arXiv:2610.31625v1 Announce Type: cross Abstract: We consider the offline generation of coordinated paths for N interchangeable robots and N goals in planar environments with polygonal obstacles. After a conformal map to a circular domain, starts and goals become the poles and zeros of a rational function R, and the paths are the branches of its Generalized Root Locus (GRL), i.e. the streamlines of the logarithmic potential -log R. We show that, for every phase outside a finite set of critical phases and a set of boundary phases, the induced gradient motion drives each robot to a distinct goal along pairwise disjoint paths that never reach the obstacles; the robot-goal assignment is produced by the construction, and with the gain used as a clock all robots move as the closed-loop roots of the GRL and arrive together at a prescribed time. Obstacles become streamlines through successive Milne-Thomson reflections, and for a finite number of reflections a finite test certifies safety for the potential that is actually computed. Reciprocal root-locus planning is recovered as a special case. The method is illustrated on two polygonal workspaces, 100 random instances compared with independent harmonic planning and artificial potentials, and a scene with ten robots and ten obstacles.