STUDY OF THE SMOOTH DEGREE OF THE CLOSED C-B（E）ZIER AND B-TYPE SPLINE CURVE

本文研究了与多边形相切的样条曲线的构造方法和基本属性问题,给出了曲线光顺度的一般定义和计算方法.利用该方法对分段C-Bézier曲线、4-5-5-4次交错B-样条曲线和3阶B样条曲线的光顺度进行计算,获得了3阶B样条曲线最为光顺的结果.     

C-Bézier和B-型样条闭曲线光滑性研究

本文研究了与多边形相切的样条曲线的构造方法和基本属性问题,给出了曲线光顺度的一般定义和计算方法.利用该方法对分段C-Bézier曲线、4-5-5-4次交错B-样条曲线和3阶B样条曲线的光顺度进行计算,获得了3阶B样条曲线最为光顺的结果.     

高阶C-B样条曲线曲面的显式形式

1引言基于多项式空间span{1,t,t~2,…,t~k}的B样条和Bézier曲线(面)是构造自由曲线、曲面强有力的工具,但是它们不能精确表示圆弧、椭圆等,也不能精确地表示正弦曲线和二次曲面,于是文献[1]提出了一种新的三次曲线(面)模型,称为C曲线(面),它们是低次多项式样条曲线(面)的拓广,具有很多B样条的良好性质,如对称性、保凸性等,不仅能精确表示二次曲线和曲面和某些超越曲线,而且克服了NURBS求导求积复杂的困难,因此引起了国内外广泛的关注,近年来涌现了大量的文献.文[5]将其三次C-B样条推广到了高阶的情形,给出了任意阶C-B样条曲线(面)的积分递推公式,并发展了许多诸     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

数控系统中封闭曲线三次均匀B样条插补不平滑处理研究

数控系统中经常会用到样条插值,对于不封闭曲线,一般的样条插值,就能解决,但对于封闭曲线,首尾控制点的处理不当,传统的增加控制点的方法经常会出现不平滑,甚至锐角,本文就应用埃尔米特插值解决这一问题,并在Visual C++上进行仿真实现.     

圆弧样条的单圆弧逼近

<正> 长期以来奇次样条在实践中有广泛的应用,在生产上有显著的成效,而偶次样条在某种程度上受到了忽视.实际上偶次样条有着自己的特点。对此有许多文章做了详细的论述.例如,在生产实践中(如用数控线切割机进行金属切割),往往需拟合一些非圆曲线或拟合一些离散点,当然三次(或三次以上的)多项式可以拟合非圆曲线及离散点,但三次曲线本身却不能直接用简单的工具准确地绘出,数控线切割机的基本加工轨迹也仅是圆与直线,因此在加工非圆曲线时,往往还沿用原始的方法——用直线段(一次样条)逼近.其缺点是显然的.(1)易产生角点,即在两条直线段的衔接处(节点处)不光滑;(2)在     

一类C^3—连续的B—型样条曲线及其插值与逼近性

本文提出一类Ｃ３－连续的带有因子的Ｂ－型参数样条曲线,它的每一段只要四个 控制点就能生成,可用它直接插值或逼近于任意控制点或对控制边多边形作局部或整体逼 近。利用因子间的某些关系可将其次数降到最低．与普通的四次Ｂ－样条曲线相比,这类 曲线更加方便灵活。     

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.     

射线跟踪技术中物体外形精确建模算法

为了对复杂物体外形进行快速精确建模,利用多面体样条方法进行物体表面的设计及应用研究.通过对样条函数节点进行分类,给出空间中任意拓扑结构网格上样条函数组的具体构造方法.研究曲线曲面控制点与生成样条函数的节点的相互依赖关系,提出在无单位分解条件下构造具有几何不变性曲线曲面的合理化方法.通过构造控制点处多面体样条函数,生成了任意拓扑结构的连续多面体样条曲面.这样的多面体样条曲面在同样性质要求下曲面次数低,只有张量积NURBS曲面的一半;并且与细分曲面相比,所获取的物体外形表面具有坐标计算准确,中间数据较少等优点,可以用于精确构造电磁射线追踪中物体的外形.     

基于函数值的有理三次插值样条曲线的区域控制

将插值曲线约束于给定的区域之内是曲线形状控制中的重要问题.构造了一种基于函数值的分母为三次的C~1连续有理三次插值样条.这种有理三次插值样条中含有二个调节参数,因而给约束控制带来了方便.对该种插值曲线的区域控制问题进行了研究,给出了将其约束于给定的折线、二次曲线之上、之下或之间的充分条件.最后给出了数值例子.     

C-Bézier曲线降阶的B网扰动和约束优化法

基于B网扰动和约束优化方法,对n次C-Bézier曲线控制多边形顶点进行扰动,并找到其退化为n-1次次C-Bézier曲线的条件.在满足退化条件的约束下,使n次C-Bézier曲线控制多边形顶点扰动量最小,由此找到降阶为n-1次的C-Bézier曲线,同时也研究了在C~0,C~1连续条件下对n次C-Bézier曲线降阶的B网扰动和约束优化方法.给出了扰动显示格式计算方法和降阶逼近的误差估计式.     

G2 composite cubic Bézier curves

We present an algorithm for creating planar G² spline curves using rational Bézier cubic segments. The splines interpolate a sequence of points, tangents and curvatures. In addition each segment has two more geometric shape handles. These are obtained from an analysis of the singular point of the curve. The individual segments are convex, but zero curvature can be assigned at a junction point, hence inflection points can be placed where desired but cannot occur otherwise.     

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.     

整体最优双圆弧拟合

提出了用整体最优双圆弧样条拟合离散数据点的算法．首先分析了双圆弧逼近的误差分布,并且根据这个误差分布调整端点、切向和双圆弧插值时的惟一自由变量,使得数据点的误差分布均匀、圆弧的段数尽可能少,由此得到G^1连续的整体最优双圆弧样条．这个方法在数值控制刀具的运动路线的设计和机器人的移动路线设计上非常有用。     

Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form

The problems of determining the B–spline form of a C ² Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C ² quintic bases on uniform triple knots are constructed for both open and closed C ² curves, and are used to derive simple explicit formulae for the B–spline control points of C ² PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C ² to C ¹. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.     

Motion by curvature of planar curves with end points moving freely on a line

This paper deals with the motion by curvature of planar curves having end points moving freely along a line with fixed contact angles to this line. We first prove the existence and uniqueness of self-similar shrinking solution. Then we show that the curve shrinks to a point in a self-similar manner, if initially the curve is a graph.     

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.     

OnG 2 continuous cubic spline interpolation

In this paper the interpolation byG ² continuous planar cubic Bézier spline curves is studied. The interpolation is based upon the underlying curve points and the end tangent directions only, and could be viewed as an extension of the cubic spline interpolation to the curve case. Two boundary, and two interior points are interpolated per each spline section. It is shown that under certain conditions the interpolation problem is asymptotically solvable, and for a smooth curvef the optimal approximation order is achieved. The practical experiments demonstrate the interpolation to be very satisfactory. Supported in prat by the Ministry of Science and Technology of Slovenjia, and in part by the NSF and SF of National Educational Committee of China.