Wikipedia

File:Hyperboloid ruled surface animation v2.gif

Hyperboloid_ruled_surface_animation_v2.gif (370 × 470 pixels, file size: 9.87 MB, MIME type: image/gif, looped, 181 frames, 16 s)

Summary

 
This diagram was created with Mathematica.
Description
English: An illustration of the the generation of a hyperboloid of revolution as the surface of revolution of a slanted line, also featuring all possible angles at which the line can be slanted to create a unique hyperboloid (thus including its two degenerate forms - a cone and an open cylinder).

This is the second version - the code has been cleaned up slightly and is now parallelized, and the bottom grid markings in the circle now spin as the slant changes as inspired by File:Cylinder - hyperboloid - cone.gif.

This may not necessarily be superior to the first version - the motion of the circle grid markings might distract from the change in the line's slant.
Date
Source Own work
Author Lemondoge
Other versions
  • Without spinning bottom circle
    Without spinning bottom circle
Source code
InfoField

Mathematica code

(* 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/a/10958/89865; 
function by user Sjoerd C. de Vries; CC BY-SA 4.0 *)
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 = 
  ParallelTable[
   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]];
(* Show transition into cone *)
frames2 = 60;
AppendTo[hyperbList, 
  Splice @ParallelTable[
    ImagePad[Module[{p = 3 (j/frames2)^2 - 2 (j/frames2)^3},
      Show[
       ParametricPlot3D[
        f[u, v, (2 + p) Pi/3], {u, 0, 2 Pi}, {v, 0, 1}, 
        ColorFunction -> 
         Function[{x, y, z, u}, 
          RGBColor[k[0.880722, 0, u], 0.611041, k[0.142051, 1, u], 
           1 - (u)^3]], Boxed -> False, Axes -> False, 
        ViewPoint -> {1.3, -3, 2}, 
        PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}},
        PlotPoints -> 127,
        MeshFunctions -> {#5 &}],
       Graphics3D[{
         (* custom-bake the v mesh, and add other thingamabobbers *)
         GrayLevel[0.2],
         
         Table[Line[{f[2 Pi/16*i, 0, (2 + p) Pi/3], 
            f[2 Pi/16*i, 1, (2 + p) Pi/3]}], {i, 1, 16}],
         
         Thick, Blue,
         splineCircle[{0, 0 , 1}, 1],
         splineCircle[{0, 0, -1}, 1],
         
         Line[{f[2 Pi, 1, (2 + p) Pi/3], {0, 0, 
            1}, {Cos[offset + (2 + p) Pi/3], 
            Sin[offset + (2 + p) Pi/3], 1}, f[2 Pi, 0, (2 + p) Pi/3]}],
         
         splineCircle[{0, 0, 1}, 
          1/4, {2 Pi + offset, 2 Pi + offset + (2 + p) Pi/3}],
         
         Black,
         Line[{f[0, 0, (2 + p) Pi/3], f[0, 1, (2 + p) Pi/3]}],
         Line[{f[2 Pi, 0, (2 + p) Pi/3], f[2 Pi, 1, (2 + p) Pi/3]}],
         PointSize[0.03],
         Point[{f[0, 0, (2 + p) Pi/3], f[0, 1, (2 + p) Pi/3]}],
         Point[{f[2 Pi, 0, (2 + p) Pi/3], f[2 Pi, 1, (2 + p) Pi/3]}],
         
         GrayLevel[0.2],
         
         Line[{{Cos[p (Pi/3)], 
            Sin[p (Pi/3)], -1}, {-Cos[p (Pi/3)], -Sin[p (Pi/3)], -1}}],
         
