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Front. Inform. Technol. Electron. Eng.  2014, Vol. 15 Issue (12): 1098-1105    DOI: 10.1631/jzus.C1400076
    
Degree elevation of unified and extended spline curves
Xiao-juan Duan, Guo-zhao Wang
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
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Abstract  Unified and extended splines (UE-splines), which unify and extend polynomial, trigonometric, and hyperbolic B-splines, inherit most properties of B-splines and have some advantages over B-splines. The interest of this paper is the degree elevation algorithm of UE-spline curves and its geometric meaning. Our main idea is to elevate the degree of UE-spline curves one knot interval by one knot interval. First, we construct a new class of basis functions, called bi-order UE-spline basis functions which are defined by the integral definition of splines. Then some important properties of bi-order UE-splines are given, especially for the transformation formulae of the basis functions before and after inserting a knot into the knot vector. Finally, we prove that the degree elevation of UE-spline curves can be interpreted as a process of corner cutting on the control polygons, just as in the manner of B-splines. This degree elevation algorithm possesses strong geometric intuition.

Key words： Degree elevation      Unified and extended splines (UE-splines)      Bi-order UE-splines      Corner cutting      Geometric explanation     
Received: 06 March 2014      Published: 05 December 2014
CLC:  TP391.7  
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Xiao-juan Duan
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Xiao-juan Duan, Guo-zhao Wang. Degree elevation of unified and extended spline curves. Front. Inform. Technol. Electron. Eng., 2014, 15(12): 1098-1105.

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http://www.zjujournals.com/xueshu/fitee/10.1631/jzus.C1400076     OR     http://www.zjujournals.com/xueshu/fitee/Y2014/V15/I12/1098


UE样条曲线的升阶

针对UE样条悬而未决的升阶问题，给出UE样条的升阶方法，并揭示此方法的几何意义。引入一种新的样条基函数-双阶UE样条基函数。在原始节点向量中逐个插入互异节点，将UE样条函数按区间逐段升阶，最终使UE样条在整个定义域内达到升阶效果，并给出这种升阶方法的几何意义。由于曲线在节点处的连续性保持不变，低阶的UE样条曲线可由高阶UE样条曲线表示。首先，引入一种新的样条基函数-双阶UE样条基函数。这种样条基在整个节点区间有两种阶数。其中，前一段节点区间的次数比后一段节点区间的次数高1次（图1）。然后，通过往节点向量中插入节点，双阶UE样条基的某特定区间次数升高1次，从而得到双阶UE样条在节点插入前后的基函数关系（图2）继而得到节点插入前后双阶UE样条函数控制顶点之间的关系。通过逐个插入互异节点，可使UE样条逐段升阶。最后，根据节点插入前后的新旧控制顶点关系，证明UE样条的升阶可以理解为其控制多边形的割角过程（图3、4）。通过在节点向量中逐个插入互异节点，解决了UE样条的升阶问题，并证明了UE样条的升阶可以解释为其控制多边形的割角过程。

关键词： 升阶,  UE样条,  双阶UE样条,  割角,  几何解释 
[1] Juan Cao, Guo-zhao Wang. Non-uniform B-spline curves with multiple shape parameters[J]. Front. Inform. Technol. Electron. Eng., 2011, 12(10): 800-808.