带有形状参数的八次Bezier曲线

Eighth Times Bezier Curve with a Shape Parameter

摘要: 为了有效地改进Bezier曲线的形状，给出了一组带有形状参数β的八次多项式基函数，是七次Bernstein基函数的拓展；分析了这组基的性质，基于该组基定义了带有形状参数β的八次多项式曲线。该曲线不仅具有七次Bezier曲线的特性，如：端边相切、凸包性、保凸性、变差缩减性等，而且具有形状的可调性和更好的逼近性。在控制多边形顶点不变的情况下，通过调整形状参数β，可以产生逼近多边形的不同曲线。当β=0时，曲线退化为七次Bezier曲线。应用实例表明，本文定义的曲线为曲线/曲面的设计提供了一种可行的方法。

Abstract: In order to effectively improve the shape of Bezier curve. In this paper, a set of eighth times polynomial basis function with shape parameter β is given, this is an extension of the seventh times Bernstein basis function. Next, properties of thebasis are analyzed, and a polynomial curve with a shape parameter β is defined based on it. This curve not only has the characteristics of the seventh times Bezier curve, such as tangent of end-edge, convex envelope, convexity-preserving and variation-reducing, but also has the shape adjustability and better approximation. Under the condition that the vertices of the control polygon are invariable, different curves approximating the polygon can be produced by adjusting the shape parameter β. When β=0, the curve degenerates into a seventh times Bezier curve. At last, some examples illustrate that the curves defined in this paper provide a feasible method for the design of curves/surfaces.