Bottleneck Bichromatic Non-crossing Matchings using Orbits

Let $R$ and $B$ be sets of $n$ red and $n$ blue points in the plane, respectively, with $P = R \cup B$. Let $M$ be a perfect matching between points from $R$ and $B$, using $n$ straight line segments to match the points, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. We forbid line segments to cross. Denote the length of a longest line segment in $M$ with $bn(M)$, which we also call the \intro[~]{value, bn} of $M$. We aim to find a matching under given constraints that minimizes $bn(M)$. Any such matching is called a \intro[~]{bottleneck matching} of $P$.