Degree elevation of unified and extended spline curves

Xiao-juan DUAN; Guo-zhao WANG

doi:10.1631/jzus.C1400076

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Degree elevation of unified and extended spline curves

Computer & Automation | Updated：2023-11-23

- Degree elevation of unified and extended spline curves
- UE样条曲线的升阶
- Frontiers of Information Technology & Electronic Engineering Vol. 15, Issue 12, Pages: 1098-1105(2014)
- Affiliations：
  
  Department of Mathematics, Zhejiang University, Hangzhou 310027, China
- Author bio：
  
  Xiao-juan DUAN, E-mail: heng-juan0721@163.com
  [ "Guo-zhao WANG, E-mail: wanggz@zju.edu.cn" ]
- Funds：
- DOI：10.1631/jzus.C1400076
  CLC： TP391.7
- Received：06 March 2014，
  
  Revised：2014-11-13，
  
  Published：2014-12
- Accepted：
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Xiao-juan DUAN, Guo-zhao WANG. Degree elevation of unified and extended spline curves[J]. Frontiers of Information Technology & Electronic Engineering, 2014, 15(12): 1098-1105.
DOI：

Xiao-juan DUAN, Guo-zhao WANG. Degree elevation of unified and extended spline curves[J]. Frontiers of Information Technology & Electronic Engineering, 2014, 15(12): 1098-1105. DOI： 10.1631/jzus.C1400076.

摘要

研究目的

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。

创新要点

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。

研究方法

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）。

重要结论

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。

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.

关键词

Keywords

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