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Journal of Zhejiang University SCIENCE C 2010 Vol.11 No.4 P.278-289

http://doi.org/10.1631/jzus.C0910148


Representing conics by low degree rational DP curves


Author(s):  Qian-qian Hu, Guo-jin Wang

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China, College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

Corresponding email(s):   qianqian_hu@163.com, wanggj@zju.edu.cn

Key Words:  Conic sections, Bernstein basis, DP basis, Rational low degree Bé, zier curves, Rational low degree DP curves


Qian-qian Hu, Guo-jin Wang. Representing conics by low degree rational DP curves[J]. Journal of Zhejiang University Science C, 2010, 11(4): 278-289.

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author="Qian-qian Hu, Guo-jin Wang",
journal="Journal of Zhejiang University Science C",
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year="2010",
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doi="10.1631/jzus.C0910148"
}

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%A Guo-jin Wang
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%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.C0910148

TY - JOUR
T1 - Representing conics by low degree rational DP curves
A1 - Qian-qian Hu
A1 - Guo-jin Wang
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VL - 11
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SP - 278
EP - 289
%@ 1869-1951
Y1 - 2010
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.C0910148


Abstract: 
A DP curve is a new kind of parametric curve defined by Delgado and Peña (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 Bé;zier 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.

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Reference

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