travel_along_mobius.nb

x[u_, v_] := (1 + (v/2) Cos[u/2]) Cos[u]
y[u_, v_] := (1 + (v/2) Cos[u/2]) Sin[u]
z[u_, v_] := (v/2) Sin[u/2]

r[u_, v_] := {x[u, v], y[u, v], z[u, v]}

normalVecTube[f_, {u_, u0_}, {v_, v0_}] := 
 Block[{f0, g0}, f0 = f /. {u -> u0, v -> v0};
  g0 = Transpose[D[f, {{u, v}}] /. {u -> u0, v -> v0}];
  Arrow[Tube[{f0, f0 + Normalize[Cross @@ g0]}, .02]]]



normVertVec[f_, {u_, u0_}, {v_, v0_}] := 
  Block[{f0, g0}, f0 = f /. {u -> u0, v -> v0};
   g0 = Transpose[D[f, {{u, v}}] /. {u -> u0, v -> v0}];
   .3 Normalize[Cross @@ g0]];

normVec[f_, {u_, u0_}, {v_, v0_}] := 
  Block[{f0, g0}, f0 = f /. {u -> u0, v -> v0};
   g0 = Transpose[D[f, {{u, v}}] /. {u -> u0, v -> v0}];
   f0 + .3 Normalize[Cross @@ g0]];

tanVec[u_, 
   v_] := (r[U, V] + .2 (D[r[U, 0], U])/Norm[(D[r[U, 0], U])] /. {U ->
       u, V -> v} );

tanArrowArg[u_, v_, 
   scale_: .2] := {r[U, V], 
     r[U, V] + scale (D[r[U, 0], U])/Norm[(D[r[U, 0], U])]} /. {U -> 
      u, V -> v} // N ;


perpVecTube[u_] := Arrow[Tube[{ r[u, 0] , r[u, .5] }]]

Clear[surfacePlot];

surfacePlot[r_, color_: Green, opacity_: .9, 
  pltRange_: 1.8*{{-1, 1.2}, {-1.2, 1}, {-1, 1}}, 
  boundaryStyle_: {Transparent}] := 
 ParametricPlot3D[r @@ {u, v}, {u, 0, 2 Pi}, {v, -1, 1}, 
  Boxed -> False, Axes -> False, 
  PlotStyle -> {color, Opacity[opacity]}, PlotPoints -> 50,  
  Background -> Black, PlotRange -> pltRange, AspectRatio -> 1, 
  SphericalRegion -> True, ImageSize -> 4 {150, 100}
  , BoundaryStyle -> boundaryStyle, Mesh -> None]


Clear[plo]

plo = Table[Show[
    
    surfacePlot[r],
     Graphics3D[{Transparent, Opacity[0],
      normalVecTube[r[u, v], {u, t }, {v, 0}] }],
    Graphics3D[{Arrowheads[.02], Opacity[1], Green, Thickness[.01], 
      perpVecTube[t]}], 
    Graphics3D[{Thickness[.01], Arrowheads[.03], Green, 
      Arrow[Tube[tanArrowArg[t, 0, .3]]]}]    ,
    Graphics3D[{Red, PointSize[.1], Point[r[0, 1.1]]}],
    Graphics3D[{Lighter[Blue, .5], PointSize[.1], 
      Point[r[0, -1.1]]}],
    ViewVector -> {normVec[r[u, v], {u, t}, {v, 0}] , tanVec[t, 0]},
    ViewVertical -> normalVecViewing[r[u, v], {u, t}, {v, 0}],
    ViewAngle -> \[Pi]/1.5,
    Epilog -> {Text[
       Framed[Style[
         "\[Theta] = 2\[Pi] \[Cross] " <> 
          ToString[NumberForm[N[t/(2 \[Pi])], {3, 2}]], 14, White, 
         FontFamily -> "Georgia"], 
        Background -> Opacity[.5, Gray]], {-.1, .95}] , 
      Text[Style["@InertialObservr", 12, White, 
        FontFamily -> "Georgia" ], {1.1, .05}] }
    
     ], {t, 0, 4 \[Pi], 4 \[Pi]}];