  We consider mean trajectories of multiple tracking points on a rigid body
that conducts Brownian motion in the absence and presence of an external force
field. Based on a na\"{\i}ve representation of rigid body - polygon and
polyhedron where hydrodynamic interactions are neglected, we study the Langevin
dynamics of these Brownian polygons and polyhedra. Constant force, harmonic
force and an exponentially decaying force are investigated as examples. In two
dimensional space, depending on the magnitude and form of the external force
and the isotropy and anisotropy of the body, mean trajectories of these
tracking points can exhibit three regimes of interactions: convergence, where
the mean trajectories converge to either a point or a single trajectory;
alignment, where the mean trajectories juxtapose in parallel; twist, where the
mean trajectories twist and intertwine, forming a plait structure. Moreover, we
have shown that in general a rigid body can sample from these regimes and
transit between them. And its Brownian behavior could be modified during such
transition. Notably, from a polygon in two dimensional space to a polyhedron in
three dimensional space, the alignment and twist regimes disappear and there is
only the convergence regime survived, due to the two more rotational degrees of
freedom in three dimensional space.
