二次带参均匀Ｂ样条的控制点生成及其最优插值

Control point generation and optimal interpolation for quadratic uniform B-splines with parameters

摘要

在特定数据点上进行Ｂ样条曲线插值建模时，通常需要逆向求解其控制点，并在确定控制点后绘制出符合工程要求的曲线形状。提出一种基于曲线内能最小化的插值型二次Ｂ样条控制点反求方法，该方法首先依据内能最小化原则选取关键控制顶点，随后利用递推关系式逐步确定其余所有控制顶点的位置。在控制点确定之后，结合具体的工程需求，通过最小化内能以及最优逼近策略，构造出最优插值曲线。最后，本文通过数值实例验证所提方法的有效性。

Abstract

In the modeling of B-spline curves that interpolates specific data points, it is often necessary to inversely solve for their control points. Once the control points are determined, a curve shapes that satisfy engineering requirements can be generated. An inverse method for determining the control points of interpolating quadratic B-spline curves is proposed based on the minimization of internal energy. The method selects a control vertex with minimal internal energy and then determines all control vertices step by step using a recursive relationship. After the control points are established, the optimal interpolating curve is determined by minimizing internal energy and achieving optimal approximation, tailored to engineering needs. Finally, the effectiveness of the proposed method is demonstrated through numerical examples.

关键词

二次均匀Ｂ样条 / 插值曲线 / 内能极小 / 最优插值

Key words

quadratic uniform B-Spline / interpolation curve / minimization of internal energy / optimal interpolation

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基金资助

国家自然科学基金天元基金资助项目(12426519)