Message Boards Message Boards

GROUPS:

[GIF] Sweep Out

Posted 5 years ago
2867 Views
|
2 Replies
|
8 Total Likes
|

Sweep Out

Sweep Out

This is a great 2-sphere inside a rotating 3-sphere, viewed by first stereographically projecting to 3-dimensional space, and then orthogonally projecting to the yz-plane. I enjoy these kinds of projections, where rotations end up looking nothing like a rotation.

Here's the code for the corresponding Manipulate object. Multiplying by the rotation matrix makes this pretty slow (though not as slow as using RotationMatrix[\[Theta], {{1, 0, 0, 0}, {0, 0, 0, 1}}] would have been), so the Manipulate may be super-laggy unless you have a pretty high-powered computer.

Stereo3D[{x1_, y1_, x2_, y2_}] := {x1/(1 - y2), y1/(1 - y2), x2/(1 - y2)};
Manipulate[
 ParametricPlot[
  Table[Stereo3D[{{Cos[t], 0, 0, -Sin[t]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sin[t], 0, 0, Cos[t]}}.
       {Cos[\[Theta]] Sin[\[Phi]], Sin[\[Theta]] Sin[\[Phi]], Cos[\[Phi]], 0}][[2 ;;]],
       {\[Theta], 0, 2 \[Pi], 2 \[Pi]/20}], {\[Phi], 0, \[Pi]}, PlotRange -> 2.5, 
  Background -> RGBColor["#172940"], 
  PlotStyle -> Directive[Thickness[.004], RGBColor["#acf0f2"]], 
  ImageSize -> 540, Axes -> False], {t, 0, \[Pi]}]

And here's the same thing, but without projecting to the yz-plane:

3D stereographic sphere

Manipulate[
 ParametricPlot3D[
  Table[Stereo3D[{{Cos[\[Theta]], 0, 0, -Sin[\[Theta]]}, {0, 1, 0, 0}, {0, 0, 1, 0}, {Sin[\[Theta]], 0, 0, Cos[\[Theta]]}}.
      {Cos[t] Sin[\[Phi]], Sin[t] Sin[\[Phi]], Cos[\[Phi]], 0}], {t, 0, 2 \[Pi], 2 \[Pi]/20}], {\[Phi], 0, \[Pi]}, PlotRange -> 10, 
  Background -> RGBColor["#172940"], 
  PlotStyle -> Directive[Thickness[.004], RGBColor["#acf0f2"]], 
  ImageSize -> 540, Boxed -> False, Axes -> False, 
  ViewAngle -> \[Pi]/40, ViewPoint -> {1.3, -2.4, 2.}], {\[Theta], 0, \[Pi]}]
2 Replies

Nice, first one reminds me of electric field lines of to charges; I wonder if it is coincidentally the same 2D formula.

enter image description here

enter image description here -- you have earned Featured Contributor Badge enter image description here Your exceptional post has been selected for our editorial column Staff Picks http://wolfr.am/StaffPicks and Your Profile is now distinguished by a Featured Contributor Badge and is displayed on the Featured Contributor Board. Thank you!

Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract