Manipulate with arguments points on a circle

Manipulate[Column[{Row[{Graphics[{Red, PointSize[Large], Circle[], 
      Locator[Dynamic[pt1, (pt1 = Normalize[#]) &]], Text[Round[pt1, .01], {0, 1.5}]}],
   Graphics[{Blue, PointSize[Large], Circle[], 
    Locator[Dynamic[pt2, (pt2 = Normalize[#]) &]], Text[Round[pt2, 0.01], {0, 1.5}]}]}, 
   Spacer[20]], 
   Show[ParametricPlot3D[{Cos[t] (R + Cos[u]), Sin[t] (R + Cos[u]), Sin[u]}, 
     {t, 0, 2 Pi}, {u, 0, 2 Pi}, 
     Mesh -> None, PlotStyle -> Opacity[.4], 
     PlotLabel -> Style[Row[{"f[{pt1, pt2}] =", f[{pt1, pt2}]}], 16], 
     ImageSize -> 400], 
    Graphics3D[{Red, PointSize[Large], Point[f[{pt1, pt2}]]}]]}, 
  Dividers -> Center, Alignment -> Center],
 {{pt1, {Cos[0], Sin[0]}}, None}, {{pt2, {Cos[Pi], Sin[Pi]}}, None}]

As an alternative to kglr's take, here is how one can use Experimental`AngularSlider[] inside Manipulate[]:

With[{R = 2, r = 1},
     Manipulate[Show[torus, 
                     Graphics3D[{Red, PointSize[Large], 
                                 Point[{(R + r Cos[ϕ]) Cos[θ],
                                        (R + r Cos[ϕ]) Sin[θ], r Sin[ϕ]}]}]], 
                Row[{Control[{{θ, 0}, Experimental`AngularSlider[Dynamic[θ]] &}],
                     Spacer[20], 
                     Control[{{ϕ, 0}, Experimental`AngularSlider[Dynamic[ϕ]] &}]}], 
                Initialization :> (torus =
                ParametricPlot3D[{(R + Cos[u]) Cos[t], (R + Cos[u]) Sin[t], Sin[u]},
                                 {t, 0, 2 π}, {u, 0, 2 π}, Mesh -> None, 
                                 PlotStyle -> Opacity[.4], ImageSize -> 400];), 
                SaveDefinitions -> True]]