gc = 
      ParametricPlot3D[{{-5, z Cos[φ], 
            -z Sin[φ]}, {-5 + 10 z, 
            Cos[φ - 
                (Max[z, .5] - .5) ( φ - Pi) - 3  Pi/4], 
            -Sin[φ - 
                  (Max[z, .5] - .5) ( φ - Pi) - 3  Pi/4]
                }, {5, z Cos[φ/2 - Pi/4], 
            -z Sin[φ/2 - Pi/4]}}, {z, 0, 1}, {φ, 0, 
          2  Pi}, Axes -> False, Boxed -> False]

Show[
      Graphics3D[{RGBColor[1, 0, 0], Thickness[.02], 
          Line[Table[{φ, 1.01  Cos[2  φ], 
                1.01  Sin[2  φ]}, {φ, -5, 5, 
                (1/10)}]], 
          Line[Table[{
                φ, .99  Cos[2  φ], .99 
                   Sin[2  φ]}, {φ, -5, 5, (1/10)}]]}, 
        Boxed -> False], gc]