The reach control problem (RCP) deals with finding a feedback control which drives the states of an affine control system to leave a simplex through a predetermined facet. In analogy with Brockett's obstruction to continuous feedback stabilization, a topological obstruction to solving the RCP by continuous feedback results in a deep necessary condition for solvability. In this paper, we interpret the problem of topological obstruction as a problem of existence of null-homotopic maps on spheres. This results in a complete and easily implementable characterization of the topological obstruction for the case of systems with two inputs.