Given a set of $n$ red and $n$ blue points in the plane, we are interested in matching red points with blue points by straight line segments so that the segments do not cross. We develop a range of tools for dealing with the non-crossing matchings of points in convex position. It turns out that the points naturally partition into groups that we refer to as orbits, with a number of properties that prove useful for studying and efficiently processing the non-crossing matchings.

Bottleneck matching is a matching that minimizes the length of the longest segment. Illustrating the use of the developed tools, we solve the problem of finding bottleneck matchings of points in convex position in $O\left(n^2\right)$ time. Subsequently, combining our tools with a geometric analysis we design an $O\left(n\right)$-time algorithm for the case where the given points lie on a circle. The best previously known running times were $O\left(n^3\right)$ for points in convex position, and $O(n\log n$) for points on a circle.

给定一组 $n$ 红色和 $n$平面中的蓝点，我们感兴趣的是通过直线段将红点与蓝点匹配，使线段不交叉。我们开发了一系列工具来处理凸位置点的非交叉匹配。事实证明，这些点自然地划分为我们称为轨道的组，具有许多被证明对研究和有效处理非交叉匹配有用的特性。

瓶颈匹配是将最长段的长度最小化的匹配。说明了开发工具的使用，我们解决了在凸位置找到点的瓶颈匹配的问题$哦\left(n^2\right)$时间。随后，将我们的工具与几何分析相结合，我们设计了一个$哦\left(n\right)$-给定点位于圆上的情况下的时间算法。以前已知的最佳运行时间是$哦\left(n^3\right)$ 对于凸位置的点，和 $哦(n\mathrm{日志}n$) 表示圆上的点。