# Is there a way to animate the plane wrapping into a cylinder (and also the curves of the plane folding into a cylinder)?

I would like to animate (continuously) a plane with any number of curves on it folding into a cylinder (and also seeing the resulting curves -arbitrary curves, really- in the cylinder - folded lines would be the geodesics of the cylinder, circles would be sort of pseudo-geodesics, and so on). I've thought a lot about this problem but haven't found any way to actually implement what I'm thinking of (I don't even know if there is really a way, so it's worth asking). I'm aware it's simple to do this without the curves part, but I haven't been able to adapt it.

• Possible duplicate: Wrapping a rectangle to form a cylinder Sep 12, 2018 at 0:35
• @VitaliyKaurov That's exactly why I wrote: "I'm aware it's simple to do this without the curves part, but I haven't been able to adapt it." Sep 12, 2018 at 0:36
• Please provide an example of what exactly you want to wrap.
– Kuba
Sep 13, 2018 at 6:05

I take the solution from the linked QA, but slightly change the parametrization:

Manipulate[
ParametricPlot3D[
{Sinc[a u] u, v, 1/a - Cos[a u]/a}, {u, 0, 2 Pi}, {v, 0, 2 Pi},
ImageSize -> {300, 300}, Mesh -> {{0.}},
MeshFunctions -> {#4 - #5 &},
BoxRatios -> {3 \[Pi], 2 \[Pi], 4},
PlotRange -> {{-\[Pi], 2 \[Pi]}, {0, 2 \[Pi]}, {0, 4}},
PlotStyle -> Directive[Opacity[0.7], Orange, Specularity[White, 50]],
ViewPoint -> {0, 1, -2}, ViewVertical -> {0, 1, 0},
BoundaryStyle -> Directive[Black, Opacity[.3]]
],
{{a, .01, "wrap"}, 0.001, 1}
]


This way we start with a 2Pi*2Pi sheet in the x-y plane with one vertex fixed at (0,0,0). Slots #4 and #5 correspond to the x-y coordinates of that sheet (not the 3-d space), I'll call them u and v. Let's say, we want to look at how the line u - v == 0 wraps into a cylinder. We pass the MeshFunction -> {#4 - #5 &} option (alternatively, we could write Function[{x,y,z,u,v}, u - v]) for the left hand side of the equation u - v == 0. and pass Mesh -> {{0.}} for the right hand side.

Check it out.

As far as the question in the comments goes: In my example the sheet is in (0,0)-(2Pi,2Pi). Let's say, we want to put a spiral in the center, in terms of u and v it would be {u[t], v[t]} == {Pi + t/10 Cos[t], Pi + t/10 Sin[t]}.

We use the very same expression as for the sheet above:

Function[{u,v}, {Sinc[a u] u, v, 1/a - Cos[a u]/a}]


and apply it to the parametric definition of the spiral like so:

Function[{u, v}, {Sinc[a u] u, v, 1/a - Cos[a u]/a}] @@ {
Pi + t/10 Cos[t], Pi + t/10 Sin[t]}

Manipulate[
Show[{
ParametricPlot3D[{Sinc[a u] u, v, 1/a - Cos[a u]/a}, {u, 0,
2 Pi}, {v, 0, 2 Pi},
ImageSize -> {300, 300}, Mesh -> {{0.}},
MeshFunctions -> {#4 - #5 &}, BoxRatios -> {3 \[Pi], 2 \[Pi], 4},
PlotRange -> {{-\[Pi], 2 \[Pi]}, {0, 2 \[Pi]}, {0, 4}},
PlotStyle -> Directive[Opacity[0.7],
Orange, Specularity[White, 50]], ViewPoint -> {0, 1, -2},
ViewVertical -> {0, 1, 0},
BoundaryStyle -> Directive[Black, Opacity[.3]]],

ParametricPlot3D[
Function[{u, v}, {Sinc[a u] u, v, 1/a - Cos[a u]/a}] @@ {
Pi + t/10 Cos[t], Pi + t/10 Sin[t]},
{t, 0, 10 Pi}]

}], {{a, .01, "wrap"}, 0.001, 1}]


Here is the generalized approach for arbitrary-sized sheets. I've included k and h as parameters that determine the size of the sheet in the Manipulate to yield a sheet sized 2 Pi k by h

Manipulate[Show@{
ParametricPlot3D[
{Sinc[a u/ k] u, v, k/a - k Cos[a u/k]/a}, {u, 0, 2 Pi k}, {v, 0, h},
ImageSize -> {300, 300}, BoxRatios -> {k 3 \[Pi], h, 4 k},
PlotRange -> {k {-\[Pi], 2 \[Pi]}, {0, h}, k {0, 4}}, Mesh -> None,
PlotStyle ->
Directive[Opacity[0.7], Orange, Specularity[White, 50]],
ViewPoint -> {0, 1, -2}, ViewVertical -> {0, 1, 0},
BoundaryStyle -> Directive[Black, Opacity[.3]]],

ParametricPlot3D[
Function[{u, v}, {Sinc[a u/ k] u, v, k/a - k Cos[a u/k]/a}] @@
{Pi + t/10 Cos[t], Pi + t/10 Sin[t]}, {t, 0, 10 Pi}]},

{{a, .01, "wrap"}, 0.001, 1},
{{k, 2, "u-length"}, 0.1, 4},
{{h, 1, "v-length"}, 0.1, 10}]
`

• Comments are not for extended discussion; this conversation has been moved to chat.
– Kuba
Sep 13, 2018 at 9:57