# More on Intersecting Cylinders and "Ambiguous Rings"

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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", 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: 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). 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 , 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"}] This GIF rotates trough all possible closed ring sets and its corresponding fit of the pentagonal cross section inside the circular. 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"]] In a GIF while rotating around a vertical (L) or horizontal (R) axis: We are now ready to print the ring: Printout3D[ring5, "Sculpteo"]  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. Many similar rings are possible. Here are 2 views of a ring resulting from the intersection of a triangular and a circular cylinder: And finally 2views of a ring from a square- circular cylinder intersection. 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"! Answer - Congratulations! This post is now a Staff Pick as distinguished by a badge on your profile! Thank you, keep it coming! Answer