Bézier曲线到AH-Bézier曲线的升阶算法

计算机工程与应用 ›› 2014, Vol. 50 ›› Issue (17): 7-11.

Bézier曲线到AH-Bézier曲线的升阶算法

沈莞蔷1，汪国昭2，3

1.江南大学理学院，江苏无锡 214122
2.浙江大学 CAD & CG国家重点实验室，杭州 310027
3.浙江大学数学系，杭州 310027

出版日期:2014-09-01 发布日期:2014-09-12

Degree elevation algorithm from Bézier curve to AH-Bézier curve

SHEN Wanqiang1, WANG Guozhao2，3

1.School of Science, Jiangnan University, Wuxi, Jiangsu 214122, China
2.State Key Lab of CAD & CG, Zhejiang University, Hangzhou 310027, China
3.Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Online:2014-09-01 Published:2014-09-12

摘要/Abstract

摘要： 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。

关键词: Bé, zier曲线, AH-Bé, zier曲线, 升阶, 基函数, 转换矩阵

Abstract: 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 algorithm from Bézier curve, defined on algebraic polynomial space, to AH-Bézier curve, defined on algebraic and hyperbolic polynomial space. The study begins with basis functions. Firstly, the transformation matrix from AH-Bézier basis to Bernstein basis is built by using the block matrix idea and the same property of Bézier and AH-Bézier 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 Bézier curve of degree n can be turned into an AH-Bézier curve of order n+3 （i.e. degree n+2） by using this algorithm. The algorithm gives an accurate transformation from Bézier to AH-Bézier curve model.

Key words: Bézier curve, AH-Bézier curve, degree elevation, basis function, transformation matrix

沈莞蔷1，汪国昭2，3. Bézier曲线到AH-Bézier曲线的升阶算法[J]. 计算机工程与应用, 2014, 50(17): 7-11.

SHEN Wanqiang1, WANG Guozhao2，3. Degree elevation algorithm from Bézier curve to AH-Bézier curve[J]. Computer Engineering and Applications, 2014, 50(17): 7-11.

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