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对可调控Bézier曲线的改进

严兰兰,韩旭里(东华理工大学理学院, 南昌 330013;中南大学数学与统计学院, 长沙 410083)

摘 要
目的 在用Bézier曲线表示复杂形状时,相邻曲线的控制顶点间必须满足一定的光滑性条件。一般情况下,对光滑度的要求越高,条件越复杂。通过改进文献中的“可调控Bézier曲线”,以构造具有多种优点的自动光滑分段组合曲线。方法 首先给出了两条位置连续的曲线Gl连续的一个充分条件,进而证明了“可调控Bézier曲线”在普通Bézier曲线的Gl光滑拼接条件下可达Gl(l为曲线中的参数)光滑拼接。然后对“可调控Bézier基”进行改进得到了一组新的基函数,利用该基函数按照Bézier曲线的定义方式构造了一种新曲线。分析了该曲线的光滑拼接条件,并根据该条件定义了一种分段组合曲线。结果 对于新曲线而言,只要前一条曲线的最后一条控制边与后一条曲线的第1条控制边重合,两条曲线便自动光滑连接,并且在连接点处的光滑度可以简单地通过改变参数的值来自由调整。由新曲线按照特殊方式构成的分段组合曲线具有类似于B样条曲线的自动光滑性和局部控制性。不同的是,组合曲线的各条曲线段可以由不同数量的控制顶点定义,选择合适的参数,可以使曲线在各个连接点处达到任何期望的光滑度。另外,改变一个控制顶点,至多只会影响两条曲线段的形状,改变一条曲线段中的参数,只会影响当前曲线段的形状,以及至多两个连接点处的光滑度。结论 本文给出了构造易于拼接的曲线的通用方法,极大简化了曲线的拼接条件。此基础上,提出的一种新的分段组合曲线定义方法,无需对控制顶点附加任何条件,所得曲线自动光滑,且其形状、光滑度可以或整体或局部地进行调整。本文方法具有一般性,为复杂曲线的设计创造了条件。
关键词
Improvement of the modifiable Bézier curves

Yan Lanlan,Han Xuli(College of Science, East China Institute of Technology, Nanchang 330013, China;School of Mathematics and Statistics, Central South University, Changsha 410083, China)

Abstract
Objective When a Bézier curve is used to describe complex shapes, the problem of joining curve segments smoothly has to be solved. To maintain the continuity of the whole curve, adjacent curve segments must meet strict continuity conditions. A higher the requirement for continuity usually causes conditions to become more complex and involves a larger number of control points. This study improves the modifiable Bézier curve in the literature to achieve smooth connection between curves automatically and to construct a piecewise composite curve with numerous merits. Method We first present a sufficient condition of Gl continuity for two curves with continuous position.On the basis of the sufficient condition Gl, we prove that the modifiable Bézier curve can achieve a smooth connection under conditions that usually guarantee Gl continuity only for the usual Bézier curve and most Bézier-like curves in the literature. We then use a transition matrix to convert the modifiable Bézier basis to a new set of basis functions. We employ this set of basis functions to define a new kind of curve according to the definition mode of the standard Bézier curve. We then analyze the smooth connection conditions of the new curve.Considering theses smooth connection conditions and by using a special definition mode, we construct a kind of piecewise composite curve. The connection of the control points between adjacent curve segmentsis apparently similar to that of the classical B-spline curve. However, the connections are actually different. The B-spline curve only has one edge between the control polygons of two adjacent curve segments, that is, only one control point is different. Nevertheless, the composite curve defined in this study only has one edge. Thus, only two control points are the same. Result The new curve defined by the new basis function has relatively simple and special smooth connection conditions. Two neighboring curves can be smoothly joined automatically as long as the last control edge of the former coincides with the first control edge of the latter. Furthermore, the degree of smoothness at the meeting point can be freely adjusted by simply changing the value of the parameter. The piecewise composite curve of the new curve possesses numerous desirable properties, such as geometric invariance, symmetry, automatic smoothness property, and local control capability similar to that of the classical B-spline curve. However, the main difference between the newly defined piecewise curve and the standard B-spline curve is that in the new composite curve, each segment can be defined by different numbers of control points. By contrast, the number of segments must be equal forthe usual B-spline curve.This difference is the main reason that the new piecewise curve has a stronger local control capability than the B-spline curve. Furthermore, a suitable parameter can enable the new composite curve to achieve the expected smoothness at each connection point. In addition, changing one control point can alter the shape of two curve segments. When the parameter of one curve segment is changed, only the shape of the current curve and the degree of smoothness at two connection points will change. Conclusion This study presents the general Method to construct curves. This Method can easily achieve a smooth connection. We further present the general Method to construct composite curves. This Method can achieve a smooth connection automatically.
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