[vtkusers] Proving a surface mesh of closeness

[Dominik Szczerba]

vtkCleanUnstructuredGrid will act like duplicate triangle filter you refer to. 
It is not in vtk but you can 'steal' it from Paraview source.

Dominik

On Saturday 11 October 2008 11:35:31 pm Michael Jackson wrote:
> 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
>
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-- 
Dominik Szczerba, Ph.D.
Computational Physics Group
Foundation for Research on Information Technologies in Society
http://www.itis.ethz.ch