My donut has holes in it!

Question

More than one hole, I mean…

I’m trying to export from Mathematica into the X3D format, with the longer term goal of generating 3D figures for PDF inclusion. But I'm stuck at the first step:

p = ParametricPlot3D[{(2 + Cos[v]) Sin[u], (2 + Cos[v]) Cos[u], 
    Sin[v]}, {u, 0, 2 Pi}, {v, 0, 2 Pi}, PlotStyle -> Red];
Export["test.x3d", p];

The donut thus generated has holes in it (as visualized here with FreeWRL):

and the same is true for every 3D surface I've tried to export. MeshLab complains that the file contains “4 degenerated faces”, but I doubt it's the same issue, as there are a lot more than 4 holes!

I am not a 3D format expert, so I don't really know where to go. Exporting to VRML gives the same issue, so I suspect something generic is going on, but I don't really know how to investigate. I tried importing back the files into Mathematica, but 3D graphics formats are apparently write-only.

So, how do you advise me to tackle the issue? Do you have any experience in this kind of export?

Answer 1

Answer

Apparently, the mesh lines generate points which are too close to the triangle vertices, and VRML is not being able to handle them correctly.

To prove the theory, try the example without meshes:

p = ParametricPlot3D[{(2 + Cos[v]) Sin[u], (2 + Cos[v]) Cos[u], 
   Sin[v]}, {u, 0, 2 Pi}, {v, 0, 2 Pi}, PlotStyle -> Red, Mesh -> None];

Export["test.x3d", p];

It should look OK using FreeWRL:

So the possible solution is to isolate meshes from the surface, i.e. generate them separately and let them be two different graphics complex so that they wouldn't share any points.

We already know how to generate the surface without mesh. To generate only meshes, this would do it (plotting with PlotStyle->None):

p2 = ParametricPlot3D[{(2 + Cos[v]) Sin[u], (2 + Cos[v]) Cos[u], 
   Sin[v]}, {u, 0, 2 Pi}, {v, 0, 2 Pi}, PlotStyle -> None]

The result is:

Now, combine those two using Show and export.

Export["test.x3d", Show[p, p2]];

The result is perfect:

Now, you got your wholly donut back. Enjoy!

Note: I am using Windows version of FreeWRL so the result may be different on other platform. In that case, it may as well a bug in FreeWRL, not Mathematica's problem.

Bonus

OK. I shouldn't advocate the use of undocumented features. But if you really want more solid looking meshes, not shamble lines (many format/renderer is not so great at pure line drawing, more so with 3D printing...), this syntax may help you: MeshStyle->Tube[thickness] (thickness in user coordinate scale).

For instance:

p2 = ParametricPlot3D[{(2 + Cos[v]) Sin[u], (2 + Cos[v]) Cos[u], 
   Sin[v]}, {u, 0, 2 Pi}, {v, 0, 2 Pi}, PlotStyle -> None, MeshStyle -> Tube[.02]]

will create:

Disclaimer There is no guarantee that the syntax will work on the future version of Mathematica. So if you value compatibility, you should not use this. But the resulting 3D graphics will be always valid since it is using our Tube primitive. Tube is supported for export formats. For instance, if you export it to x3d:

Export["test.x3d", Show[p, p2]];

(It may take quite a while, since Mathematica is converting tubes into polygons for compatibility during exports), the result will be:

Again, it is not a permanent solution but if you really need better mesh lines for export or 3d printing, it will give you a temporary relief.

Answer 2

It seams that the normals of those faces are reversed for some reason. Thing you can try... 1) Increasing quality instead of performance of video card. 2) Updating GL driver. 3) If the software can render those faces double sided you can switch that option on. 4) Import to an other software or edit it and flip the normals.