带形状参数的插值曲线的构造及其应用

刘华勇; 程思捷; 李伟伟

文章摘要

刘华勇,程思捷,李伟伟.带形状参数的插值曲线的构造及其应用[J].安徽建筑大学学报,2025,33(3):56-66

带形状参数的插值曲线的构造及其应用

The Construction and Application of Interpolating Curves with Shape Parameters

DOI：

中文关键词: Catmull-Rom样条样条曲线形状参数插值曲线

英文关键词: Catmull-Rom Spline spline curve shape parameter interpolating curve

基金项目:安徽省高等学校自然科学基金重点项目（KJ2021A0630、KJ2021A0633、KJ2021A0634）

作者	单位
刘华勇	School of Mathematics and Physics，Anhui Jianzhu University，Hefei 230601，China
程思捷	School of Mathematics and Physics，Anhui Jianzhu University，Hefei 230601，China
李伟伟	School of Mathematics and Physics，Anhui Jianzhu University，Hefei 230601，China

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中文摘要:

为了提高Catmull-Rom样条曲线的灵活性，本文构造了带两个形状参数的λμ-CPI-样条曲线和三个形状参数的αλμ-CPI-样条曲线。首先，根据Catmull-Rom样条构造了λμ-CPI-样条基函数和αλμ-CPI-样条基函数，进而分析了这两个样条基函数的归一性、拟对称性和线性无关性等基本属性。随后，构造了 λμ-CPI-样条曲线和αλμ-CPI-样条曲线，并探讨了其对称性、几何不变性、仿射不变性以及端点特性。通过数值实验,验证了λμ-CPI-样条曲线和αλμ-CPI-样条曲线的形状可调性和连续性，展示了其在复杂曲线造型中的灵活性和优势。同时，与拉格朗日插值曲线的误差对比表明，样条曲线在逼近效果上更具优势。最后，提出了一种基于最小弯曲能量的最优参数选取方法，为样条曲线设计提供支持。实例表明，λμ-CPI-样条曲线和 αλμ-CPI-样条曲线提高了 CatmullRom样条曲线的灵活性，通过选取最优形状参数提高了曲线的光滑度。

英文摘要:

To enhance the flexibility of Catmull-Rom spline curves, this paper constructs λμ-CPI-spline curves with two shape parameters and αλμ-CPI-spline curves with three shape parameters. First, the λμ-CPI-spline and αλμ-CPI-spline basis functions are constructed based on the Catmull-Rom spline, and the fundamental properties of these basis functions, such as normality, quasi-symmetry, and linear independence, are analyzed. Then, λμ-CPI-spline curves and αλμ-CPI-spline curves are constructed, and their symmetry, geometric invariance, affine invariance, and endpoint characteristics are discussed. Numerical experiments verify the shape adjustability and continuity of the λμ-CPI-spline and αλμ-CPI-spline curves, demonstrating their flexibility and advantages in modeling complex curves. Additionally, a comparison with the error of Lagrange interpolation curves shows that spline curves have su‑ perior approximation effects. Finally, an optimal parameter selection method based on minimal bending energy is proposed for spline curve design. Examples illustrate that λμ - CPI-spline and αλμ - CPI-spline curves enhance the flexibility of Catmull-Rom spline curves and improve the smoothness of the curves by selecting the optimal shape parameters.

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