圆锥曲线的三次有理多项式参数化

圆锥曲线重新参数化可以提高曲线参数的均匀性,且增强在拼接点处的光滑性.常用的参数化方法是采用一次有理多项式或二次有理多项式.采用三次有理多项式对圆锥曲线重新参数化,使曲线的次数由二次升到六次.以圆弧为例所得的实验结果袁明,在两段圆弧的公共点处的连续性为C~3,而且三次有理多项式参数化与弧长参数化的弦长偏差相比二次有理多项式参数化减小两个数量级.     

确定空间曲线参数方程的一般方法

确定空间曲线参数方程的一般方法杨孝先，尹业富（中国科技大学数学系２３００２６）在计算曲线的弧长和第一型曲线积分时，如果曲线的方程用两张曲面的交线给出时，计算往往难于下手．本文介绍一个用两张曲面相交表示的空间曲线化为参数方程的方法．在多年的教学中，总感...     

曲线造型的本征离散法

本文提出了一种基于弧长和转角这两个本质几何参数的曲线离散造型方法．该方法计算简单，几何意义明显，适用于逼近（拟合）和插值（补角），还可作出一类分形图形．其特点是一切细分操作都在转角关于弧长的对应关系图上进行．     

常见参数法求轨迹时参数的选取

在通常求轨迹问题中所用的参数有几何参数(角、斜率、弧长、距离、有向线段、点的坐标等)和物理量参数(时间、路程、速度等)两种,而选用哪个量为参数又与动点运动的条件有关,常见的有三种情况.一、固定在动曲线上的点的轨迹这一类点在曲线上的相对位置不变,它的运动是由于曲线的移动、滚动、摆动等造成的。因此,常选用曲线运动时某个变动的角(它刻划动曲线的瞬时位     

导数振荡与应变能极小的平面分段三次参数Hermite插值

由于分段三次参数Hermite插值的切矢往往被作为变量,故可对其进行优化以使得构造的插值曲线满足特定的要求.为了构造兼具保形性与光顺性的平面分段三次参数Hermite插值曲线,给出了一种通过同时极小化导数振荡和应变能来确定切矢的方法.首先以导数振荡函数和应变能函数为双目标建立了切矢满足的方程系统;然后证明了方程系统存在唯一解,并给出了解的具体表达式;最后给出了误差分析,并通过数值算例表明方法的有效性.结果表明,相对于导数振荡极小化方法和应变能极小化方法,所提出的导数振荡和应变能极小化方法同时兼顾了平面分段三次参数Hermite插值曲线的保形性和光顺性.     

一类含参数均匀三角样条插值曲线

基于一类C3连续的三角样条基函数,首先分别构造了含参数α的C2和C3连续的三角样条插值曲线,然后通过在基函数中引入参数λ,构造了含两个参数α,λ的形状可调控插值曲线,通过α,λ的不同取值,可得到一类有较好保凸和保单调效果的插值曲线,最后用图例验证了理论的有效性和正确性.     

正多边形控制的曲线段及其性质

本文给出了由多边形控制曲线段的方法，并依据多边长的延长量的性质，讨论了相应曲线段的性质，并给出了数值例子和对应图象，文末还给出了以曲线段为切线多边形的B－样条曲线方程和以曲线段端点为插值点的插值函数。     

用分段仿射变换生成C^1参数样条曲线

利用分段仿射变换，可将任意函数类型的Ｃ＾１曲线段生成一条Ｃ＾１的参数样条曲线，分而段的保持所选曲线的仿射不变的几何性质。     

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.     

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。     

C~2 CONTINUOUS NEARLY ARC-LENGTH PARAMETERIZED CURVE

An efficient method for C~2 nearly arc-length parameterized curve is presented. An idea of approximation for the arc-length function of parametric curve which interpolates CAD data points is discussed. The parameterization is implemented by using parameter transformation. Finally, two numerical examples are given..     

可调形三次三角Cardinal插值样条曲线

在三次Cardinal插值样条曲线的基础上,引入了三角函数多项式,得到一组带调形参数的三次三角Cardinal样条基函数,以此构造一种可调形的三次三角Cardinal插值样条曲线.该插值样条可以精确表示直线、圆弧、椭圆以及自由曲线,改变调形参数可以调控插值曲线的形状.该插值样条避免了使用有理形式,其表达式较为简洁,计算量也相对较少,从而为多种线段的构造与处理提供了一种通用与简便的方法.     

Arc Length Estimation and the Convergence of Polynomial Curve Interpolation

When fitting parametric polynomial curves to sequences of points or derivatives we have to choose suitable parameter values at the interpolation points. This paper investigates the effect of the parameterization on the approximation order of the interpolation. We show that chord length parameter values yield full approximation order when the polynomial degree is at most three. We obtain full approximation order for arbitrary degree by developing an algorithm which generates more and more accurate approximations to arc length: the lengths of the segments of an interpolant of one degree provide parameter intervals for interpolants of degree two higher. The algorithm can also be used to estimate the length of a curve and its arc-length derivatives. AMS subject classification (2000) 65D05, 65D10     

On parametric polynomial circle approximation

In the paper, the uniform approximation of a circle arc (or a whole circle) by a parametric polynomial curve is considered. The approximant is obtained in a closed form. It depends on a parameter that should satisfy a particular equation, and it takes only a couple of tangent method steps to compute it. For low degree curves, the parameter is provided exactly. The distance between a circle arc and its approximant asymptotically decreases faster than exponentially as a function of polynomial degree. Additionally, it is shown that the approximant could be applied for a fast evaluation of trigonometric functions too.     

Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines

Summary A method is presented for fitting a function defined on a general smooth spherelike surfaceS, given measurements on the function at a set of scattered points lying onS. The approximating surface is constructed by mapping the surface onto a rectangle, and using a tensor-product of polynomial splines with periodic trigonometric splines. The use of trigonometric splines allows a convenient solution of the problem of assuring that the resulting surface is continuous and has continuous tangent planes at all points onS. Two alternative algorithms for computing the coefficients of the tensor fit are presented; one based on global least-squares, and the other on the use of local quasi-interpolators. The approximation order of the method is established, and the numerical performance of the two algorithms is compared.Supported in part by the National Science Foundation under Grant DMS-8902331 and by the Alexander von Humboldt Foundation     

A control point based curve with two exponential shape parameters

A generalization of a recently developed trigonometric Bézier curve is presented in this paper. The set of original basis functions are generalized also for non-trigonometric functions, and essential properties, such as linear independence, nonnegativity and partition of unity are proved. The new curve—contrary to the original one—can be defined by arbitrary number of control points meanwhile it preserves the properties of the original curve.     

A planar cubic Bézier spiral

A planar cubic Bézier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.     

On the curvatures of midcurve and gable curve

Introducing in plane affine differential geometry a gable curve to a convex arc as the set of points of intersection for pairs of tangents to the convex arc at endpoints of parallel chords it is shown, that the gable curve and the midcurve for the same chords combined form a curve with a point of inflection at their meeting point P and such that the ratio of their curvatures tends to 3 at P, independent of the convex arc.     

Circular spline fitting using an evolution process

We propose a new method to approximate a given set of ordered data points by a spatial circular spline curve. At first an initial circular spline curve is generated by biarc interpolation. Then an evolution process based on a least-squares approximation is applied to the curve. During the evolution process, the circular spline curve converges dynamically to a stable shape. Our method does not need any tangent information. During the evolution process, the number of arcs is automatically adapted to the data such that the final curve contains as few arc arcs as possible. We prove that the evolution process is equivalent to a Gauss-Newton-type method.