Level sets of a parametric plot (3D)

Question

I am having difficulty getting the level sets of a 3D parametric plot. I have a parametrized 3D surface that I suspect is self-intersecting and one way to see this visually is by plotting the level sets (intersection with planes say z=-4,-2,0,4.. etc) and these isocurves will show me visually that there is self-intersection. This is the code I use (apologies for the long equations!):

wspace[t2_, t3_] := 
 Module[{l1 = 1/2, l2 = 1/3, d2 = 2, a2 = 3, v2, v3}, v2 = Tan[t2/2];
  v3 = Tan[t3/2];
  R = (-l1*v2 + l2*v2 - (l1 + l2)*v3)^2 + (v2 + 
       l1*l2*v2 + (1 - l1*l2) v3)^2 + (l1 + 
       l2 + (-l1*v2 + l2*v2) v3)^2 + (1 - 
       l1*l2 - (v2 + l1*l2*v2) v3)^2;
  {(1 + l1^2) (1 + l2^2) (-1 - a2 - v2^2 + a2*v2^2) (1 + v3^2), 
    2*a2*l1^2*l2^2*v2*v3^2 + 2*a2*l1^2*l2^2*v2 - 
     a2*l1^2*l2*v2^2*v3^2 + 2*a2*l1^2*l2*v2*v3 - a2*l1^2*l2 + 
     a2*l1^2*v2^2*v3 + a2*l1^2*v2*v3^2 + a2*l1^2*v2 + a2*l1^2*v3 + 
     a2*l1*l2^2*v2^2*v3^2 - a2*l1*l2^2 - 2*a2*l1*l2*v2^2*v3 + 
     2*a2*l1*l2*v3 - a2*l1*v2^2*v3^2 + a2*l1 + a2*l2^2*v2^2*v3 - 
     a2*l2^2*v2*v3^2 - a2*l2^2*v2 + a2*l2^2*v3 + a2*l2*v2^2*v3^2 + 
     2*a2*l2*v2*v3 + a2*l2 - 2*a2*v2*v3^2 - 2*a2*v2 + 
     d2*l1^2*l2*v2^2*v3^2 - d2*l1^2*l2 + d2*l1^2*v2^2*v3 + 
     d2*l1^2*v2*v3^2 - d2*l1^2*v2 - d2*l1^2*v3 + 
     d2*l1*l2^2*v2^2*v3^2 + 2*d2*l1*l2^2*v2^2 - 2*d2*l1*l2^2*v2*v3 + 
     2*d2*l1*l2^2*v3^2 + d2*l1*l2^2 - 2*d2*l1*l2*v2*v3^2 - 
     2*d2*l1*l2*v2 + d2*l1*v2^2*v3^2 + 2*d2*l1*v2^2 + 2*d2*l1*v2*v3 + 
     2*d2*l1*v3^2 + d2*l1 - d2*l2^2*v2^2*v3 + d2*l2^2*v2*v3^2 - 
     d2*l2^2*v2 + d2*l2^2*v3 + d2*l2*v2^2*v3^2 - d2*l2 + 
     l1^2*l2*v2^2*v3^2 - l1^2*l2 - l1^2*v2^2*v3 - l1^2*v2*v3^2 + 
     l1^2*v2 + l1^2*v3 - l1*l2^2*v2^2*v3^2 - 2*l1*l2^2*v2*v3 - 
     l1*l2^2 + 2*l1*l2*v2^2*v3 + 2*l1*l2*v3 + l1*v2^2*v3^2 - 
     2*l1*v2*v3 + l1 - l2^2*v2^2*v3 + l2^2*v2*v3^2 - l2^2*v2 + 
     l2^2*v3 - l2*v2^2*v3^2 + l2, 
    2*a2*l1^2*l2*v2^2*v3 + 2*a2*l1^2*l2*v2*v3^2 - 2*a2*l1^2*l2*v2 - 
     2*a2*l1^2*l2*v3 - 2*a2*l1*l2^2*v2^2*v3 - 2*a2*l1*l2^2*v2*v3^2 - 
     2*a2*l1*l2^2*v2 - 2*a2*l1*l2^2*v3 + 2*a2*l1*v2^2*v3 - 
     2*a2*l1*v2*v3^2 - 2*a2*l1*v2 + 2*a2*l1*v3 - 2*a2*l2*v2^2*v3 + 
     2*a2*l2*v2*v3^2 - 2*a2*l2*v2 + 2*a2*l2*v3 + 
     d2*l1^2*l2^2*v2^2*v3^2 + d2*l1^2*l2^2*v2^2 + d2*l1^2*l2^2*v3^2 + 
     d2*l1^2*l2^2 + d2*l1^2*v2^2*v3^2 - d2*l1^2*v2^2 - 
     4*d2*l1^2*v2*v3 - d2*l1^2*v3^2 + d2*l1^2 - d2*l2^2*v2^2*v3^2 + 
     d2*l2^2*v2^2 - 4*d2*l2^2*v2*v3 + d2*l2^2*v3^2 - d2*l2^2 - 
     d2*v2^2*v3^2 - d2*v2^2 - d2*v3^2 - d2 - 2*l1^2*l2*v2^2*v3 - 
     2*l1^2*l2*v2*v3^2 - 2*l1^2*l2*v2 - 2*l1^2*l2*v3 + 
     2*l1*l2^2*v2^2*v3 - 2*l1*l2^2*v2*v3^2 + 2*l1*l2^2*v2 - 
     2*l1*l2^2*v3 - 2*l1*v2^2*v3 - 2*l1*v2*v3^2 + 2*l1*v2 + 2*l1*v3 + 
     2*l2*v2^2*v3 - 2*l2*v2*v3^2 - 2*l2*v2 + 2*l2*v3}/R]

