Easy way to add another parameter to Manipulate

Question

Vector definitions:

I'm sorry that I didn't include the Vector definitions. I forgot because I had them in initialization cells. Here they are:

Vector[{ip : {_, _}, vec : {_, _}}, styles___] := 
  Graphics[{Arrowheads[Medium], Thickness[Medium], styles, 
    Arrow[{ip, ip + vec}]}];
Vector[vec : {_, _}, styles___] := Vector[{{0, 0}, vec}, styles];
Vector[{ip : {_, _, _}, vec : {_, _, _}}, styles___] := 
  Graphics3D[{GrayLevel[.4], CapForm["Butt"], Arrowheads[Small], 
    styles, Arrow[Tube[{ip, ip + vec}]]}];
Vector[vec : {_, _, _}, styles___] := Vector[{{0, 0, 0}, vec}, styles]

I have:

DynamicModule[{f, plt, curve, k, r},
 f[x_, y_] = 3 Exp[-x^2 - y^2];
 plt = Plot3D[3 Exp[-x^2 - y^2], {x, -2, 2}, {y, -2, 2},
   Mesh -> None,
   BoxRatios -> Automatic];
 k = .5;
 kk = Sqrt[Log[3/k]];
 r[s_] = {kk Cos[s/kk], kk Sin[s/kk], k};
 curve = ParametricPlot3D[r[s], {s, 0, 2 \[Pi] kk},
   PlotStyle -> {Blue, Thick}];
 Manipulate[
  Show[plt, curve,
   Vector[{r[tt], r'[tt]}, Thick, Red],
   Graphics3D[{
     Red, PointSize[Large], Point[r[tt]]
     }]
   ],
  {{tt, 0.1}, 0.1, 2 \[Pi] kk}
  ]
 ]

Which produces:

Then suddenly I decide I want to add a slider for k. What's the easy way to adapt my code to do this?

Reason for Dynamic Module:

This technique I use is due to a long discussion some time ago: Look Here

For example, if I take m_goldberg's example (which is nice, and sorry I did this before he edited it---I also put f[x,y] in the Plot3D command to demonstrate what happens) and place two copies in one notebook.

This in one cell.

Manipulate[
 Show[Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, Mesh -> None, 
   BoxRatios -> Automatic], 
  ParametricPlot3D[r[s, k], {s, 0, 2 \[Pi] kk[k]}, 
   PlotStyle -> {Blue, Thick}], 
  Graphics3D[{Red, Thick, Arrow[{r[tt, k], r[tt, k] + rds[tt, k]}]}], 
  Graphics3D[{Red, Sphere[r[tt, k], .1]}]], {{tt, 0.1}, 0.1, 
  2 \[Pi] kk[k], Appearance -> "Labeled"}, {{k, .5}, .1, 2.8, .1, 
  Appearance -> "Labeled"}, 
 Initialization :> (f[x_, y_] = 3 Exp[-x^2 - y^2];
   kk[k_] := Sqrt[Log[3/k]];
   r[s_, k_] := {kk[k] Cos[s/kk[k]], kk[k] Sin[s/kk[k]], k};
   rds[s_, k_] := {-Sin[s/Sqrt[Log[3/k]]], Cos[s/Sqrt[Log[3/k]]], 0})]

This in a second cell in the same notebook

Manipulate[
 Show[Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, Mesh -> None, 
   BoxRatios -> Automatic], 
  ParametricPlot3D[r[s, k], {s, 0, 2 \[Pi] kk[k]}, 
   PlotStyle -> {Blue, Thick}], 
  Graphics3D[{Red, Thick, Arrow[{r[tt, k], r[tt, k] + rds[tt, k]}]}], 
  Graphics3D[{Red, Sphere[r[tt, k], .1]}]], {{tt, 0.1}, 0.1, 
  2 \[Pi] kk[k], Appearance -> "Labeled"}, {{k, .5}, .1, 2.8, .1, 
  Appearance -> "Labeled"}, 
 Initialization :> (f[x_, y_] = 3 Exp[x^2 - y^2];
   kk[k_] := Sqrt[Log[3/k]];
   r[s_, k_] := {kk[k] Cos[s/kk[k]], kk[k] Sin[s/kk[k]], k};
   rds[s_, k_] := {-Sin[s/Sqrt[Log[3/k]]], Cos[s/Sqrt[Log[3/k]]], 0})]

