形状可调二次Bezier曲线

The Quadratic Bezier Curves that Shape can Adjust

给出了一组含两个参数的三次多项式基函数,它是二次Bernstein基函数的扩展,分析了这组基函数的性质。基于这组基定义了带有两个形状参数的多项式曲线,它不仅保留了Bezier曲线的一些实用的几何特征,而且具有形状的可调性。在控制多边形不变的情况下,随着参数α和β的改变,可以生成不同的逼近该控制多边形的曲线。当α和β取某些特殊值时,所定义的曲线可退化为二次Bezier曲线和文献中的一些曲线。通过分析该曲线与三次Bezier曲线之间的关系,给出了α和β明确的几何意义。并利用Bezier曲线的递归分割算法给出了这种曲线的几何作图法,同时还讨论了曲线间的拼接问题。

First a set of polynomial function of degree three with two parameters is presented. It is an extension of the quadratic Bernstein basis function. The quality of the new basis is analyzed. Based on the new basis, the polynomial curve with shape parameters is defined. The new curve not only holds many applied geometrical qualities of Bezier curve, but also can rectify the shape. When the control polygon is fixed, different curves approaching the control polygon with the changing of shape parameters are given. When take α and β some especial values, the new curve can degenerate to the quadratic Bezier curve and the curves in the literatures. The geometrical meaning of α and β are discovered by analyzing the connection of the new curve and the Bezier curve. Meanwhile, the geometrical drawing method of the curve is given with the recursion segmentation algorithm of Bezier curve and the continuity condition of the curve is discussed.