Bézier曲线到AH-Bézier曲线的升阶算法

Degree elevation algorithm from Bézier curve to AH-Bézier curve

关于曲线升阶,已有的结论往往限于同类曲线之间。为了突破这一限制,考虑不同类曲线间的升阶,关注代数多项式空间中的Bézier曲线到代数双曲多项式空间中的AH-Bézier曲线的升阶。研究从基函数入手,利用Bézier和AH-Bézier共有的求导降阶的特点,结合矩阵分块的思想,先给出AH-Bézier基到Bernstein基的转换矩阵,进而推出控制顶点的升阶公式,最后给出升阶算法。结果表明,任意n次Bézier曲线可以通过该算法升到n+3阶(等同于n+2次)的AH-Bézier曲线。算法实现了Bézier到AH-Bézier曲线模型的精确转换。

The existing results about curve degree elevation are mainly limited to the same type of curves. In order to push the limit and consider degree elevation between different types of curves, this paper focuses on degree elevation algo-rithm from B＆#233;zier curve, defined on algebraic polynomial space, to AH-B＆#233;zier curve, defined on algebraic and hyperbolic polynomial space. The study begins with basis functions. Firstly, the transformation matrix from AH-B＆#233;zier basis to Bern-stein basis is built by using the block matrix idea and the same property of B＆#233;zier and AH-B＆#233;zier that the order of basis is reduced for derivative. Secondly, the degree elevation formula of control points is obtained. Lastly, the degree elevation algorithm is given. Results show that any B＆#233;zier curve of degree n can be turned into an AH-B＆#233;zier curve of order n＋3（i.e. degree n＋2）by using this algorithm. The algorithm gives an accurate transformation from B＆#233;zier to AH-B＆#233;zier curve model.