Just for the logs and the bug tracker: in this case $\lambda=4, \mu=5$ are minimal to invoke the error

Clear[pts]
With[{\[Lambda] = 4, \[Mu] = 5},
 pts = Flatten[Table[{x, Im[BesselJ[\[Pi] o I, x + o I]]}, {o, 0, \[Lambda]}, {x, 0, \[Mu], 0.1}], 1];
 VoronoiMesh[pts, Epilog -> Point[pts], PlotRange -> {{-1/10, 5/6}, {-1/2, 1/2}}]
]

which imposes now on VoronoiMesh[] that way

Note

In[122]:= MinMax /@ Transpose[pts]
Out[122]= {{0., 5.}, {-853.437, 3364.88}}

This is $\frac{33188.819 - (-23917.866)}{3364.88 - (-853.437)}=13.5378$ more than one order of magnitude smaller as in the previous example and still a failure.

Interesting, separation on a diagonal can also be a problem! Modifying your code into

lower = 1.0; upper = 10^9; n = 40; offset =(* 1000; *) 0;
m = 1; 
While[n-- > 0, test = (upper + lower)/2; 
  pts = {offset + {0.0, 0.0}, offset + m {1.0, 1.0}, {test, test}}; 
  vm = VoronoiMesh[pts]; 
  cells = MeshCellCount[vm, 2];
  (* Print["ult: ", {upper,lower,test},"| n: ",n, "| cells = ",cells]; *)
  If[cells == 2,
   Print["pts: ", pts];
   upper = test,(* else *)
   lower = test]
 ];

then

m = 0.1:   pts: {{0., 0.}, {0.1,0.1}, {1394.5, 1394.5}}
m = 1:     pts: {{0., 0.}, {1., 1.},  {13944.9, 13944.9}}
m = 5:     pts: {{0., 0.}, {5., 5.},  {69724.6, 69724.6}}

all are defective and the defect scales linear

In[22]:= 50 1394.5
Out[22]= 69725.

Well, with a trustworthy MeshCellCount the error is straightforward to detect, given it is suspected - which it is now, after your original observation.

Hi all,

regarding the example from Udo I guess the problem arises from the wide range where the points lay in:

Clear[pts]
pts = Flatten[Table[{x, Re[BesselJ[\[Pi] o I, x + o I]]}, {o, 1, 8}, {x, 0, 5, 0.1}], 1];
MinMax /@ Transpose[pts]
(*  {{0.`,5.`},{-23917.819047547382`,33188.76634356813`}}  *)

As one can see the y-range is just huge. Playing around with a limitation of that y-range a gradual occurrence of that problem can be observed:

pts1000 = Select[pts, Abs[Last[#]] < 1000 &];
pts2000 = Select[pts, Abs[Last[#]] < 2000 &];
pts3000 = Select[pts, Abs[Last[#]] < 3000 &];
pts5000 = Select[pts, Abs[Last[#]] < 5000 &];
VoronoiMesh[#, Epilog -> Point[pts], PlotRange -> {{-1/2, 1}, {-1, 1}}, ImageSize -> 250] & /@ {pts1000, pts2000, pts3000, ts5000}

giving:

The effect also could be provoced by the sheer number of points. In any case this cannot be the reason for the original problem.

Regards -- Henrik

Giving a domspec without selecting from the point set does not work either

Clear[pts]
pts = Flatten[Table[{x, Re[BesselJ[\[Pi] o I, x + o I]]}, {o, 1, 8}, {x, 0, 5, 0.1}], 1];
VoronoiMesh[pts, {{-1/2, 1}, {-1, 1}}, Epilog -> Point[pts], PlotRange -> {{-1/2, 1}, {-1, 1}}]

selecting from the point set works with and without domspec, but must not necessarily give the same set of cells the whole point set gives:

Clear[pts, domspec]
pts = Flatten[Table[{x, Re[BesselJ[\[Pi] o I, x + o I]]}, {o, 1, 8}, {x, 0, 5, 0.1}], 1];
domspec = {{-1/2, 1}, {-1, 1}};
VoronoiMesh[#, domspec, Epilog -> Point[#], PlotRange -> domspec]&[Select[pts, Norm[#] <= Norm[domspec] &]]

To select the right way one had roughly spoken to take into account all pairs of points whose perpendicular bisector has a non-empty intersection with the domspec area, a thing VoronoiMesh should do on it's own if it is given a domspec.

O NO!!! That's BAD! How did you find this case? just trial and error? (trial and terror!)

I'll submit it, so they get inundated by me about the same issue, and will have a good look at it! Thanks a lot!

Well... it does illustrate that it is not just 1 weird case, apparently it shows up in many more cases... VoronoiMesh should at least spawn a error message that ****something**** went wrong!

@Udo Krause thanks for checking! All 3 operating systems and various versions of Mathematica... I guess a bug...

Hi Udo,

I indeed found the same. The results can be domain-dependent, which it shouldn't be, unless point fall outside the domain of course. The more worrying is that the number of mesh-polygons returned is not equal to the number of points! I would not care so much if the angle of the polygon was slightly wrong because of rescaling, it is just very strange the number of points change! It should at least show a warning message!

Rescaling seem also strange to me though; the distance is still relatively large compared to the common double precision, and even single precision...

well.... I send it to Wolfram bug-support, let's see what they see, hope they fix it, not so much for me, more for other people who got frustrated!

Thanks for checking! I think VoronoiMesh is not implemented in the kernel but through Tetgen-library. But apparently also doesn't work on Linux.

I was driving me mad today! The main problem was that I need the mesh ordered in the original order of the points (by default it isn't!), and I needed this periodic Voronoi. So I had a lot of data, and one of the files did not work. So I thought it was my custom PeriodicVoronoi function that had an error. I had number of point xyz-million and number of mesh-polygons was xyz-million minus one! But after long digging down I found the error....very frustrating! lost one day on it... oops!