Geometric construction of energy-minimizing Bézier curves  

作　　者：XU Gang 1,4,5,WANG GuoZhao 2 & CHEN WenYu 2,3 1 Institute of Graphics and Image,Hangzhou Dianzi University,Hangzhou 310018,China 2 Department of Mathematics,Zhejiang University,Hangzhou 310027,China 3 School of Computer Engineering,Nanyang Technological University,639798,Singapore 4 Galaad,INRIA Sophia-Antipolis,2004 Route des Lucioles,06902,France 5 State Key Lab of CAD & CG,Zhejiang University,Hangzhou 310027,China 

出　　处：《Science China(Information Sciences)》2011年第7期1395-1406,共12页中国科学（信息科学）（英文版）

基　　金：supported by the National Basic Research Program of China (Grant No.G2004CB318000);the National Natural Science Foundation of China (Grant Nos.61004117,60970079,60773179 );the Zhejiang Provincial Natural Science Foundation (Grant No.Y1090718);the Open Project Program of the State Key Lab of CAD & CG,Zhejiang University (Grant No.A1105)

摘　　要：Modeling energy-minimizing curves have many applications and are a basic problem of Geometric Modeling. In this paper, we propose the method for geometric design of energy-minimizing Bezier curves. Firstly, the necessary and sufficient condition on the control points for Bezier curves to have minimal internal energy is derived. Based on this condition, we propose the geometric constructions of three kinds of Bezier curves with minimal internal energy including stretch energy, strain energy and jerk energy. Given some control points, the other control points can be determined as the linear combination of the given control points. We compare the three kinds of energy-minimizing Bezier curves via curvature combs and curvature plots, and present the collinear properties of quartic energy-minimizing Bezier curves. We also compare the proposed method with previous methods on efficiency and accuracy. Finally, several applications of the curve generation technique, such as curve interpolation with geometric constraints and modeling of circle-like curves are discussed.Modeling energy-minimizing curves have many applications and are a basic problem of Geometric Modeling. In this paper, we propose the method for geometric design of energy-minimizing Bezier curves. Firstly, the necessary and sufficient condition on the control points for Bezier curves to have minimal internal energy is derived. Based on this condition, we propose the geometric constructions of three kinds of Bezier curves with minimal internal energy including stretch energy, strain energy and jerk energy. Given some control points, the other control points can be determined as the linear combination of the given control points. We compare the three kinds of energy-minimizing Bezier curves via curvature combs and curvature plots, and present the collinear properties of quartic energy-minimizing Bezier curves. We also compare the proposed method with previous methods on efficiency and accuracy. Finally, several applications of the curve generation technique, such as curve interpolation with geometric constraints and modeling of circle-like curves are discussed.

关 键 词：curve design minimal energy geometric construction interpolation with constraints circle-like curves 

分 类 号：TU201.5[建筑科学—建筑设计及理论]