[vtkusers] Proving a surface mesh of closeness

[Marie-Gabrielle Vallet]

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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