Constrained Bézier curves' best multi-degree reduction in the L_2-norm  

作　　者：ZHANG Renjiang WANG Guojin 

机构地区：[1]Institute of Computer Images and Graphics, Zhejiang University, Hangzhou 310027, China [2]State Key Laboratory of CAD＆CG,Zhejiang University, Hangzhou 310027, China [3]China Institute of Metrology, Zhejiang, Hangzhou 310034, China

出　　处：《Progress in Natural Science:Materials International》2005年第9期843-850,共8页自然科学进展·国际材料（英文版）

基　　金：SupportedbyNationalNaturalScienceFoundationofChina(GrantNos.60373033and60333010),theNationalNaturalScienceFoundationforInnovativeResearchGroups(GrantNo.60021201),andtheMajorStateBasicResearchDevelopmentProgramofChina(GrantNo.2004CB719400)

摘　　要：In computer aided geometric design, the degree reduction of the parameter curve is a key technique in data exchange and data compression. The various existing methods of degree reduction cannot decide whether the degree reduction curve satisfying the given tolerance exists beforehand, cannot give approximation of the best multi-degree reduction, or cannot provide explicit expression and error formula of the degree reduction curve. In this paper, we propose an entirely new method, which can hurdle the above flaws completely. For a given Beézier curve of degree n, we can easily decide whether a Bézier reduction curve of degree m exists, which has equal, derivatives with the given curve up to （ r - 1）-th and （ s - 1） th orders （ r, s≤ m 〈 n ） respectively at the endpoints, so the approximating error is less than the given tolerance e in the L2-norm. If the curve exists, the explicit expression can be given.In computer aided geometric design, the degree reduction of the parameter curve is a key technique in data exchange and data compression. The various existing methods of degree reduction cannot decide whether the degree reduction curve satisfying the given tolerance exists beforehand, cannot give approximation of the best multi-degree reduction, or cannot provide explicit expression and error formula of the degree reduction curve. In this paper, we propose an entirely new method, which can hurdle the above flaws completely. For a given Bézier curve of degree n, we can easily decide whether a Bézier reduction curve of degree m exists, which has equal derivatives with the given curve up to （r-1）-th and （s-1）-th orders （r,s≤m） respectively at the endpoints, so the approximating error is less than the given tolerance ε in the L2-norm. If the curve exists, the explicit expression can be given.

关 键 词：computer aided design Bézier curve  degree reduction  L2-norm  tolerance 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]