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Planar \({G^{3}}\) Hermite interpolation by quintic Bézier curves

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Abstract

To achieve \(G^3\) Hermite interpolation with a lower degree curve, this paper studies planar \(G^3\) Hermite interpolation using a quintic Bézier curve. First, the first and second derivatives of the quintic Bézier curve satisfying \(G^2\) condition are constructed according to the interpolation conditions. Four parameters are introduced into the construction. Two of them are set as free design parameters, which represent the tangent vector module length of the quintic Bézier curve at the two endpoints, and the other two parameters are derived from \(G^3\) condition. Then, to match \(G^3\) condition, it is necessary to ensure that the first derivative of curvature with respect to arc length is equal. Nevertheless, the direct calculation of the derivative of curvature involves the calculation of square root. Alternatively, an equivalent condition is derived by investigating the first derivative of curvature square. Based on this condition, the two parameters can be computed as the solutions of linear systems. Finally, the control points of the quintic Bézier curve are obtained. Several comparative examples are provided to demonstrate the effectiveness of the proposed method. A variety of complex shape curves can be obtained by adjusting the two free design parameters. Applications to shape design are also shown.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant No. 51705469), the National Key Project (Grant No. GJXM92579) and the Key Scientific Research Projects of Colleges and Universities in Henan Province (Grant No. 19A460028).

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Authors and Affiliations

  1. School of Mechanical and Power Engineering, Zhengzhou University, Zhengzhou, 450001, China

    Jiong Yang

  2. School of Mechanical Engineering and Automation, Beihang University, Beijing, 100191, China

    Tao Ning

  3. Aero Engine Corporation of China Sichuan Gas Turbine Research Establishment, Chengdu, 610500, China

    YunChao Shen

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  1. Jiong Yang
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  2. Tao Ning
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  3. YunChao Shen
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Correspondence to Tao Ning.

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Yang, J., Ning, T. & Shen, Y. Planar \({G^{3}}\) Hermite interpolation by quintic Bézier curves. Vis Comput 38, 4319–4328 (2022). https://doi.org/10.1007/s00371-021-02298-w

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