Geodesics in the configuration spaces of two points in Rn

D. M. Davis

We determine explicit formulas for geodesics (in the Euclidean metric) in the configuration space of ordered pairs (x,x') of points in Rn which satisfy d(x,x') ≥ ε. 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 ε. For this space, with a Euclidean-related metric, we show that geodesic motion-planning rules correspond to ordinary motion-planning rules on RPn-1.

Tbilisi Mathematical Journal, Vol. 14(1) (2021), pp. 149-162