单位圆弧上复二次Bézier曲线

实空间中的Bezier曲线在计算机辅助设计和制造(CAD/CAM)中起着重要的作用,尤其二次和三次Bezier曲线的应用十分广泛.将复样条函数作为逼近工具的研究工作已有[1]—[4],但几何性质的研究尚罕见,难以在CAD/CAM中得到应用.本文先对单位圆弧上的复二次Bezier曲线的几何性质(特别是凸性)作了一些较深入的讨论,再以它们为基本曲线段给出一种构造一阶几何连续(GC~1)的插值复样条曲线的方法.此样     

封闭曲线上的奇异积分方程的样条间接逼近解法

利用复插值样条函数，给出了定义于光滑封闭曲线上一般的正则型奇异积分方程的样条间接逼近解法，证明了一致收敛性．对于其中的一类奇异积分方程，还给出了近似解和误差估计．     

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。     

带有给定切线多边形的β样条曲线

对于给定的多边形，本讨论了与它相切的β样条曲线，其方法易懂且有效。     

一种带参数的有理三次样条函数及其应用

构造了一种带参数的有理三次样条函数,它是标准三次样条函数的推广.选择合适的参数,该样条曲线比标准三次插值曲线更加逼近被插值曲线.参数还能局部调节曲线的形状,这给约束控制带来了方便.研究了该种插值曲线的区域控制问题.给出了将其约束于给定的二次曲线之上、之下或之间的充分条件.文中给出了两个数值例子.     

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．     

一种四次有理插值样条及其逼近性质

1引言有理样条函数是多项式样条函数的一种自然推广,但由于有理样条空间的复杂性,所以有关它的研究成果不象多项式样条那样完美,许多问题还值得进一步的研究．近几十年来,有理插值样条,特别是有理三次有理插值样条,由于它们在曲线曲面设计中的应用,已有许多学者进行了深入研究,取得了一系列的成果(见[1]-[7])．但四次有理插值样条由于其构造所花费的计算量太大以及在使用上很不方便而让人们忽视了其重要的应用价值,因此很少有人研究他们．实际上,在某些情况下四次有理插值样条有其独特的应用效果,如文[8]建立的一种具有局部插值性质的分母为二次的四次有理样条,即一个剖分     

分片代数曲线交点的结式求法

本文对确定分片代数曲线的二元样条函数的整体表达式中的截断引入参数表示,给出了分片代数曲线交点的结式求法．理论与实例表明,这种算法是有效的．     

样条的机械化求解

本文在逐次分解法的基础上，给出一种样条机械化求解方法．该方法对多项式样条，有理样条乃至更一般样条的研究都是十分有效的．它适用于三角剖分，矩形剖分乃至更一般的代数曲线剖分     

NURBS曲线的一个插值性质

本介绍了非均匀有理B样条曲线，并给出了非均匀有理B样条曲线的一个插值性质。     

圆弧曲线的有理五次Bézier表示

给出了有理五次Bézier曲线精确表示0<2θ<2π(θ为圆心角)的圆弧及整圆的充要条件,加以了证明并给出了图例..     

Drawing planar graphs with circular arcs

In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area. We present a lower bound on the area of drawings in which edges are drawn using exactly one circular arc. We also give an algorithm for drawing n -vertex planar graphs such that the edges are sequences of two continuous circular arcs. The algorithm runs in O(n) time and embeds the graph on the O(n)\times O(n) grid, while maintaining Θ(1/d(v)) angular resolution, where d(v) is the degree of vertex v . Since in this case we use circular arcs of infinite radius, this is also the first algorithm that simultaneously achieves good angular resolution, small area, and at most one bend per edge using straight-line segments. Finally, we show how to create drawings in which edges are smooth C ¹ -continuous curves, represented by a sequence of at most three circular arcs. Received September 30, 1999, and in revised form March 27, 2000. Online publication October 26, 2000.     

Approximation of an open polygonal curve with a minimum number of circular arcs and biarcs

We present an algorithm for approximating a given open polygonal curve with a minimum number of circular arcs. In computer-aided manufacturing environments, the paths of cutting tools are usually described with circular arcs and straight line segments. Greedy algorithms for approximating a polygonal curve with curves of higher order can be found in the literature. Without theoretical bounds it is difficult to say anything about the quality of these algorithms. We present an algorithm which finds a series of circular arcs that approximate the polygonal curve while remaining within a given tolerance region. This series contains the minimum number of arcs of any such series. Our algorithm takes O(n²logn) time for an original polygonal chain with n vertices. Using a similar approach, we design an algorithm with a runtime of O(n²logn), for computing a tangent-continuous approximation with the minimum number of biarcs, for a sequence of points with given tangent directions.     

Pointed drawings of planar graphs

We study the problem how to draw a planar graph crossing-free such that every vertex is incident to an angle greater than π. In general a plane straight-line drawing cannot guarantee this property. We present algorithms which construct such drawings with either tangent-continuous biarcs or quadratic Bézier curves (parabolic arcs), even if the positions of the vertices are predefined by a given plane straight-line drawing of the graph. Moreover, the graph can be drawn with circular arcs if the vertices can be placed arbitrarily. The topic is related to non-crossing drawings of multigraphs and vertex labeling.     

On the length preserving approximation of plane curves by circular arcs

A technique for the length preserving approximation of plane curves by two circular arcs is analyzed. The conditions under which this technique can be applied are extended, and certain consequences of the proved results unrelated to the approximation problem are discussed. More precisely, inequalities for the length of a convex spiral arc subject to the given boundary conditions are obtained. Conjectures on curve closeness conditions obtained using computer simulation are discussed.     

Reflections on Reflections Computer Math Snapshots

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review.This issue’s snapshot explores some generalizations of the definition of geometric reflection. Dynamic geometry tools can facilitate generalizations such as those obtained by relaxing the requirement that the reflection be through a straight line. The author compares the families of curves obtained by reflecting thru circular arcs with the curves generated in response to a physical problem proposed by Wittgenstein. He suggests that the strategy of generalizing definitions is a good avenue for bringing students quickly to the activity of doing mathematics.     

圆弧和球面上的广义Ball多项式

本文以Ｂéｚｉｅｒ多项式理论为基础,引进了圆弧上的广义Ｂａｌ曲线及球面三角剖分上的广义Ｂａｌ曲面及其递归算法．     

Approximation of a planar cubic Bézier spiral by circular arcs

A planar cubic Bézier curve that is a spiral, i.e., its curvature varies monotonically, does not have internal cusps, loops, and inflection points. It is suitable as a design tool for applications in which fair curves are important. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS. When machining objects, it is desirable that as much as possible of a curved toolpath be approximated by a sequence of circular arcs rather than straight-line segments. Such an arc-spline approximation of a planar cubic Bézier spiral is presented.