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Geometric construction of energy-minimizing Béezier curves

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Abstract

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 Bézier curves. Firstly, the necessary and sufficient condition on the control points for Bézier curves to have minimal internal energy is derived. Based on this condition, we propose the geometric constructions of three kinds of Bézier 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 Bézier curves via curvature combs and curvature plots, and present the collinear properties of quartic energy-minimizing Bézier 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.

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Authors and Affiliations

  1. Institute of Graphics and Image, Hangzhou Dianzi University, Hangzhou, 310018, China

    Gang Xu

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    GuoZhao Wang & WenYu Chen

  3. School of Computer Engineering, Nanyang Technological University, Singapore, 639798, Singapore

    WenYu Chen

  4. Galaad, INRIA Sophia-Antipolis, 2004 Route des Lucioles, 06902, Sophia Antipolis, France

    Gang Xu

  5. State Key Lab of CAD & CG, Zhejiang University, Hangzhou, 310027, China

    Gang Xu

Authors
  1. Gang Xu
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  2. GuoZhao Wang
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  3. WenYu Chen
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Correspondence to Gang Xu.

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Xu, G., Wang, G. & Chen, W. Geometric construction of energy-minimizing Béezier curves. Sci. China Inf. Sci. 54, 1395–1406 (2011). https://doi.org/10.1007/s11432-011-4294-8

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  • DOI: https://doi.org/10.1007/s11432-011-4294-8

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