  The Lusternik-Schnirelmann category cat and topological complexity TC are
related homotopy invariants. The topological complexity TC has applications to
the robot motion planning problem. We calculate the Lusternik-Schnirelmann
category and topological complexity of the ordered configuration space of two
distinct points in the product $G\times\mathbb{R}^n$ and apply the results to
the planar and spatial motion of two rigid bodies in $\mathbb{R}^2$ and
$\mathbb{R}^3$ respectively.
