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[GIF] Ruled (doubly ruled surface)

Posted 5 years ago
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One-parameter family of hyperboloids

Ruled

I made this for this month's MCCC. The theme is "lines", which immediately made me think about ruled surfaces, so this is just a family of ruled hyperboloids.

Here's the code for the corresponding Manipulate[]:

MCCCcols2 = {RGBColor["#F07B3F"], RGBColor["#FFD460"], 
  RGBColor["#EA5455"], RGBColor["#2D4059"]}

Module[{n},
 n = 99;
 Manipulate[
  Graphics3D[{Thickness[.004], 
    Table[{MCCCcols2[[Mod[t, 3, 1]]], 
      InfiniteLine[{{Cos[2 \[Pi] t/n], Sin[2 \[Pi] t/n], 
         0}, {Cos[2 \[Pi] t/n], Sin[2 \[Pi] t/n], 
          0} + {-Cos[\[Theta]] Sin[2 \[Pi] t/n], 
          Cos[\[Theta]] Cos[2 \[Pi] t/n], Sin[\[Theta]]}}]}, {t, 1, 
      n}]}, PlotRange -> 2, Boxed -> False, ImageSize -> 540, 
   ViewAngle -> \[Pi]/6, ViewCenter -> {.46, .5, 1/2}, 
   Background -> MCCCcols2[[4]]], {\[Theta], 0, \[Pi]}]
 ]

The fact that the hyperboloid is squared-off when \[Theta] is small is due to the bounding box. This bothered me for a while, but I made a version where the line segment endpoints always formed a circle and found it a little too boring. The transition from squared-off to round adds visual interest that I think enhances the animation.

(Incidentally, to make alternate round version I ended up converting to ParametricPlot3D so that I could use RegionFunction, but it would have been nicer to be able to do it in Graphics3D. Is there any way to mimic the behavior of RegionFunction in a Graphics3D object?)

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