Comparing unit normal definition in calculus with FrenetSerretSystem

Question

I'm trying to compare the unit normal definition in calculus texts (i.e., $\vec N=\vec T'/\|\vec T'\|$) where $\vec T$ is the unit tangent vector, with the unit normal vector returned by FrenetSerretSystem. Using the calculus definition, this works:

Manipulate[
 
 Module[{r, unitT, unitN},
  
  r[x_] = {x, Sin[x]};
  unitT[x_] = r'[x]/Sqrt[r'[x].r'[x]];
  unitN[x_] = unitT'[x]/Sqrt[unitT'[x].unitT'[x]];
  
  ParametricPlot[r[x], {x, 0, 2 Pi},
   Epilog -> { 
     Orange, Thick, Arrow[{r[tt], r[tt] + unitN[tt]}],
     Red, PointSize[Large], Point[r[tt]]
     }, PlotRange -> {{0, 2 Pi}, {-4, 4}}]],
 
 {tt, 0.01, 2 Pi - 0.01}]

But trying FrenetSerretSystem, this doesn't.

Manipulate[
 
 Module[{r, unitT, unitN, sys},
  
  r[x_] = {x, Sin[x]};
  sys = FrenetSerretSystem[r[x], x];
  unitT[x_] = sys[[2, 1]];
  unitN[x_] = sys[[2, 2]];
  
  ParametricPlot[r[x], {x, 0, 2 Pi},
   Epilog -> { 
     Orange, Thick, Arrow[{r[tt], r[tt] + unitN[tt]}],
     Red, PointSize[Large], Point[r[tt]]
     }, PlotRange -> {{0, 2 Pi}, {-4, 4}}]
  
  ],
 
 {tt, 0.01, 2 Pi - 0.01}]

I get an error such as this:

Coordinate {0.01 - Cos[$CellContext`x] (1 + 
Cos[$CellContext`x]^2)^Rational[-1, 2], 0.009999833334166664 +
(1 + Cos[$CellContext`x]^2)^Rational[-1, 2]} should be a pair of numbers,
or a Scaled or Offset form.

And this gives a similar error.

DynamicModule[{r, unitT, unitN, sys},
 
 r[x_] = {x, Sin[x]};
 sys = FrenetSerretSystem[r[x], x];
 unitT[x_] = sys[[2, 1]];
 unitN[x_] = sys[[2, 2]];
 
 Manipulate[
  
  ParametricPlot[r[x], {x, 0, 2 Pi},
   Epilog -> { 
     Orange, Thick, Arrow[{r[tt], r[tt] + unitN[tt]}],
     Red, PointSize[Large], Point[r[tt]]
     }, PlotRange -> {{0, 2 Pi}, {-4, 4}}],
  
  {tt, 0.01, 2 Pi - 0.01}]]

Once I get this repaired, I am going to see the unit normal vector (calculus definition) pointing in the opposite direction of the unit normal vector (FrenetSerretSystem) from 0 to $\pi$ and I am wondering why. I think it might have something to do with the fact that the calculus books tend to use the curvature, whereas the FrenetSerretSystem uses the signed curvature, but I am still trying to figure out the different directions of the unit normal.

Thanks for the help.

Update due to m_goldberg help:

This works:

Manipulate[
 
 Module[{r, k, unitT, unitN},
  
  r[x_] = {x, Sin[x]};
  {{k[x_]}, {unitT[x_], unitN[x_]}} = FrenetSerretSystem[r[x], x];
  
  ParametricPlot[r[x], {x, 0, 2 Pi},
   Epilog -> { 
     Orange, Thick, Arrow[{r[tt], r[tt] + unitN[tt]}],
     Red, Thick, Arrow[{r[tt], r[tt] + unitT[tt]}],
     Red, PointSize[Large], Point[r[tt]]
     }, PlotRange -> {{0, 2 Pi}, {-4, 4}}]
  
  ],
 
 {tt, 0.01, 2 Pi - 0.01}]

And this works:

