Sinusoid modelling

Question

There are lots of great demonstrations that show how to get periodic functions in the unit circle. I'm wondering how hard it would be to simulate a "moving grid" similar to this video on youtube (at about 1 minute)

The circle and moving points are easy , but I don't know how to simulate that grid and sinusoid moving under the moving points. Any help would be appreciated.

Here is my start, lots of possibilities to make this look better cosmetically, but the moving "paper" is something that would really help to illustrate the idea.

Manipulate[Graphics[{
   Circle[],
   PointSize[0.012],
   Point[{Cos[t], Sin[t]}],
   Point[{0, Sin[t]}]
   }, PlotRange -> {{-1.1, 11}, {-1.1, 1.1}}, 
  ImageSize -> {500, 100}], {t, 0, 10}]

Answer 1

Here's a "compositional" approach. If you take things piece-by-piece it is not too hard to build up more complicalated demonstrations.

Animate[Module[
  {spazzyP, scrollingPaper, scrollingSine, circle,
   pCoords, yCoords, yellowDot, blueLine,
   offset = 1, range = 2 Pi, padding = 1, fmin = Floor[min]},

  pCoords = {min + offset - Cos[min + offset], Sin[min + offset]};
  spazzyP = Graphics[{
     {Red, Opacity[.8], Disk[pCoords, .2]},
     {Style[Text["P", pCoords], Bold]}}];

  yCoords = {min + offset, Sin[min + offset]};
  yellowDot = Graphics[{Yellow, Point[yCoords]}];

  blueLine = Graphics[{{Blue, Opacity[.6], Line[{pCoords, yCoords}]}}];

  circle = Graphics[{Darker@Green, Circle[{min + offset, 0}, 1]}];

  scrollingPaper = Graphics[{},
    GridLines -> {Range[fmin - padding, fmin + range + padding + 1], 
      Range[-1 - padding, 1 + padding, .5]},
    PlotRange -> {{min, min + range}, {-1, 1}},
    PlotRangePadding -> padding,
    GridLinesStyle -> Lighter[Gray, .6]];

  scrollingSine = 
   Plot[Sin[x], {x, min + offset, min + range + padding}, 
    PlotStyle -> {Thick, Red}];

  elements = {
    scrollingPaper,
    circle,
    scrollingSine,
    blueLine,
    yellowDot,
    spazzyP (* note: spazzyP fails to live up to its name *)
    };

  (* elements *) (* useful for 'debugging' individual components, if necessary *)
  Magnify@Show[elements]
  ]

 , {min, 2 Pi, -2 Pi}]

I encourage others to fix/build on this if they want to.

A general note: One of the powerful things about Mathematica is its symbolic evaluation scheme. In this code I save individual components in the form of Graphics objects, but I could have just as easily kept things as lists like this:

circle = {Darker@Green, Circle[{min + offset, 0}, 1]};

And then later wrapped them in Graphics or Show. The point is you're completely free to build things up piecemeal and compose them later on.

Answer 2

I first create the plot with GridLines -> Automatic:

plot = Plot[-Sin[x], {x, -10, 0},
  PlotRange -> {{-10, 1}, {-1.1, 1.1}},
  ImageSize -> {500, 100},
  Axes -> False,
  GridLines -> Automatic]

Then I combine your graphics object with plot using Inset:

Manipulate[
 Graphics[{Circle[], PointSize[0.012], Point[{Cos[t], Sin[t]}],
   ColorData[1, "ColorList"][[1]], Point[{0, Sin[t]}],
   Inset[plot, {0, 0}, {-t, 0}, {12.1, 2.2}]},
  PlotRange -> {{-1.1, 11}, {-1.1, 1.1}}, 
  ImageSize -> {500, 100}], {t, 0, 10}]

I also changed the second Point to match with the color in the plot. Notice how I use the thrid argument of Inset to get the plot exactly aligned with the graphic.

Edit: Of course you can also use your own GridLines. In this case it makes sense to place vertical lines at fractions of Pi:

...
  GridLines -> {Table[i Pi/4, {i, -12, 4}], 
    Table[i, {i, -1, 1, 1/4}]}
...

Edit 2: Since it was asked: Export can handle list of images/graphics.

plotlist=Table[
 Graphics[{Circle[], PointSize[0.012], Point[{Cos[t], Sin[t]}],
   ColorData[1, "ColorList"][[1]], Point[{0, Sin[t]}],
   Inset[plot, {0, 0}, {-t, 0}, {12.1, 2.2}]},
  PlotRange -> {{-1.1, 11}, {-1.1, 1.1}}, 
  ImageSize -> {500, 100}], {t, 0, 10, 0.2}];

Export["foo.gif", plotlist];

Likewise you can export to .avi using Export["foo.avi", plotlist].

Answer 3

Same kind of approach as @einbandi's here but without insetting and the grid:

Manipulate[
    plot = Plot[-Sin[x - t], {x, 0, 10},
        PlotRange -> {{-10, 1}, {-1.1, 1.1}},
        ImageSize -> {500, 100},
        Axes -> False,
        PlotStyle -> Blue];
    line = Graphics@Line[{{0, Sin[t]}, {Cos[t], Sin[t]}}];
    circle = Graphics[{Circle[],
        {PointSize[0.02], Black, Point[{Cos[t], Sin[t]}]}, 
        {PointSize[0.02], Blue, Point[{0, Sin[t]}]}},
            PlotRange -> {{-1.1, 11}, {-1.1, 1.1}},
            ImageSize -> {500, 100}];
    Show[{line, circle, plot}]
, {t, 0, 2 \[Pi], \[Pi]/80, Animator}]

----EDIT----

Apologies, I only just saw the requirement for a grid (which I subconsciously skipped as I didn't like it) but the following modification seems to work and the grid gives the illusion of moving:

Manipulate[
    plot = Plot[-Sin[x - t], {x, 0, 10},
        PlotRange -> {{-10, 1}, {-1.1, 1.1}},
        ImageSize -> {500, 100},
        Axes -> False,
        PlotStyle -> Blue];
    line = Graphics@Line[{{0, Sin[t]}, {Cos[t], Sin[t]}}];
    circle = Graphics[{Circle[],
        {PointSize[0.02], Black, Point[{Cos[t], Sin[t]}]}, 
        {PointSize[0.02], Blue, Point[{0, Sin[t]}]}},
            PlotRange -> {{-1.1, 11}, {-1.1, 1.1}},
            ImageSize -> {500, 100}];
    gridx = Graphics[Table[{Line[{{0, i}, {10, i}}]}, {i, -1, 1, .5}]];
    gridy = Graphics[Table[{Line[{{i + Mod[t, π/4], -1}, {i + Mod[t, π/4], 1}}]}, {i, 0, 10, π/4}]];
    Show[{gridx, gridy,line, circle, plot}, PlotRange ->{{-1.1, 11}, {-1.1, 1.1}}]
, {t, 0, 2 π, π/80, Animator}]