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Staff Picks Mathematics Visual Arts Geometry Graphics and Visualization Wolfram Language Publication Materials

Probability Theory of Random Polygons from the Quaternionic Viewpoint

Clayton Shonkwiler

Clayton Shonkwiler, Colorado State University

Posted 10 years ago

POSTED BY: Clayton Shonkwiler

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EDITORIAL BOARD

EDITORIAL BOARD, WOLFRAM

Posted 3 years ago

-- you have earned *Featured Contributor Badge* 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!

POSTED BY: EDITORIAL BOARD

Clayton Shonkwiler

Clayton Shonkwiler, Colorado State University

Posted 10 years ago

By the way, @Vitaliy Kaurov, I encountered a strange issue when writing the second paragraph above. The parser for the preview window wouldn't convert "[somewhat surprising][4]" into a link if the "O(n)" and "n" after the parentheses had dollar signs around them, even though it comes out fine in the published post. This seems to be due to some sort of issue with the interplay between parentheses, dollar signs, and square brackets. Here's the minimal example I could come up with: Writing `$On$ ([link][1]) $O(n)$` produces what you'd expect, but writing `$O(n)$ ([link][1]) $O(n)$` gives unexpected behavior, in that the intended link isn't actually a link in the post preview window (though it is in the final post).

POSTED BY: Clayton Shonkwiler

Vitaliy Kaurov

Vitaliy Kaurov, WOLFRAM Research

Posted 10 years ago

Thanks for reporting this, @Clayton Shonkwiler we will add this to the list of small issues we have with editor and address at a Community update. I by the way quite enjoyed this GIF, I love building objects on trajectories - I call them "rails". I find that for nice visual feel sometimes it is worth to show the rails: ToComplex[{x_, y_}] := x + I y; ToReal[z_] := {Re[z], Im[z]}; FrameToEdges[frame_] := ToReal[ToComplex[#]^2] & /@ Transpose[frame]; FrameToVertices[frame_] := Accumulate[FrameToEdges[frame]]; data = Table[ Module[{mat, frame, rotatedframe, verts, com}, mat = RotationMatrix[t, {Cos[θ] Sin[φ], Sin[θ] Sin[φ], Cos[φ]}] /. {θ -> 3 π/4, φ -> 3 π/4}; frame = Transpose[ToReal[Sqrt[#]] & /@ (E^(I #)2/3 & /@ Range[0, 4 π/3, 2 π/3])]; rotatedframe = frame.Transpose[mat]; verts = FrameToVertices[rotatedframe]; com = Mean[verts]; verts = RotationMatrix[t].# & /@ (# - com & /@ verts) ], {t, 0, 2 π, .03}]; rail = Graphics[{Dashed, Thick, RandomColor[], Line[#]} & /@ Transpose[data]] Manipulate[ Module[{mat, frame, rotatedframe, verts, com}, mat = RotationMatrix[t, {Cos[θ] Sin[φ], Sin[θ] Sin[φ], Cos[φ]}] /. {θ -> 3 π/4, φ -> 3 π/4}; frame = Transpose[ToReal[Sqrt[#]] & /@ (E^(I #)2/3 & /@ Range[0, 4 π/3, 2 π/3])]; rotatedframe = frame.Transpose[mat]; verts = FrameToVertices[rotatedframe]; com = Mean[verts]; verts = RotationMatrix[t].# & /@ (# - com & /@ verts); Show[ rail, Graphics[{FaceForm[None], EdgeForm[ Directive[RGBColor["#383a40"], Thickness[.009], JoinForm["Round"]]], Polygon[verts]}, PlotRange -> .8, ImageSize -> 540, Background -> RGBColor["#F9F9F9"]] ]], {t, 0, 2 π}]

Thanks for reporting this, @Clayton Shonkwiler we will add this to the list of small issues we have with editor and address at a Community update. I by the way quite enjoyed this GIF, I love building objects on trajectories - I call them "rails". I find that for nice visual feel sometimes it is worth to show the rails:

enter image description here

ToComplex[{x_, y_}] := x + I y;
ToReal[z_] := {Re[z], Im[z]};
FrameToEdges[frame_] := ToReal[ToComplex[#]^2] & /@ Transpose[frame];
FrameToVertices[frame_] := Accumulate[FrameToEdges[frame]];

data = Table[
   Module[{mat, frame, rotatedframe, verts, com}, 
    mat = RotationMatrix[t, {Cos[θ] Sin[φ], Sin[θ] Sin[φ], Cos[φ]}] /. {θ -> 3 π/4, φ -> 3 π/4};
    frame = Transpose[ToReal[Sqrt[#]] & /@ (E^(I #)*2/3 & /@ Range[0, 4 π/3, 2 π/3])];
    rotatedframe = frame.Transpose[mat];
    verts = FrameToVertices[rotatedframe];
    com = Mean[verts];
    verts = RotationMatrix[t].# & /@ (# - com & /@ verts)
    ], {t, 0, 2 π, .03}];

rail = Graphics[{Dashed, Thick, RandomColor[], Line[#]} & /@ Transpose[data]]

Manipulate[
 Module[{mat, frame, rotatedframe, verts, com}, 
  mat = RotationMatrix[t, {Cos[θ] Sin[φ], Sin[θ] Sin[φ], Cos[φ]}] /. {θ -> 3 π/4, φ -> 3 π/4};
  frame = Transpose[ToReal[Sqrt[#]] & /@ (E^(I #)*2/3 & /@ Range[0, 4 π/3, 2 π/3])];
  rotatedframe = frame.Transpose[mat];
  verts = FrameToVertices[rotatedframe];
  com = Mean[verts];
  verts = RotationMatrix[t].# & /@ (# - com & /@ verts);
  Show[
   rail,
   Graphics[{FaceForm[None], 
     EdgeForm[
      Directive[RGBColor["#383a40"], Thickness[.009], 
       JoinForm["Round"]]], Polygon[verts]}, PlotRange -> .8, 
    ImageSize -> 540, Background -> RGBColor["#F9F9F9"]]
   ]], {t, 0, 2 π}]

POSTED BY: Vitaliy Kaurov

Clayton Shonkwiler

Clayton Shonkwiler, Colorado State University

Posted 10 years ago

@Vitaliy Kaurov Very cool, I like the rails a lot. By the way, how do you get the actual Greek letters (rather than just `\[Phi]` etc.) to show up in the source code in your post?