DynamicModule[{r, unitT, unitN, k},
 
 r[x_] = {x, Sin[x]};
 {{k[x_]}, {unitT[x_], unitN[x_]}} = FrenetSerretSystem[r[x], x];
 
 Manipulate[ParametricPlot[r[x], {x, 0, 2 Pi},
   Epilog -> {
     Orange, Thick, Arrow[{r[tt], r[tt] + unitN[tt]}],
     Red, Thick, Arrow[{r[tt], r[tt] + unitT[tt]}],
     Red, PointSize[Large], Point[r[tt]]}, 
   PlotRange -> {{0, 2 Pi}, {-4, 4}}], {tt, 0.01, 2 Pi - 0.01}]]

Both protect the functions in the modules from being changed when functions with the same name are entered in the same notebook in the global workspace.

Answer 1

This much simpler implementation seems to work just fine.

Clear[k, n, t]
r[x_] := {x, Sin[x]};
{{k}, {t[x_], n[x_]}} = FrenetSerretSystem[r[x], x];

Manipulate[
  ParametricPlot[r[x], {x, 0, 2 Pi},
    Epilog -> {
      Orange, Thick, 
      Arrow[{r[tt], r[tt] + n[tt]}],
      Arrow[{r[tt], r[tt] + t[tt]}],
      Red, PointSize[Large], Point[r[tt]]}, 
      PlotRange -> {{-Pi/4, 2 Pi}, {-2, 2}}],
  {tt, 0., 2 Pi}]

Update

The reason your code produces the error message is because you have a scoping problem

If you were to insert a debugging code into your Manipulate expression like so

Manipulate[
  Module[{r, unitT, unitN, sys},
    ...
    unitN[x_] = sys[[2, 2]];
    Print[Definition[unitN]];
    ParametricPlot[...],
  {tt, 0.01, 2 Pi - 0.01}]

you would see

Note that on the lhs the independent variable is x$_ and the rhs it is x, so your function definitions for the tangent and normal don't work. There several ways you can fix this. Here are three.

1. Define the functions for r, t and n at top-level as I did in the code given above. This is not a bad solution. It's only real drawback is that the Manipulate expression is not self-contained.

2. Define the functions in an Initialization clause of the Manipulate expression. The functions are still defined globally but, at least, they are only defined once and the Manipulate expression is self-contained. This is my recommendation.

3. Fix the scoping problem with a variable injection trick I learned form Mr.Wizard (I am not sure he invented it). Although this will fix the scoping, I do not recommend it because it still leaves the problem that the expressions defining r, t and n are re-evaluated each time the front-end updates the Manipulate's content pane -- a big performance hit.

My recommendation is to use the following code.

Manipulate[
  ParametricPlot[r[x], {x, 0, 2 Pi},
    Epilog -> {
      Orange, Thick,
      Arrow[{r[tt], r[tt] + n[tt]}],
      Arrow[{r[tt], r[tt] + t[tt]}],
      Red, PointSize[Large], Point[r[tt]]}, 
    PlotRange -> {{-Pi/4, 2 Pi}, {-2, 2}}],
  {tt, 0., 2. Pi},
  Initialization :> (
    Clear[k, n, t];
    r[x_] := {x, Sin[x]}; 
    {{k}, {t[x_], n[x_]}} = FrenetSerretSystem[r[x], x];)]

The code that only addresses the scoping problem is:

Manipulate[
  Module[{r, unitT, unitN, sys},
    r[x_] = {x, Sin[x]};
    sys = FrenetSerretSystem[r[x], x];
    Function[unitT[x_] = #][sys[[2, 1]]];
    Function[unitN[x_] = #][sys[[2, 2]]];
    ParametricPlot[r[x], {x, 0, 2 Pi},
      Epilog -> {
        Orange, Thick,
        Arrow[{r[tt], r[tt] + unitN[tt]}],
        Arrow[{r[tt], r[tt] + unitT[tt]}],
        Red, PointSize[Large], Point[r[tt]]}, 
      PlotRange -> {{-Pi/4, 2 Pi}, {-2, 2}}]],
  {tt, 0., 2. Pi}]