Datei:Hyperboloid ruled surface animation v2.gif
Originaldatei (370 × 470 Pixel, Dateigröße: 9,87 MB, MIME-Typ: image/gif, Endlosschleife, 181 Bilder, 16 s)
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Beschreibung
| Beschreibung |
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. |
| Datum | |
| Quelle | Eigenes Werk |
| Urheber | Lemondoge |
| Andere Versionen |
|
| Quelltext | 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}]
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| Version vom | Vorschaubild | Maße | Benutzer | Kommentar | |
|---|---|---|---|---|---|
| aktuell | 01:30, 17. Mai 2024 | Kein Vorschaubild vorhanden | 370 × 470 (9,87 MB) | wikimediacommons>Lemondoge | Uploaded own work with UploadWizard |
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