目的 随着科学技术的快速发展,曲线的几何造型技术开始成为近来的热点研究方向.为了获得更多的变形效果,面向2维、3维参数曲线和自由曲线变形,提出一种带平台伸缩函数的变形方法。方法 有别于现有的大多数自由变形算法,首先构造了一种形式简洁的多项式形式伸缩函数;其次借助于伸缩函数,构造了含有伸缩参数与光滑参数的新型伸缩因子,算法表明,这种新型伸缩因子具有单点峰值性、区间峰值性、对称性等优良性质;最后将伸缩因子所构造变形矩阵作用于待变形的曲线,通过控制变形区间、伸缩参数、光滑参数以及变形方向,可以获得整体的、局部的、周期的、伸缩的等各类丰富的图形效果。结果 此变形操作对造型系统中的主流参数曲线(Bézier和NURBS)具有封闭性;通过大量数值实例表明了该方法计算量小,可控性强,重复使用可以得到形状多样、具有艺术效果的轮廓线等效果。结论 与其他方法相比,本文算法不仅可以用于一般的平面与空间参数曲线,也可以用于自由型曲线,扩大了多数自由变形算法的适用范围;由于伸缩函数具备单点峰值性、区间峰值性、对称性等性质,从而能够产生以前变形方法无法产生各类角点、尖点的特殊曲线,在一定程度上极大丰富了曲线的变形效果。

Objective With the rapid development of science and technology, geometric modeling techniques for curves have recently gained significant research interest. According to the 2D and 3D freeform deformation of parametric curves and free curves, a new algorithm based on polynomial extension function with platform is proposedto obtain several deformation effects. The new deformation method possesses perfect properties such as interval peak, peak value and symmetries. Furthermore, it has the closure property for deformations of Bézier and NURBS curves which are the mainFstream curves of present modeling system. Method A simple extension function is first presented. Unlike most existing freeform deformation methods, the proposed extension function has an ordinary polynomial form. Hence, this extension function does not include transcendental functions and complex functions. With this extension function, one new extension factor with extension and smoothing parameters is also established. This new extension factor constructed with the extension function possesses perfect properties, including interval peaks, peak values, and symmetries. Finally, the extension matrixes are applied to the parametric and freeform curves to obtain the global, local, periodic, and elastic deformation effects. This objective is achieved by adjusting the deformation intervals, extension parameters, smoothing parameters, and deformation directions. Result The deformation method remains self-enclosed when implemented on Bezier and NURBS curves, which are mainstream curves in CAGD. Several numerical examples show that the proposed method is easy to calculate and control and that it can be used repeatedly to achieve a large number of contour lines. Conclusion Compared with many freeform deformation methods, the proposed algorithm can be used for not only general 2D and 3D parametric curves but also freeform curves. Thus, it demonstrates extensive application scope. Because the extension function has perfect properties such as interval peak, peak value and symmetries, it can produce different kinds of special curves with angular points and cuspidal points which the other deformation method can't achieve. In sum, the proposed algorithm greatly enriches the deformation effects of curves.