This　paper　addresses　the　problem　of　determining　time-optimal　trajectories,　between　two　specified　configurations,　for　a　nonholonomic　bidirectional　steered　robot.　It　presents　an　original　geometric　reasoning　that　is　grounded　on　Pontryagin＇s　maximum　principle,　which　provides　analytical　solutions　of　this　problem　in　a　visually　clear　way　and　allows　for　an　effective　algorithm　to　compute　the　exact　optimal　trajectories　between　two　arbitrarily　specified　configurations.　The　proposed　geometric　reasoning　is　based　on　the　analysis　of　the　switching　functions　of　the　optimal　controller　and　the　definition　of　a　switching　vector　from　which　it　is　able　to　determine　a　unit　vector　rotating　along　a　unit　circle　of　an　appropriate　coordinate　system.　It　is　shown　that　simple　geometric　rules　are　sufficient　to　determine　all　possible　rotations　of　this　unit　vector,　from　which　the　time-optimal　trajectories　can　be　uniquely　determined.　The　proposed　algorithm,　which　is　based　on　this　geometric　reasoning,　is　guaranteed　to　be　complete　and　has　a　low　computational　cost.　Moreover,　the　proposed　geometric　representation　provides　an　interesting　insight　into　the　structure　of　this　class　of　nonholonomic　systems,　thereby　offering　a　model　for　further　studies.