Douglas--Rachford is the best projection method

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.