基于迭代算法的Bézier曲线求交方法研究

Intersection Computation of Bézier Curves Based on Iterative Methods

摘要: 在计算机图形学及相关领域中，Bézier曲线求交问题备受关注。本文着重探究基于迭代算法的Bézier曲线求交方法。研究发现，当Bézier曲线的控制多边形不相交时，曲线本身必定不相交；而当控制多边形相交时，采用De Casteljau作为核心迭代手段，通过不断对曲线区间进行二分处理，逐步逼近曲线交点，有效提升求交的准确性与效率。此方法为解决Bézier曲线求交难题提供了新的思路与途径，在图形设计、计算机辅助几何设计等方面具有潜在应用价值。

Abstract: In the field of computer graphics and related areas, the problem of intersection of Bézier curves has received much attention. This paper focuses on exploring the method of finding the intersection of Bézier curves based on an iterative algorithm. The study reveals that when the control polygons of Bézier curves do not intersect, the curves themselves must not intersect; while when the control polygons intersect, the De Casteljau algorithm is adopted as the core iterative method. By continuously performing bisection processing on the curve intervals, the intersection points of the curves are gradually approached, effectively improving the accuracy and efficiency of intersection calculation. This method provides new ideas and approaches for solving the difficult problem of finding the intersection of Bézier curves, and has potential application values in aspects such as graphic design and computer-aided geometric design.