支德佳 王国瑾. Bézier曲线合并的区间逼近[J]. 计算机研究与发展, 2011, 48(4): 675-682.
 引用本文: 支德佳 王国瑾. Bézier曲线合并的区间逼近[J]. 计算机研究与发展, 2011, 48(4): 675-682.
Zhi Dejia and Wang Guojin. Bézier Approximate Merging by Interval Curves[J]. Journal of Computer Research and Development, 2011, 48(4): 675-682.
 Citation: Zhi Dejia and Wang Guojin. Bézier Approximate Merging by Interval Curves[J]. Journal of Computer Research and Development, 2011, 48(4): 675-682.

## Bézier Approximate Merging by Interval Curves

• 摘要: 从区域逼近的全新角度来研究几何逼近的核心问题之一：曲线的近似合并.给出了将两条或多条平面Bézier曲线合并为一条尽量细窄的区间Bézier曲线的两种方法:一是基于求已知Bézier样条曲线的上下边界直接得到区间控制顶点的值，从而诱导出一条区间合并Bézier曲线；二是基于最小二乘法求出原多段Bézier曲线合并结果的最佳一致逼近曲线作为区间Bézier曲线的中心曲线,再取区间Bézier点为常值域或变值域来得出两种误差曲线.给出大量实例来展示上述算法的逼近效果，并进行分析与比较.结果表明，算法在实现外形信息的几何逼近及数据转换方面有明显的应用前景，并可推广于空间Bézier曲线、圆域Bézier曲线、有理Bézier曲线的合并.

Abstract: This paper mainly deals with the approximate merging problem of multiple adjacent Bézier curves with different degrees by new perspective—a single interval Bézier curve, which is a frequently seen problem in modeling, geometric approximation and data transformation. At first, two methods are given in this paper as theoretical models. One gives the result of the interval Bézier curve by one-sided approximation, which can directly get the values of Bézier control points. The other uses the unified matrix representation for precise merging. The approximate merging center curve is further derived based on matrix operation and the error curve is given by both constant and unconstant interval. Then, several examples are provided to demonstrate the algorithms, which show that the methods in this paper both achieve satisfying merging results and have a good prospect in geometric approximation and data transformation. These methods will be chosen for different results and reasons. More and more, it can be proved that these methods can also be used in the merging process of multiple adjacent rational Bézier curves, in the merging process of multiple adjacent 3D Bézier curves, in the approximate merging problem by a disk Bézier curve and in the approximate merging problem of multiple adjacent Bézier surfaces.

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