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Front. Inform. Technol. Electron. Eng.  2010, Vol. 11 Issue (4): 278-289    DOI: 10.1631/jzus.C0910148
    
Representing conics by low degree rational DP curves
Qian-qian Hu1,2, Guo-jin Wang*,1
1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China 2 College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
Representing conics by low degree rational DP curves
Qian-qian Hu1,2, Guo-jin Wang*,1
1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China 2 College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
 全文: PDF(341 KB)  
摘要： A DP curve is a new kind of parametric curve defined by Delgado and Pe?a (2003); it has very good properties when used in both geometry and algebra, i.e., it is shape preserving and has a linear time complexity for evaluation. It overcomes the disadvantage of some generalized Ball curves that are fast for evaluation but cannot preserve shape, and the disadvantage of the Bézier curve that is shape preserving but slow for evaluation. It also has potential applications in computer-aided design and manufacturing (CAD/CAM) systems. As conic section is often used in shape design, this paper deduces the necessary and sufficient conditions for rational cubic or quartic DP representation of conics to expand the application area of DP curves. The main idea is based on the transformation relationship between low degree DP basis and Bernstein basis, and the representation theory of conics in rational low degree Bézier form. The results can identify whether a rational low degree DP curve is a conic section and also express a given conic section in rational low degree DP form, i.e., give positions of the control points and values of the weights of rational cubic or quartic DP conics. Finally, several numerical examples are presented to validate the effectiveness of the method.
关键词： Conic sections;  Bernstein basis;  DP basis;  Rational low degree Bézier curves;  Rational low degree DP curves    
Abstract: A DP curve is a new kind of parametric curve defined by Delgado and Pe?a (2003); it has very good properties when used in both geometry and algebra, i.e., it is shape preserving and has a linear time complexity for evaluation. It overcomes the disadvantage of some generalized Ball curves that are fast for evaluation but cannot preserve shape, and the disadvantage of the Bézier curve that is shape preserving but slow for evaluation. It also has potential applications in computer-aided design and manufacturing (CAD/CAM) systems. As conic section is often used in shape design, this paper deduces the necessary and sufficient conditions for rational cubic or quartic DP representation of conics to expand the application area of DP curves. The main idea is based on the transformation relationship between low degree DP basis and Bernstein basis, and the representation theory of conics in rational low degree Bézier form. The results can identify whether a rational low degree DP curve is a conic section and also express a given conic section in rational low degree DP form, i.e., give positions of the control points and values of the weights of rational cubic or quartic DP conics. Finally, several numerical examples are presented to validate the effectiveness of the method.
Key words: Conic sections    Bernstein basis    DP basis    Rational low degree Bézier curves    Rational low degree DP curves
收稿日期: 2009-03-13 出版日期: 2010-03-22
CLC:  TP391.72  
基金资助: Project  supported  by  the  National  Natural  Science  Foundation  of China (Nos. 60873111 and 60933007), and the Natural Science Foun-
dation of Zhejiang Province, China (No. Y6090211)
通讯作者: Guo-jin WANG     E-mail: wanggj@zju.edu.cn
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引用本文:

Qian-qian Hu, Guo-jin Wang. Representing conics by low degree rational DP curves. Front. Inform. Technol. Electron. Eng., 2010, 11(4): 278-289.

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http://www.zjujournals.com/xueshu/fitee/CN/10.1631/jzus.C0910148    或    http://www.zjujournals.com/xueshu/fitee/CN/Y2010/V11/I4/278

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