File:Hyperboloid ruled surface animation (short).gif

(* config *)
frames = 60;  (*frame count for first generation loop *)
offset = -Pi + 0.8; (* ensure favorable "alignment" *)
(* From the Mathematica stack exchange: 
https://mathematica.stackexchange.com/questions/10957/an-efficient-circular-arc-primitive-for-graphics3d; 
function by user Sjoerd C.de Vries; CC BY-SA *)
splineCircle[m_List, r_, angles_List : {0, 2 \[Pi]}] := 
 Module[{seg, \[Phi], start, end, pts, w, k}, {start, end} = 
    Mod[angles // N, 2 \[Pi]];
   If[end <= start, end += 2 \[Pi]];
   seg = Quotient[end - start // N, \[Pi]/2];
   \[Phi] = Mod[end - start // N, \[Pi]/2];
   If[seg == 4, seg = 3; \[Phi] = \[Pi]/2];
   pts = 
    r RotationMatrix[start] . # & /@ 
     Join[Take[{{1, 0}, {1, 1}, {0, 1}, {-1, 1}, {-1, 
         0}, {-1, -1}, {0, -1}}, 2 seg + 1], 
      RotationMatrix[seg \[Pi]/2] . # & /@ {{1, 
         Tan[\[Phi]/2]}, {Cos[\[Phi]], Sin[\[Phi]]}}];
   If[Length[m] == 2, pts = m + # & /@ pts, 
    pts = m + # & /@ 
      Transpose[
       Append[Transpose[pts], ConstantArray[0, Length[pts]]]]];
   w = Join[
     Take[{1, 1/Sqrt[2], 1, 1/Sqrt[2], 1, 1/Sqrt[2], 1}, 
      2 seg + 1], {Cos[\[Phi]/2], 1}];
   k = Join[{0, 0, 0}, Riffle[#, #] &@Range[seg + 1], {seg + 1}];
   BSplineCurve[pts, SplineDegree -> 2, SplineKnots -> k, 
    SplineWeights -> w]] /; Length[m] == 2 || Length[m] == 3

k[a_, b_, t_] := a + (b - a)*t^3
f[u_, v_, 
  skew_ : 2 Pi/3] := {Cos[u + offset], Sin[u + offset], 
    1} (v) + {Cos[u + skew + offset], 
    Sin[u + skew + offset], -1} (1 - v)

borderWidth = -BorderDimensions[
     ParametricPlot3D[f[u, v, 2 Pi/3], {u, 0, 2 Pi}, {v, 0, 1}, 
      PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}, Axes -> False, 
      Boxed -> False, ViewPoint -> {1.3, -3, 2}]] + 2;
(*generate table*)
hyperbList = 
  Table[ImagePad[Module[{p = 3 (j/frames)^2 - 2 (j/frames)^3},
     Show[
      ParametricPlot3D[f[u, v], {u, 0, p*2 Pi}, {v, 0, 1}, 
       ColorFunction -> 
        Function[{x, y, z, u}, 
         RGBColor[k[0.880722, 0, u*p], 0.611041, k[0.142051, 1, u*p], 
          1 - (u*p)^3]], Boxed -> False, Axes -> False, 
       ViewPoint -> {1.3, -3, 2}, 
       PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}, 
       PlotPoints -> Ceiling[125 j/frames + 2],
       MeshFunctions -> {#5 &}],
      Graphics3D[{
        (* custom-bake the v mesh, and add other thingamabobbers *)
        GrayLevel[0.2],
        Table[
         Line[{f[2 Pi/16*i, 0], f[2 Pi/16*i, 1]}], {i, 1, 
          Floor[p*16]}],
        Thick, Blue,
        splineCircle[{0, 0 , 1}, 1],
        splineCircle[{0, 0, -1}, 1],
        Black,
        Line[{f[0, 0], f[0, 1]}],
        Line[{f[p*2 Pi, 0], f[p * 2 Pi, 1]}],
        PointSize[0.03],
        Point[{f[0, 0], f[0, 1]}],
        Point[{f[p*2 Pi, 0], f[p * 2 Pi, 1]}],
        GrayLevel[0.2],
        Line[{f[p*2 Pi, 1], {0, 0, 
           1}, {Cos[p*2 Pi + offset + 2 Pi/3], 
           Sin[p*2 Pi + offset + 2 Pi/3], 1}, f[p*2 Pi, 0]}],
        splineCircle[{0, 0, 1}, 
         1/4, {p*2 Pi + offset, p*2 Pi + offset + 2 Pi/3}],
        Line[{{0, 1, -1}, {0, -1, -1}}],
        Line[{{1, 0, -1}, {-1, 0, -1}}],
        Line[{{0, 0, 1}, {0, 0, -1}}]
        }]
      ]
     ], borderWidth], {j, 1, frames}];

(* add frame of u = 0 manually *)

PrependTo[hyperbList, ImagePad[Graphics3D[
    {Thick, Blue,
     splineCircle[{0, 0 , 1}, 1],
     splineCircle[{0, 0, -1}, 1],
     Black,
     Line[{f[0, 0], f[0, 1]}],
     PointSize[0.03],
     Point[{f[0, 0], f[0, 1]}],
     GrayLevel[0.2],
     Line[{f[0, 1], {0, 0, 1}, {Cos[offset + 2 Pi/3], 
        Sin[offset + 2 Pi/3], 1}, f[0, 0]}],
     splineCircle[{0, 0, 1}, 1/4, {offset, offset + 2 Pi/3}],
     Line[{{0, 1, -1}, {0, -1, -1}}],
     Line[{{1, 0, -1}, {-1, 0, -1}}],
     Line[{{0, 0, 1}, {0, 0, -1}}]
     },
    Boxed -> False, Axes -> False, 
    PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}}, 
    ViewPoint -> {1.3, -3, 2}],
   borderWidth]];

(* Export, with first and last frames lengthened *)
Export["hyperboloidAnimShort.gif", hyperbList, 
 "DisplayDurations" -> {1, Splice@ConstantArray[1/15, frames - 1], 1}]