and the other one may naturally degenerate a sextic PH curve to a quintic PH curve. In the third identification, we use some odd PH curves to construct a family of sextic PH curves with convexity-preserving property.     

CAN A CUBIC SPLINE CURVE BE G3

This paper proposes a method to construct an G³cubic spline curve from any given open control polygon.For any two inner Bezier points on each edge of a control polygon,we can de ne each Bezier junction point such that the spline curve is G2-continuous.Then by suitably choosing the inner Bezier points,we can construct a global G³spline curve.The curvature combs and curvature plots show the advantage of the G³cubic spline curve in contrast with the traditional C2 cubic spline curve.     

A nonalgebraic patchwork

Itenberg and Shustin’s pseudoholomorphic curve patchworking is in principle more flexible than Viro’s original algebraic one. It was natural to wonder if the former method allows one to construct nonalgebraic objects. In this paper we construct the first examples of patchworked real pseudoholomorphic curves in Σ _n whose position with respect to the pencil of lines cannot be realized by any real algebraic curve of the same bidegree. Both authors are very grateful to the Max Planck Institute für Mathematik in Bonn for its financial support and excellent working conditions.     

First order hermite interpolation with spherical Pythagorean-hodograph curves

The general stereographic projection which maps a point on a sphere with arbitrary radius to a point on a plane stereographically and its inverse projection have the Pythagorean-hodograph (PH) preserving property in the sense that they map a PH curve to another PH curve. Upon this fact, for given spatialC ¹ Hermite data, we construct a spatial PH curve on a sphere that is aC ¹ Hermite interpolant of the given data as follows: First, we solveC ¹ Hermite interpolation problem for the stereographically projected planar data of the given data in ?³ with planar PH curves expressed in the complex representation. Second, we construct spherical PH curves which are interpolants for the given data in ?³ using the inverse general stereographic projection.     

关于平面四次Bézier曲线的拐点与奇点

在计算几何中,已给出了三次Bezier曲线的保凸性的充要条件,并进行了几何解释。本文则是导出形式简洁的拐点和奇点方程并对四次Bezier曲线的拐点和奇点的分布进行讨论。按Bezier曲线的拐点个数进行分类,还得到了四次Bezier曲线有奇点的充分必要条件,并给出几个数值实例,实例说明,不但非凸的单纯特征多角形可以有凸的Bezier曲线段,而且非单纯特征多角形也可以有凸的Bezier曲线段。四次Bezier曲线的奇点和拐点是可以共存的。     

在拼接点达到GC~2连续的双二次曲线与样条的构造

本文研究了连续的双二次曲线 ,给出双二次曲线样条的构造方法 .本方法适用在一些特定点上给定相对曲率并要求整个样条曲线 GC2连续的一类问题 ,采用双二次曲线样条得到的样条曲线具有较好保凸性能 .     

On 2D generalization of Higuchi’s fractal dimension

We propose a new numerical method for calculating 2D fractal dimension (DF) of a surface. This method represents a generalization of Higuchi’s method for calculating fractal dimension of a planar curve. Using a family of Weierstrass–Mandelbrot functions, we construct Weierstrass–Mandelbrot surfaces in order to test exactness of our new numerical method. The 2D fractal analysis method was applied to the set of histological images collected during direct shoot organogenesis from leaf explants. The efficiency of the proposed method in differentiating phases of organogenesis is proved.     

A Theorem on the Convexity of Planar Bezier Curves of Degree n

The puppose of this paper is to prove the following Theorem. If the polygon $\[{P_0}{P_1} \cdots {P_n}{P_0}\]$ formed by the characteristic polygon $\[{P_0}{P_1} \cdots {P_n}{P_0}\]$ of a planar Bezier curve is convex,then so is the Bezier curve. In the case that the angle of rotation from $\[\mathop {{P_0}P{}_1}\limits^ \to \]$ to $\[\mathop {{P_{n - 1}}P{}_n}\limits^ \to \]$ is not larger than \pi,we obtained the theorem by using certain properties of Bernstein polynomials.On the contrary,if the above angle of rotation is larger than \pi,then we cut the oringinal Bezier curve into two new Bezier curves,and prove that the new corresponding characteristic polygons are convex and angles of rotation betweenthe first edge and last edge of the both polygons are not larger than \pi,so that we reduce the latter case into the former discussed case.The theoremis proved. In the present paper we also discuss the distribution of the singular points and inflection points of a planar cubic Bezier curve in detais,and thence give a classification of planar cubic Bezier curves. This paper is prepared under the guidance of Professor Su Buchin.