中文摘要：

当2 条曲线重合或几乎重合时, 基于曲线分裂的求交算法或因为过多次数的分裂而导致内存不足而系统奔溃, 或最后的计算结果因分裂次数的不足而未能满足精度要求. 2 条曲线重合检测技术可以帮助求交算法来避开上述问题. 本文以2 条有理三次Bézier 曲线为例, 提出并证明了重合检测基于曲线控制多边形的如下判定方法, 即2 条有理三次Bézier 曲线重合的条件为或者两条曲线退化为同一条一或二次的曲线, 或者在首末权因子为1 的限制下, 2 条曲线的控制多边形重合且对应的权因子相等. 当2 条曲线部分重合时, 本文给出了简便的方法来确定相应的重合位置, 从而将部分重合的判定问题转化为完全重合的判定问题. 实例表明了本文方法的正确性及简单有效性.

英文摘要：

When two curves are coincident or almost coincident, the corresponding intersection algorithm based on curve splitting will either run out of memory and lead to system crash because of too many times of divi-sion, or the final results can not meet the accuracy requirement due to insufficient divisions. Taking two ra-tional cubic Bézier curves for instance, this paper proposes and proves the coincidence condition based on the two control polygons. Firstly, it judges whether or not the two Bézier curves can be degenerated into ra-tional Bézier curves of degree 1 or 2. If both of the two curves are not degenerated, they are represented in the form that their first and last weights are equal to 1; and then deciding whether their control polygons are coincident and their corresponding weights are the same. Finally, it discusses the coincidence condition that two rational cubic Bézier curves are partially coincident, and gives a simple method to determine the corre-sponding coincidence position, which converts the partially coincidence detection problem into the complete coincidence detection. Numerical examples demonstrate the effectiveness and validity of the proposed algo-rithm.