Notice, the only difference between the two is in the Initialization section, where the function f is different. Other than that, they are identical. Now, save the notebook with both pieces of code in separate cells.

Evaluate the notebook. Then put your cursor in the first cell and evaluate that cell. You will note that you get the same graph in both Manipulate outputs. Now, put your cursor in the second cell and evaluate that cell. You will see both Manipulate outputs change to that function.

The dynamic module prevents this. When I am writing a notebook to help my students, I might have several Manipulate demos. If I am teaching the position vector in calculus, typically title something like $\vec r(t)=\cos t\ \hat{i}+\sin t\ \hat{j},$ then I'll be using a function r[t_] in the Manipulate demos and it has to be protected.

Two Additional Questions:

1. In my dynamic module, can the lines before the Manipulate command have any access to the variable, such as tt, in the Manipulate body?

2. Can anyone duplicate my demonstration with a complete Dynamic Module and no use of Manipulate? I might be at a level where I might understand if I can get an example.

Answer 1

How about something like this?

Clear[f, plt, kk, curve]

f[x_, y_] = 3 Exp[-x^2 - y^2];
plt = Plot3D[3 Exp[-x^2 - y^2], {x, -2, 2}, {y, -2, 2}, Mesh -> None,  BoxRatios -> Automatic];

Manipulate[
 kk = Sqrt[Log[3/k]];
 curve = ParametricPlot3D[{kk Cos[s/kk], kk Sin[s/kk], k}, {s, 0, 2 Pi kk}, PlotStyle -> {Blue, Thick}];

 Show[
  plt,
  curve,
  (*Vector[{r[tt],r'[tt]},Thick,Red],*)
  Graphics3D[{Red, Ball[{kk Cos[tt/kk], kk Sin[tt/kk], k}, 0.15]}]
  ],

 {{tt, 0.1}, 0.1, 2 Pi kk},
 {{k, 0.5}, 0.1, 2.5, 0.1}
]

Unfortunately you didn't include the definition of Vector, so I wasn't able to test that one out. I also swapped out your 2D Point for a 3D Ball object that looks better in the 3D plot, at least in my opinion. Since the f and plt are not dynamic, they can also be taken outside of the Manipulate body.

Answer 2

This isn't very different from MarcoB's solution, but I did implement the tangent vector, :-)

Clear[k];
With[{kk = Sqrt[Log[3/k]]},
  Manipulate[
    Show[
      Plot3D[3 Exp[-x^2 - y^2], {x, -2, 2}, {y, -2, 2},
        Mesh -> None, BoxRatios -> Automatic],
      ParametricPlot3D[r[s, k], {s, 0, 2 π kk},
        PlotStyle -> {Blue, Thick}],
      Graphics3D[{Red, Thick, Arrow[{r[tt, k], r[tt, k] + rds[tt, k]}]}],
      Graphics3D[{Red, Sphere[r[tt, k], .1]}]],
    {{tt, 0.1}, 0.1, 2 π kk, Appearance -> "Labeled"},
    {{k, .5}, .1, 2.8, .1, Appearance -> "Labeled"},
    Initialization :> (
      r[s_, k_] := {kk Cos[s/kk], kk Sin[s/kk], k};
      rds[s_, k_] := {-Sin[s/kk], Cos[s/kk], 0})]]

Introducing k as a controlled variable isn't exactly easy because changing it from a parameter to a variable makes r a function of both tt and k.