[vtkusers] Proving a surface mesh of closeness

[Michael Jackson]

I am not a mathematician or theorist so I only have other people to go  
on for this.

Wikipedia - Take it or leave it:
http://en.wikipedia.org/wiki/Euler_characteristic

Other sources:
http://www.ics.uci.edu/~eppstein/junkyard/euler/
http://mathworld.wolfram.com/PolyhedralFormula.html
http://mathworld.wolfram.com/EulerCharacteristic.html
http://www2.in.tu-clausthal.de/~hormann/papers/Hormann.2002.AEW.pdf


I think I agree for a mesh that does not consist of ALL triangles.  
Then you are probably correct but for a triangular mesh that is closed  
the formula has been proven. It is important to understand those  
conditions and thanks for the heads up.

Mike

On Oct 9, 2008, at 4:48 AM, Dominik Szczerba wrote:

> That can be true accidently for an arbitrary mesh.
>
> Dominik
>
> On Friday 03 October 2008 07:52:05 pm Michael Jackson wrote:
>> T=2V-4
>>
>> Where T is the number of triagles and V is the number of vertices.
>> Something about Euler's Polyhedra equation...
>>
>>
>> Mike
>>
>> On Oct 3, 2008, at 1:14 PM, Marie-Gabrielle Vallet wrote:
>>> I think there is a easier way of checking the surface closeness. I
>>> mean using vtk facilities, instead of writing a (yet another) new
>>> algorithm.
>>>
>>> VTK library has algorithms to extract a mesh boundary, i.e. the set
>>> of faces (in 3D) or edges (in 2D) that are not shared by two cells.
>>> See vtkFeatureEdges. The mesh is close if and only if this set is
>>> empty. If it not, you can visualize the holes that must still be
>>> closed.
>>>
>>> Pamela is trying to do the same thing today. Have a look at the
>>> thread "get boundary triangles from a mesh" on this mailing list.
>>>
>>> By the way, Charles, are you sure you are not re-inventing the  
>>> wheel ?
>>>
>>> Marie-Gabrielle
>>>
>>>> Date: Fri, 3 Oct 2008 08:17:31 +0200
>>>> From: Dominik Szczerba <dominik at itis.ethz.ch>
>>>> Subject: Re: [vtkusers] Proving a surface mesh of closeness
>>>> To: vtkusers at vtk.org
>>>> Message-ID: <200810030817.31796.dominik at itis.ethz.ch>
>>>> Content-Type: text/plain;  charset="utf-8"
>>>>
>>>> If it is manifold then pick the 1st element and make sure each
>>>
>>> one it has the
>>>
>>>> proper number of neighbors (for triangles: 3). Mark the element
>>>
>>> as 'visited'
>>>
>>>> and visit all his neighbors, repeating the procedure. At the end,
>>>
>>> if number
>>>
>>>> of visited elements equals to number of elements in the mesh and
>>>
>>> all have
>>>
>>>> their expected neighbors the mesh is closed.
>>>>
>>>> DS
>>>>
>>>> On Friday 03 October 2008 02:48:59 am Charles Monty Burns wrote:
>>>>> Hello,
>>>>>
>>>>> I repaired a surface mesh and want to prove whether the mesh is
>>>
>>> totally
>>>
>>>>> closed or not. Save is save ...
>>>>>
>>>>> How can I do this?
>>>>>
>>>>> Greetings
>>>>
>>>> --
>>>> Dominik Szczerba, Ph.D.
>>>> Computational Physics Group
>>>> Foundation for Research on Information Technologies in Society
>>>> http://www.itis.ethz.ch
>>>
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>
>
>
> -- 
> Dominik Szczerba, Ph.D.
> Computational Physics Group
> Foundation for Research on Information Technologies in Society
> http://www.itis.ethz.ch