This　paper　provides　a　new　geometric　method　for　achieving　the　sufficient　family　of　the　time-optimal　trajectories　to　connect　any　two　configurations　of　the　robot　in　a　3-dimensional　manifold　based　on　the　geometric　optimal　control　theory.　We　provide　a　new　perspective　for　analyzing　this　special　type　of　nonlinear　problems.　Based　on　the　structural　characteristics　of　the　switching　functions　and　their　derivatives　from　the　Pontryagin＇s　minimum　principle　(PMP)　and　the　Lie　algebra,　we　build　a　special　coordinate　system　and　introduce　a　new　vector.　We　discover　the　one-to-one　mapping　between　the　rotation　trajectory　of　this　new　vector　and　the　optimal　control　trajectory.　Furthermore,　we　define　a　switching　vector　that　denotes　the　position　and　rotation　direction　of　this　vector,　and　reach　a　conclusion　that　the　specified　initial　and　final　switching　vectors　can　uniquely　determine　an　optimal　trajectory.　In　addition,　it　is　the　first　time　a　condition　that　can　be　used　directly　for　selecting　a　time-optimal　trajectory　is　provided.