Why can't Mathematica draw a consistent mesh on a torus?

Question

All 3 images are produced using ParametricPlot3D The left example is produced with no extra options. Note that the black lines, which is all I really want to see, cut in and out and are not connected. In the right example, the colors are gone, but the mesh lines are still broken up and the picture looks terrible. In the middle frame, I've tried changing the PlotStyle so the faces are only 0.8 opaque. This helps, but I don't want the surface to be transparent. How can I draw a simple black and white mesh picture of a torus?

Code for middle picture:

SetOptions[ParametricPlot3D,
    Axes->False,
    Boxed->False,
    Mesh->20,
    Lighting -> {White},
    PlotStyle->FaceForm[{White,Opacity[0.8]}]
    ]
 ParametricPlot3D[
    {Cos[u]*(Sqrt[2]+Cos[t]), Sin[u]*(Sqrt[2]+Cos[t]),Sin[t]
    },
    {u,0,2Pi},{t,0,2Pi}]

Answer 1

I think you can do this:

ParametricPlot3D[{Cos[u]*(Sqrt[2] + Cos[t]), 
  Sin[u]*(Sqrt[2] + Cos[t]), Sin[t]}, {u, 0, 2 Pi + Pi/10}, {t, 0, 
  2 Pi + Pi/10}, Axes -> False, Boxed -> False, Mesh -> 20, 
  PlotStyle -> Glow[White], MeshStyle -> Thick, 
  PlotPoints -> 100]

The main point is is to use Glow in the PlotStyle option to make the white surface independent of the lighting.

I also added a fix for another annoying shortcoming: the Mesh lines are drawn at subdivisions of the periodicity interval that leave a gap, unless you extend the parameter ranges by one interval size ($2\pi/n$ where $n$ is the number specified in the Mesh option).

Finally, I increased PlotPoints to make the lines smoother.

Instead of extending the parameter intervals, you can also fix the gaps in the mesh lines by adding BoundaryStyle:

ParametricPlot3D[{Cos[u]*(Sqrt[2] + Cos[t]), 
  Sin[u]*(Sqrt[2] + Cos[t]), Sin[t]}, {u, 0, 2 Pi}, {t, 0, 2 Pi}, 
 Axes -> False, Boxed -> False, Mesh -> 20, PlotStyle -> Glow[White], 
 MeshStyle -> Thick, PlotPoints -> 100, BoundaryStyle -> Thick]

The ends of the torus surface in the given parametrization are considered a boundary curve and therefore don't get a mesh line. The BoundaryStyle -> Thick option adds lines where the mesh is missing.