Rational cubic/quartic Said-Ball conics  

作　　者：HU Qian-qian WANG Guo-jin 

机构地区：[1]Department of Mathematics, Zhejiang University, Hangzhou 310027 [2]College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018.

出　　处：《Applied Mathematics(A Journal of Chinese Universities)》2011年第2期198-212,共15页高校应用数学学报（英文版）（B辑）

基　　金：Supported by the National Natural Science Foundations of China(61070065, 60933007);the Zhejiang Provincial Natural Science Foundation of China(Y6090211)

摘　　要：In CAGD, the Said-Ball representation for a polynomial curve has two advantagesover the B′ezier representation, since the degrees of Said-Ball basis are distributed in a step type.One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomialcurve runs twice as fast as the de Casteljau algorithm of B′ezier curve. Another is that theoperations of degree elevation and reduction for a polynomial curve in Said-Ball form are simplerand faster than in B′ezier form. However, Said-Ball curve can not exactly represent conics whichare usually used in aircraft and machine element design. To further extend the utilizationof Said-Ball curve, this paper deduces the representation theory of rational cubic and quarticSaid-Ball conics, according to the necessary and su？cient conditions for conic representation inrational low degree B′ezier form and the transformation formula from Bernstein basis to Said-Ballbasis. The results include the judging method for whether a rational quartic Said-Ball curve is aconic section and design method for presenting a given conic section in rational quartic Said-Ballform. Many experimental curves are given for confirming that our approaches are correct ande？ective.In CAGD, the Said-Ball representation for a polynomial curve has two advantagesover the B′ezier representation, since the degrees of Said-Ball basis are distributed in a step type.One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomialcurve runs twice as fast as the de Casteljau algorithm of B′ezier curve. Another is that theoperations of degree elevation and reduction for a polynomial curve in Said-Ball form are simplerand faster than in B′ezier form. However, Said-Ball curve can not exactly represent conics whichare usually used in aircraft and machine element design. To further extend the utilizationof Said-Ball curve, this paper deduces the representation theory of rational cubic and quarticSaid-Ball conics, according to the necessary and su？cient conditions for conic representation inrational low degree B′ezier form and the transformation formula from Bernstein basis to Said-Ballbasis. The results include the judging method for whether a rational quartic Said-Ball curve is aconic section and design method for presenting a given conic section in rational quartic Said-Ballform. Many experimental curves are given for confirming that our approaches are correct ande？ective.

关 键 词：Rational Said-Ball curve Rational B′ezier curve conics. 

分 类 号：O174.41[理学—数学] TP391.72[理学—基础数学]