Geodesics in the configuration spaces of two points in $\mathbb R^n$

Abstract

We determine explicit formulas for geodesics (in the Euclidean metric) in the configuration space of ordered pairs $(x,x')$ of points in $\mathbb R^n$ which satisfy $d(x,x')\ge \varepsilon$. We interpret this as two or three (depending on the parity of $n$) geodesic motion-planning rules for this configuration space. In the associated unordered configuration space, we need not prescribe that the points stay apart by $\varepsilon$. For this space, with a Euclidean-related metric, we show that geodesic motion-planning rules correspond to ordinary motion-planning rules on $RP^{n-1}$.