         Line[{{Cos[p (Pi/3) + Pi/2], 
            Sin[p (Pi/3) + Pi/2], -1}, {-Cos[p (Pi/3) + Pi/2], -Sin[
              p (Pi/3) + Pi/2], -1}}],
         Line[{{0, 0, 1}, {0, 0, -1}}]
         }]
       ]
      ], borderWidth], {j, 1, frames2}]];
(* Show transition to cylinder *)
frames3 = 60;
AppendTo[hyperbList, 
  Splice @ParallelTable[
    ImagePad[Module[{p = 3 (j/frames3)^2 - 2 (j/frames3)^3},
      Show[
       ParametricPlot3D[
        f[u, v, (3 - 3 p) Pi/3], {u, 0, 2 Pi}, {v, 0, 1}, 
        ColorFunction -> 
         Function[{x, y, z, u}, 
          RGBColor[k[0.880722, 0, u], 0.611041, k[0.142051, 1, u], 
           1 - (u)^3]], Boxed -> False, Axes -> False, 
        ViewPoint -> {1.3, -3, 2}, 
        PlotRange -> {{-1, 1}, {-1, 1}, {-1, 1}},
        PlotPoints -> 127,
        MeshFunctions -> {#5 &}],
       Graphics3D[{
         (* custom-bake the v mesh, and add other thingamabobbers *)
         GrayLevel[0.2],
         
         Table[Line[{f[2 Pi/16*i, 0, (3 - 3 p) Pi/3], 
            f[2 Pi/16*i, 1, (3 - 3 p) Pi/3]}], {i, 1, 16}],
         
         Thick, Blue,
         splineCircle[{0, 0 , 1}, 1],
         splineCircle[{0, 0, -1}, 1],
         
         Line[{f[2 Pi, 1, (3 - 3 p) Pi/3], {0, 0, 
            1}, {Cos[offset + (3 - 3 p) Pi/3], 
            Sin[offset + (3 - 3 p) Pi/3], 1}, 
           f[2 Pi, 0, (3 - 3 p) Pi/3]}],
         
         splineCircle[{0, 0, 1}, 
          1/4, {offset, offset + (3 - 3 p) Pi/3}],
         
         Black,
         Line[{f[0, 0, (3 - 3 p) Pi/3], f[0, 1, (3 - 3 p) Pi/3]}],
         
         Line[{f[2 Pi, 0, (3 - 3 p) Pi/3], 
           f[2 Pi, 1, (3 - 3 p) Pi/3]}],
         PointSize[0.03],
         Point[{f[0, 0, (3 - 3 p) Pi/3], f[0, 1, (3 - 3 p) Pi/3]}],
         
         Point[{f[2 Pi, 0, (3 - 3 p) Pi/3], 
           f[2 Pi, 1, (3 - 3 p) Pi/3]}],
         
         GrayLevel[0.2],
         
         Line[{{Cos[(1 - 3 p) (Pi/3)], 
            Sin[(1 - 3 p) (Pi/
                3)], -1}, {-Cos[(1 - 3 p) (Pi/3)], -Sin[(1 - 3 p) (Pi/
                 3)], -1}}],
         
         Line[{{Cos[(1 - 3 p) (Pi/3) + Pi/2], 
            Sin[(1 - 3 p) (Pi/3) + 
              Pi/2], -1}, {-Cos[(1 - 3 p) (Pi/3) + 
               Pi/2], -Sin[(1 - 3 p) (Pi/3) + Pi/2], -1}}],
         Line[{{0, 0, 1}, {0, 0, -1}}]
         }]
       ]
      ], borderWidth], {j, 1, frames3}]];
Export["hyperboloidAnim2.gif", hyperbList, 
 "DisplayDurations" -> {1, Splice@ConstantArray[1/15, frames - 1], 1, 
   Splice@ConstantArray[1/15, frames2 - 1], 1, 
   Splice@ConstantArray[1/15, frames3 - 1], 1}]

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public ___domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

Captions

An illustration of the the generation of a hyperboloid of revolution as the surface of revolution of a slanted line, also featuring all possible angles at which the line can be slanted to create a unique hyperboloid

Items portrayed in this file

depicts

hyperboloid of revolution

inception

16 May 2024

media type

image/gif

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current23:30, 16 May 2024Thumbnail for version as of 23:30, 16 May 2024370 × 470 (9.87 MB)LemondogeUploaded own work with UploadWizard

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