How do I plot the unit normal field for a surface?

Question

The question is pretty much in the title; I'm about to teach my multivariable calculus students about orientations on surfaces, and I would like to be able to show them pictures. Any ideas?

Answer 1

Let's start with a parametrized surface. Any one will do, but I guess being orientable helps in this case. Then calculate the unit normal, and then create a Manipulate object that lets you see how the normal behaves:

σ[u_, v_] := {(2 + Cos[v]) Cos[u], (2 + Cos[v]) Sin[u], Sin[v]}
n[u_, v_] := Evaluate[Normalize[
   Cross[D[σ[u, v], u], D[σ[u, v], v]]
   ]]
surfacePlot = 
  ParametricPlot3D[σ[u, v], {u, -Pi, Pi}, {v, -Pi, Pi},
   PlotRangePadding -> 1];
normalPlot = 
  ParametricPlot3D[n[u, v], {u, -Pi, Pi}, {v, -Pi, Pi}, 
   PlotStyle -> Opacity[0.5]];

Manipulate[
 {Show[{
    surfacePlot,
    Graphics3D[{Thick, Red, 
      Arrow[{σ @@ pt, σ @@ pt + n @@ pt}]}]
    }],
  Show[{
    normalPlot,
    Graphics3D[{Thick, Red, Arrow[{{0, 0, 0}, n @@ pt}]}]
    }]
  }
 ,
 {pt, {-Pi, -Pi}, {Pi, Pi}}]

To show the entire vector field for a surface on parametric form you can use a bunch of Arrow's:

Show[{
  surfacePlot,
  Graphics3D[
   Table[
    Arrow[{σ[u, v], σ[u, v] + n[u, v]}],
    {u, -Pi, Pi, 0.4}, {v, -Pi, Pi, 0.4}]
   ]
  }]

As you can see, the arrows are equally spaced in the parameter space, which leads to uneven distribution of arrows on the surface; it might be worth transforming the parametrization into $f(x,y,z)=0$ form to get a nicer result.

Answer 2

It isn't too hard to roll your own routine, of course:

UnitNormalVector[f_, {u_, u0_}, {v_, v0_}] := Block[{f0, g0},
                 f0 = f /. {u -> u0, v -> v0};
                 g0 = Transpose[D[f, {{u, v}}] /. {u -> u0, v -> v0}];
                 Arrow[{f0, f0 + Normalize[Cross @@ g0]}]]

(* Möbius strip *)
mobius[u_, v_] :=
      {(3 + (1/2 - v) Cos[u/2]) Cos[u], (3 + (1/2 - v) Cos[u/2]) Sin[u], (1/2 - v) Sin[u/2]}

Show[ParametricPlot3D[mobius[u, v], {u, 0, 2 π}, {v, 0, 1},
                      Mesh -> False, PlotPoints -> 55],
     Graphics3D[Table[UnitNormalVector[mobius[u, v], {u, u0}, {v, v0}],
                      {u0, 0, 2 π, 2 π/20}, {v0, 0, 1, 1/5}]], PlotRange -> All]

It's not too hard to give the arrows depicting the normals some style:

% /. Arrow[stuff__] :> {Blue, Arrow[Tube[stuff, 0.05]]}

Answer 3

The easiest thing to do is differentiate the field and use VectorPlot3D and ContourPlot3D to show orthogonality of these. This is from Documentation Center. I will change this a bit from original to polish graphics for your lecture. These are not unit vectors though - do u really need unit ones? It can be done too if you need.

Use a contour plot to visualize the region of a vector plot:

scalarField = x^2 - y^2 - z;
vectorField = D[scalarField, {{x, y, z}}]

Plot a vector field over a particular region:

v = VectorPlot3D[vectorField, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
  VectorPoints -> 20, VectorScale -> {0.1, Scaled[0.5]}, 
  RegionFunction -> Function[{x, y, z}, -0.1 <= scalarField <= 0.1], 
  VectorStyle -> "Arrow3D", VectorColorFunction -> "Rainbow"]

Create a contour plot of the vector plot's region:

c = ContourPlot3D[
  scalarField == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, Mesh -> 20, 
  MeshStyle -> Opacity[.1], 
  ContourStyle -> Directive[Green, Opacity[0.3], Specularity[White, 30]]]

Combine the vector and contour plots:

Show[v, c]