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More on Intersecting Cylinders and "Ambiguous Rings"

Posted 5 months ago
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Ambiguous rings are composite space curves that can be viewed as either a circle, a polygon, a crown-like shape or an S-like shape, depending on the view direction. In a former community contribution, I discussed the design and 3D printing of a ring that is the cross section of a square and a circular cylinder and can be seen simultanuously as a square or a circle by means of a mirror.

As can be seen from my Wolfram Demonstration "Ambiguous Rings Based on a Polygon"[1], I worked further with these ideas and expanded the "ambiguous rings" design and printing to regular polygons other than squares.

Let us start with the intersection of a pentagonal and a circular cylinder:

3 views of cylinders

To create a ring or set of rings, we need a closed intersection curve. There needs to be an exact fit of the pentagon inside the longitudinal cross-section of the circular cylinder (two parallel lines).

tight fit

This requires a radius and axial offset of the circular cylinder adapted to the axial rotation of the pentagonal cylinder. This is done by the two functions fittedRadius and fittedOffset

fittedRadius[\[Theta]0_] := (Cos[
     Mod[(\[Theta]0 + \[Pi]/10), \[Pi]/5]] - 
    Cos[Mod[(\[Theta]0 + \[Pi]/10), \[Pi]/5] + 4 \[Pi]/5])/2
fittedOffset[\[Theta]0_] := 
 TriangleWave[{- (3 - Sqrt@5)/8, (3 - Sqrt@5)/8}, 5 \[Theta]0/2/\[Pi]]

If we introduce these functions into the code supplied in [1], we get all the closed intersection curves in function of only one parameter: the axial rotation of the pentagonal cylinder:

polyRingsetCF = 
  Compile[{{\[Theta], _Real}, {\[Theta]0, _Real}, {r, _Real}, {\
\[Alpha], _Real}, {n, _Integer}, {d, _Real}},
   Module[{t, s},(*2 part composite curve*)

    t = Sec[2 ArcTan[Cot[n (\[Theta] - \[Theta]0)/2]]/n];
    s = Sec[\[Alpha]] Sqrt[-d^2 + r^2 + 
        2 d Cos[\[Pi]/n] t Sin[\[Theta]] - 
        Cos[\[Pi]/n]^2 t^2 Sin[\[Theta]]^2];
    {(*part1*){Cos[\[Pi]/n] Cos[\[Theta]] t, 
      Cos[\[Pi]/n] t Sin[\[Theta]], 
      s - Cos[\[Pi]/n] Cos[\[Theta]] t Tan[\[Alpha]]},
     {(*part 2*)Cos[\[Pi]/n] Cos[\[Theta]] t, 
      Cos[\[Pi]/n] t Sin[\[Theta]], -s - 
       Cos[\[Pi]/n] Cos[\[Theta]] t Tan[\[Alpha]]}}]];
Manipulate[Module[{r, d},
  rc = fittedRadius[\[Theta]0]; d = fittedOffset[\[Theta]0];
  ParametricPlot3D[
   polyRingsetCF[t, \[Theta]0, rc, 0., 5, d], {t, -\[Pi], \[Pi]}, 
   PlotStyle -> {{Green, Tube[.04]}}, PlotPoints -> 50, 
   PerformanceGoal -> "Quality", SphericalRegion -> True, 
   Background -> Lighter[Gray, 0.5],
   ViewAngle -> 4 \[Degree], PlotRange -> 5, Boxed -> False, 
   Axes -> False]],
 {{\[Theta]0, -1., 
   Style["axial rotation of polygonal cylinder", Bold, 10]}, -1.5708, 
  1.5708, .0001, ImageSize -> Small, Appearance -> "Labeled"}]

enter image description here

This GIF rotates trough all possible closed ring sets and its corresponding fit of the pentagonal cross section inside the circular.

enter image description here

If we add some inequalities, we can extract a single ring out of this rinset.

polyRingCF = 
  Compile[{{\[Theta], _Real}, {\[Theta]0, _Real}, {r, _Real}, {\
\[Alpha], _Real}, {n, _Integer}, {d, _Real}, {t1, _Real}, {t2, \
_Real}},(*t1 and t2 are the values of \[Theta] for switching between \
parts*)

   Module[{t},(*selection of parts of a composite curve*)

    t = Sec[2 ArcTan[Cot[1/2 n (\[Theta] - \[Theta]0)]]/n];
    {Cos[\[Pi]/n] Cos[\[Theta]] t, 
     Cos[\[Pi]/n]  Sin[\[Theta]] t,(*select part1 or part 2*)
     Piecewise[{{1, \[Theta] <= t1 \[Pi] + 2 \[Theta]0 || \[Theta] > 
            t2 \[Pi] + 2 \[Theta]0}}, -1]*
       Sec[\[Alpha]] Sqrt[-d^2 + r^2 + 
         2 d Cos[\[Pi]/n] t Sin[\[Theta]] - 
         Cos[\[Pi]/n]^2 t^2 Sin[\[Theta]]^2] - 
      Cos[\[Pi]/n] Cos[\[Theta]] t Tan[\[Alpha]]}]];
Module[{d, r, \[Theta]0, t1, 
  t2}, {d, r, \[Theta]0, t1, t2} = {1/8. (3. - Sqrt[5.]), 
   1/8. (5. + Sqrt[5.]) + .0001, \[Pi]/10., .3, 1.5}; 
 ring5 = ParametricPlot3D[
   polyRingCF[\[Theta], \[Theta]0, r, 0., 5, d, t1, 
    t2], {\[Theta], -0.01, 2 \[Pi]}, PlotStyle -> {{Red, Tube[.04]}}, 
   PerformanceGoal -> "Quality", SphericalRegion -> True, 
   PlotRange -> 6, Background -> Lighter[Gray, 0.5], ImageSize -> 400,
    ViewAngle -> 3.5 \[Degree], Boxed -> False, Axes -> False, 
   PlotTheme -> "ThickSurface"]]

enter image description here

In a GIF while rotating around a vertical (L) or horizontal (R) axis:

enter image description here

We are now ready to print the ring:

Printout3D[ring5, "Sculpteo"]

enter image description here

enter image description here

The ring can now be put in front of a mirror and we see a circle shape on the real ring while we observe a pentagon shape reflected in the mirror. This is because, the view line seen from the mirror is at a +/- 90 degree angle from the view line from eye to ring.

enter image description here

Many similar rings are possible. Here are 2 views of a ring resulting from the intersection of a triangular and a circular cylinder:

enter image description here

And finally 2views of a ring from a square- circular cylinder intersection.

enter image description here

All these rings can be printed and will show either circles or polygons depending on the view direction. Using a mirror, one can see both views simultaneously, creating the illusion of "ambiguous rings"!

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