优化端点条件的平面二次均匀B样条插值曲线

在利用反求法构造B样条插值曲线时，往往需要选取端点条件。 因此，可对端点条件进行优化选取，使得构造的B样条插值曲线满足特定要求。提出了一种利用曲线内能极小选取平面二次均匀B样条插值曲线端点条件的算法。首先给出了二次均匀B样条插值曲线分控制顶点与首个控制顶点（即端点条件）的递推关系式；然后给出了利用曲线内能极小优化选取首个控制顶点的算法，证明了利用该算法构造的C¹连续二次均匀B样条插值曲线为保形插值，并通过数值算例证明了算法的有效性；最后，为便于实际应用，基于MATLAB平台设计了算法所对应的图形用户界面，用户通过简单的操作即可获得光顺的C¹连续二次均匀B样条保形插值曲线。

Planar quadratic uniform B-spline interpolation curve with optimized endpoint condition

In the construction of B-spline interpolation curve by reverse method,it is often necessary to set the endpoint conditions. The endpoint conditions can be optimized so that the constructed B-spline interpolation curve can meet some specific requirements.An algorithm for selecting the endpoint conditions of plane quadratic uniform B-spline interpolation curve by minimizing the internal energy is proposed.Firstly,the recurrence relation between other control points and the first control point (viz.the endpoint condition) is given.Then,an algorithm for choosing the first control point by minimizing the internal energy of the curve is proposed. It is proved that the quadratic uniform B-spline interpolation curve with C¹ continuity constructed by the proposed algorithm is shape-preserving.Some numerical examples are presented to show the effectiveness of the proposed algorithm.Finally,to facilitate the practical application,the corresponding graphical user interface of the proposed algorithm is designed based on MATLAB platform. Users can generate the shape-preserving smooth C¹ quadratic uniform B-spline interpolation curve by simple operation.