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

Question

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.

Answer 1

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}]