Geometric construction of energy-minimizing Béezier curves

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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