Plotting the intersection of two surfaces with singular behaviour

Question

I would like to plot the intersection of two surfaces, as in this example: https://reference.wolfram.com/language/example/HighlightTheIntersectionOfTwoSurfaces.html

However, my surfaces are badly singular at their intersection. Is there a nice workaround? I think they should be well-behaved enough that it's not hopeless. Here is the code for the surfaces and my attempt to plot their intersection.

\[Epsilon] = 0; t = 0;
A = (x^2 + y^2 + z^2 - t^2 + 1)/2;
\[Phi] = ((A x - t z) + 
   I (A y + t (A - 1)))/(\[Epsilon] + (A z + t x) + 
   I (A (A - 1) - t y));
\[Theta] = ((A y + t (A - 1)) + 
   I (A z + t x))/(\[Epsilon] + (A x - t z) + I (A (A - 1) - t y));
ContourPlot3D[
                {Re[\[Phi]] == 1.0, Im[\[Phi]] == 1.0},
                {x, -plotRange, plotRange},
                {y, -plotRange, plotRange},
                {z, -plotRange, plotRange},
                Mesh -> None, PlotPoints -> 5,
                (*Boxed\[Rule]False,Axes\[Rule]None,*)
                PlotPoints -> 60,
                MeshFunctions -> {Function[{x, y, z, f}, 
    Re[\[Phi]] - Im[\[Phi]]]}, MeshStyle -> {{Thick, Blue}}, 
 Mesh -> {{0}}, 
 ContourStyle -> 
  Directive[Orange, Opacity[0.5], Specularity[White, 30]]
 ]

Answer 1

• In the first ContourPlot3D, we only draw the surface Re[ϕ] == 1.0 and its Mesh according to the condition Re[ϕ] - Im[ϕ].

• Then we draw the second surface Im[ϕ] == 1.0 and using Show.

ϵ = 0; t = 0;
A = (x^2 + y^2 + z^2 - t^2 + 1)/2;
ϕ = ((A x - t z) + 
     I (A y + t (A - 1)))/(ϵ + (A z + t x) + 
     I (A (A - 1) - t y));
θ = ((A y + t (A - 1)) + 
     I (A z + t x))/(ϵ + (A x - t z) + I (A (A - 1) - t y));
plotRange = 3;
surf1 = ContourPlot3D[
   Re[ϕ] == 1.0, {x, -plotRange, plotRange}, {y, -plotRange, 
    plotRange}, {z, -plotRange, plotRange}, PlotPoints -> 20, 
   MaxRecursion -> 2, 
   MeshFunctions -> {Function[{x, y, z, f}, Re[ϕ] - Im[ϕ]]},
    MeshStyle -> {{Thick, Blue}}, Mesh -> {{0}}, 
   ContourStyle -> 
    Directive[Orange, Opacity[0.5], Specularity[White, 30]]];
surf2 = ContourPlot3D[
   Im[ϕ] == 1.0, {x, -plotRange, plotRange}, {y, -plotRange, 
    plotRange}, {z, -plotRange, plotRange}, PlotPoints -> 20, 
   MaxRecursion -> 2, MeshStyle -> {{Thick, Blue}}, Mesh -> None, 
   ContourStyle -> 
    Directive[Orange, Opacity[0.5], Specularity[White, 30]]];
Show[surf2, surf1]

• In some case it looks better than draw {Re[ϕ] == 1.0, Im[ϕ] == 1.0} in the same time.

ϵ = 0; t = 0;
A = (x^2 + y^2 + z^2 - t^2 + 1)/2;
ϕ = ((A x - t z) + 
     I (A y + t (A - 1)))/(ϵ + (A z + t x) + 
     I (A (A - 1) - t y));
θ = ((A y + t (A - 1)) + 
     I (A z + t x))/(ϵ + (A x - t z) + I (A (A - 1) - t y));
plotRange = 3;
surf = ContourPlot3D[{Re[ϕ] == 1.0, 
   Im[ϕ] == 1.0}, {x, -plotRange, plotRange}, {y, -plotRange, 
   plotRange}, {z, -plotRange, plotRange}, PlotPoints -> 20, 
  MaxRecursion -> 2, 
  MeshFunctions -> {Function[{x, y, z, f}, Re[ϕ] - Im[ϕ]]}, 
  MeshStyle -> {{Thick, Blue}}, Mesh -> {{0}}, 
  ContourStyle -> {Directive[Orange, Opacity[0.5], 
     Specularity[White, 30]], 
    Directive[Green, Opacity[0.5], Specularity[White, 30]]}]