Drawing evenly spaced arrows and opaque sheet between curves?

Question

I have a question about drawing arrows/sheets between curves. Suppose I have two 3-d parametric curves like shown below:

ParametricPlot3D[{{Cos[t], Sin[t], t}, {Cos[t + .3], Sin[t + .3], 
t}}, {t, 0, 1}]

I want to draw arrows between the curves at an evenly spaced interval and also (if possible) make an opaque thin sheet that covers the surface that the arrows make.

I've made a simple 2-d analog of what I mean by this. Pretending that the arrows are at an evenly spaced interval, I'm trying to make something that looks like this, but in 3-d:

Then, adding the sheet would look something like this:

I think that the arrows can be made by generating a set of discrete points on each of the curves and using the Arrow command recursively, but I don't know where to start with the sheet.

Answer 1

Here is your general curve:

f[t_, s_] := {Cos[t + s], Sin[t + s], t};
f[t_] := f[t, 0];

Define points for arrows:

int = .1;
pts1 = Table[f[t], {t, 0, 1, int}];
pts2 = Table[f[t, .3], {t, 0, 1, int}];

Build objects separately:

surf=ParametricPlot3D[f[t,s],{t,0,1},{s,0,.3},Mesh->None,PlotStyle->Opacity[.2]];

arrows=Graphics3D[{Red,Thick,Arrowheads[0.03],Arrow[#]}&/@Transpose[{pts1,pts2}]];

curves=ParametricPlot3D[{f[t],f[t,.3]},{t,0,1}];

Then combine:

Show[curves,arrows,surf]

Note, you can offset the step int to control direction of arrows.

int=.1;
off=.15;
pts1=Table[f[t],{t,0,1,int}];
pts2=Table[f[t,.3],{t,0-off,1-off,int}];
surf=ParametricPlot3D[f[t,s],{t,0,1},{s,0,.3},Mesh->None,PlotStyle->Opacity[.2]];
arrows=Graphics3D[{Red,Thick,Arrowheads[0.03],Arrow[#]}&/@Transpose[{pts1,pts2}]];
curve=ParametricPlot3D[{f[t],f[t,.3]},{t,0,1}];
Show[curve,arrows,surf]

Answer 2

An approach using a single ParametricPlot3D using the options MeshFunctions + Mesh and the Method sub-option "BoundaryOffset" -> False (to prevent clipping of mesh lines on the boundaries):

ParametricPlot3D[(1 - v) {Cos[t], Sin[t], t} + v {Cos[t + .3], Sin[t + .3], t}, 
 {t, 0, 1}, {v, 0, 1}, 
  Method -> {"BoundaryOffset" -> False},
  PlotStyle -> Opacity[.5], 
  MeshFunctions -> {#4 &, #5 &}, 
  Mesh -> {Thread[{Subdivide[10], Red}], 
   Thread[{{0, 1}, 
       Directive[AbsoluteThickness@3, Arrowheads@0, ColorData[97]@#] & /@ {1, 2}}]},
  MeshStyle -> Thick, ImageSize -> Large] /. Line -> Arrow