[vtkusers] Proving a surface mesh of closeness

[Michael Jackson]

I do see now what can go wrong. For those playing along at home  
basically do the following:

Take an simple closed triangle mesh (like a cube for instance that has  
been meshed with triangles only).

Pop out a triangle. (Now you have a hole and the formula does NOT hold).
Take any other triangle, duplicate it.
Now you still have a hole but the T=2V-4 will be true.

    I would have to say I did not think about that scenario which I  
guess could happen if a mesh program is written and there is a bug in  
it where duplicate triangles are added then that test would in fact  
give you a false positive. I guess the "Feature Edge" filter would be  
a better way to go. Also applying a "Duplicate Triangle filter" is  
also a solution but probably not the most efficient solution in terms  
of computational terms.

So "You learn something new every day" applies here. Thanks to Marie- 
Gabrielle for sticking with me on this one.

Respectfully
_________________________________________________________
Mike Jackson


On Oct 10, 2008, at 6:40 PM, Marie-Gabrielle Vallet wrote:

> Mike,
> I'm sorry, you are right : an Icosahedron is a closed surface, even  
> if it can be nonconvexe. So it was not a good counter example.
>
> Let me justify why the formula doesn't prove anything. Starting from  
> a simple closed surface (topologically equivalent to a sphere) --  
> T=2V-4 holds.
>  - duplicate one vertex. Some of the triangles connected to it are  
> now connected to its copy. Topologically, this creates a hole (just  
> move a little the copy and you see it). The number of vertices is  
> incremented, keeping the number of faces unchanged.
>  - duplicate one face. This increase the number of faces without any  
> change to vertices.
> Then you can have a surface with duplicated faces and twice as many  
> duplicated vertices, such that T=2V-4 still hold... but there are  
> holes in it, so it's an open surface.
>
> Hope this clarify.
> Marie-Gabrielle Vallet
>
>
> 2008/10/9 Michael Jackson <mike.jackson at bluequartz.net>
> An Icosahedron isn't closed? I guess if you pop out a triangle it  
> wouldn't be close, but then the formula wouldn't work either.. I'm  
> probably just getting confused.
>
> _________________________________________________________
> Mike Jackson                  mike.jackson at bluequartz.net
> BlueQuartz Software                    www.bluequartz.net
> Principal Software Engineer                  Dayton, Ohio
>
>
> On Oct 9, 2008, at 3:09 PM, Marie-Gabrielle Vallet wrote:
>
> Mike,
>
> What can be proven is :
> for a triangular mesh that is closed, the formula F=2V-4 holds.
> But it doesn't mean :
> if a triangular mesh has F faces and V vertices with F=2V-4, then  
> the mesh is closed.
> A counter-example from the wikipedia page you pointed is the great  
> icosahedron. V=12, F=20, so F=2V-4, but this is not a closed surface.
>
> Marie-Gabrielle
>
> > Date: Thu, 9 Oct 2008 09:28:22 -0400
> > From: Michael Jackson <mike.jackson at bluequartz.net>
> > Subject: Re: [vtkusers] Proving a surface mesh of closeness
> > To: VTK Users <vtkusers at vtk.org>
> > Message-ID: <219B1FD6-F33E-475E-B6DB-9DE1E0F1C035 at bluequartz.net>
> > Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes
> >
> > 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
>
>