zfcn = Function[{t2,t3},
  l1=1/2; l2=1/3; d2=2; a2=3;
  v2=Tan[t2/2];
  v3=Tan[t3/2];
  R=(-l1*v2+l2*v2-(l1+l2)*v3)^2+(v2+l1*l2*v2+(1-l1*l2)v3)^2+(l1+l2+(-l1*v2+l2*v2)v3)^2+(1-l1*l2-(v2+l1*l2*v2)v3)^2;
  (2*a2*l1^2*l2*v2^2*v3+2*a2*l1^2*l2*v2*v3^2-2*a2*l1^2*l2*v2-2*a2*l1^2*l2*v3-2*a2*l1*l2^2*v2^2*v3-2*a2*l1*l2^2*v2*v3^2-2*a2*l1*l2^2*v2-2*a2*l1*l2^2*v3+2*a2*l1*v2^2*v3-2*a2*l1*v2*v3^2-2*a2*l1*v2+2*a2*l1*v3-2*a2*l2*v2^2*v3+2*a2*l2*v2*v3^2-2*a2*l2*v2+2*a2*l2*v3+d2*l1^2*l2^2*v2^2*v3^2+d2*l1^2*l2^2*v2^2+d2*l1^2*l2^2*v3^2+d2*l1^2*l2^2+d2*l1^2*v2^2*v3^2-d2*l1^2*v2^2-4*d2*l1^2*v2*v3-d2*l1^2*v3^2+d2*l1^2-d2*l2^2*v2^2*v3^2+d2*l2^2*v2^2-4*d2*l2^2*v2*v3+d2*l2^2*v3^2-d2*l2^2-d2*v2^2*v3^2-d2*v2^2-d2*v3^2-d2-2*l1^2*l2*v2^2*v3-2*l1^2*l2*v2*v3^2-2*l1^2*l2*v2-2*l1^2*l2*v3+2*l1*l2^2*v2^2*v3-2*l1*l2^2*v2*v3^2+2*l1*l2^2*v2-2*l1*l2^2*v3-2*l1*v2^2*v3-2*l1*v2*v3^2+2*l1*v2+2*l1*v3+2*l2*v2^2*v3-2*l2*v2*v3^2-2*l2*v2+2*l2*v3)/R
];

ParametricPlot3D[wspace[t2,t3],{t2,-Pi,Pi},{t3,-Pi,Pi},
  MeshFunctions->Function[{x,y,t2,t3},zfcn[t2,t3]], Mesh->{{-4,-2,0,2}}, MeshStyle -> {{Thick, Blue}},
  PlotPoints->60, BoxRatios->{1,1,1},AxesLabel->{x,y,z},
  PlotRange->{{-4,2},{-2,5},{-4,3}},PlotStyle->Opacity[0.5]]

This yields the image below , but the curve does not look like an intersection with a plane parallel to the xy-plane. Am I missing something? I tried simply using Mesh->{{-2}} and I still don't get a curve that looks like an intersection with a plane.

Answer 1

Your problem comes from MeshFunction. Why would your nonlinear MeshFunction produce planar curves? Try instead MeshFunctions -> {#3 &}:

ParametricPlot3D[wspace[t2,t3],{t2,-Pi,Pi},{t3,-Pi,Pi},
  MeshFunctions->{#3 &}, Mesh->{{-4,-2,0,2}}, MeshStyle -> {{Thick, Blue}},
  PlotPoints->60, BoxRatios->{1,1,1},AxesLabel->{x,y,z},
  PlotRange->{{-4,2},{-2,5},{-4,3}},PlotStyle->Opacity[0.5]]

Answer 2

You can also use the option Exclusions:

ParametricPlot3D[wspace[t2, t3], {t2, -Pi, Pi}, {t3, -Pi, Pi}, 
  PlotStyle -> Opacity[.5],  PlotPoints -> 60, Mesh -> None, BoundaryStyle -> None, 
  BoxRatios -> 1, AxesLabel -> {x, y, z}, 
  PlotRange -> {{-4, 2}, {-2, 5}, {-4, 3}},  
  Exclusions -> {zfcn[t2, t3] == -2, zfcn[t2, t3] == 0, zfcn[t2, t3] == 2}, 
  ExclusionsStyle -> Directive[Thick, Blue]]

Answer 3

Try this:

Manipulate[
 Show[{
   ParametricPlot3D[wspace[t2, t3], {t2, -Pi, Pi}, {t3, -Pi, Pi}, 
    PlotPoints -> ControlActive[10, 60], BoxRatios -> {1, 1, 1}, 
    AxesLabel -> {x, y, z}, PlotRange -> {{-4, 2}, {-2, 5}, {-4, 3}}, 
    PlotStyle -> Opacity[0.5]],
   Graphics3D[{Blue, Opacity[0.8], 
     InfinitePlane[{{0, 1, z}, {1, 0, z}, {0, 0, z}}]}]
   }], {z, -4, 4, Appearance -> "Labeled"}]

Hope it helps. Have fun!