Extraneous curve due to MeshFunctions

Question

I was using MeshFunctions to get intersection of quadratic with plane, but am getting some unexpected curves, any idea how to get rid of them, or maybe a more elegant way to achieve this effect?

genPlot[theta_] := (
   plot1 = 
    Plot3D[3 x^2 + y^2, {x, -2, 2}, {y, -2, 2}, 
     MeshFunctions -> {#3 &, If[#1 != 0, ArcTan[#1, #2], -1] &}, 
     MeshStyle -> {Automatic, Thick}, 
     Mesh -> {5, {theta, theta - Pi}}, Boxed -> False, Axes -> False, 
     RegionFunction -> Function[{x, y, z}, x^2 + y^2 < 3], 
     PlotPoints -> 25];
   point = 2 {Cos[theta], Sin[theta], 0};
   
   plot2 = 
    Graphics3D[{Opacity[.5], EdgeForm[None], 
      Polygon[{-point, point, 
        point + {0, 0, 10}, -point + {0, 0, 10}}]}];
   Show[plot1, plot2, SphericalRegion -> True]
   );
genPlot[Pi/4]

edit Brett's solution below fixes it (notebook)

Answer 1

MeshFunctions works by detecting where the function crosses your values, in this case where ArcTan becomes equal to theta. Unfortunately this is done numerically, checking sign changes, and so the branch cut along the negative $y$ axis is included. (It goes from $-\pi$ to $\pi$, so must have crossed theta somewhere in there, right?)

The cleanest solution I've found for removing it in your example is to rotate the branch cut to align along where you want to draw the mesh line by using Mod[ArcTan[#1, #2], 2 Pi, theta] for the mesh function:

genPlot[theta_] := (plot1 = 
    Plot3D[3 x^2 + y^2, {x, -2, 2}, {y, -2, 2}, 
     MeshFunctions -> {#3 &, Mod[ArcTan[#1, #2], 2 Pi, theta] &}, 
     MeshStyle -> {Automatic, Thick}, 
     Mesh -> {5, {theta, theta + Pi}}, Boxed -> False, Axes -> False, 
     RegionFunction -> Function[{x, y, z}, x^2 + y^2 < 3], 
     PlotPoints -> 25];
   point = 2 {Cos[theta], Sin[theta], 0};
   plot2 = 
    Graphics3D[{Opacity[.5], EdgeForm[None], 
      Polygon[{-point, point, 
        point + {0, 0, 10}, -point + {0, 0, 10}}]}];
   Show[plot1, plot2, SphericalRegion -> True]);
genPlot[Pi/4]