Representing conics by low degree rational DP curves  

作　　者：Qian-qian HU Guo-jin WANG 

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

出　　处：《Journal of Zhejiang University-Science C(Computers and Electronics)》2010年第4期278-289,共12页浙江大学学报C辑（计算机与电子（英文版）

基　　金：supported by the National Natural Science Foundation of China (Nos.60873111 and 60933007);the Natural Science Foundation of Zhejiang Province,China (No.Y6090211)

摘　　要：A DP curve is a new kind of parametric curve defined by Delgado and Pena （2003）; Jt 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 B6zier 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 suffi- cient 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 Bemstein basis, and the representation tbeory of conics in rational low degree B6zier 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.A DP curve is a new kind of parametric curve defined by Delgado and Pefla(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 Bezier curves Rational low degree DP curves 

分 类 号：TP391.72[自动化与计算机技术—计算机应用技术]