<div dir="ltr">Mike,<br><br>What can be proven is :<br>for a triangular mesh that is closed, the formula F=2V-4 holds.<br>But it doesn't mean :<br>if a triangular mesh has F faces and V vertices with F=2V-4, then the mesh is closed.<br>
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.<br><br>Marie-Gabrielle<br><br>> Date: Thu, 9 Oct 2008 09:28:22 -0400<br>>
From: Michael Jackson <<a href="mailto:mike.jackson@bluequartz.net">mike.jackson@bluequartz.net</a>><br>>
Subject: Re: [vtkusers] Proving a surface mesh of closeness<br>>
To: VTK Users <<a href="mailto:vtkusers@vtk.org">vtkusers@vtk.org</a>><br>>
Message-ID: <<a href="mailto:219B1FD6-F33E-475E-B6DB-9DE1E0F1C035@bluequartz.net">219B1FD6-F33E-475E-B6DB-9DE1E0F1C035@bluequartz.net</a>><br>>
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> <br>>
I am not a mathematician or theorist so I only have other people to go<br>>
on for this.<br>
> <br>>
Wikipedia - Take it or leave it:<br>
> <a href="http://en.wikipedia.org/wiki/Euler_characteristic">http://en.wikipedia.org/wiki/Euler_characteristic</a><br>
> <br>>
Other sources:<br>
> <a href="http://www.ics.uci.edu/~eppstein/junkyard/euler/">http://www.ics.uci.edu/~eppstein/junkyard/euler/</a><br>
> <a href="http://mathworld.wolfram.com/PolyhedralFormula.html">http://mathworld.wolfram.com/PolyhedralFormula.html</a><br>
> <a href="http://mathworld.wolfram.com/EulerCharacteristic.html">http://mathworld.wolfram.com/EulerCharacteristic.html</a><br>
> <a href="http://www2.in.tu-clausthal.de/~hormann/papers/Hormann.2002.AEW.pdf">http://www2.in.tu-clausthal.de/~hormann/papers/Hormann.2002.AEW.pdf</a><br>
> <br>
> <br>>
I think I agree for a mesh that does not consist of ALL triangles.<br>>
Then you are probably correct but for a triangular mesh that is closed<br>>
the formula has been proven. It is important to understand those<br>>
conditions and thanks for the heads up.<br>
> <br>>
Mike</div>