  We prove that the Douglas--Rachford method applied to two closed convex cones
in the Euclidean plane converges in finitely many steps if and only if the set
of fixed points of the Douglas--Rachford operator is nontrivial. We analyze
this special case using circle dynamics.
  We also construct explicit examples for a broad family of projection methods
for which the set of fixed points of the relevant projection method operator is
nontrivial, but the convergence is not finite. This three-parametric family is
well known in the projection method literature and includes both the
Douglas--Rachford method and the classic method of alternating projections.
  Even though our setting is fairly elementary, this work contributes in a new
way to the body of theoretical research justifying the superior performance of
the Douglas--Rachford method compared to other techniques. Moreover, our result
leads to a neat sufficient condition for finite convergence of the
Douglas--Rachford method in the locally polyhedral case on the plane, unifying
and expanding several special cases available in the literature.
