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Say that N is the number of nodes, E the number of elements and S the number of sides in a triangular 2D mesh. Is there a relationship that links these quantities, possibly taking into account that some nodes are boundary nodes (Nb) and some are interior nodes (Ni)?

The answer can be restricted to Delaunay triangulation.

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Yes there is a relationship, the Euler characteristic:

For a 2-dimensional orientable manifold with boundaries embedded in $\mathbb{R}^3$, the Euler characteristic is

$\chi = V - E + F = 2 - 2g - b$

where $V$ is the number of vertices, $E$ is the number of edges, $F$ is the number of faces, $g$ is the genus of the manifold, and $b$ is the number of borders of the manifold.

For example, a disk has no holes, and one border, so its Euler characteristic is $2 - 2(0) - 1 = 1$, thus we know that $V - E + F = 1$.

On a torus, $g = 1$ and $b = 0$, thus $\chi = 2 - 2(1) - 0 = 0$, so $V - E + F = 0$

The point being that this number is independent of the exact triangulation, it only depends on the topology of the manifold.

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  • $\begingroup$ So there are two numbers to determine: one is the Euler characteristic that is $\chi = 2 - 2g - b$ and pertains to the general topological features of the manifold in point. Then, there is a relationship $V - E + F = \chi$ that holds for all manifolds (that are 2-dimensional orientable and so forth). Is this understanding grossly right? Could you explain what is the genus of the manifold? $\endgroup$ – XavierStuvw Jan 17 '17 at 7:38
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    $\begingroup$ Technically I have seen both $\chi = V - E + F$ and $\chi = 2 - 2g - b$ used as the definition of $\chi$, but I think the way that you're thinking about it makes the most sense. The genus is the number of "holes" in the manifold, so a torus (a donut shape) has one hole, this link has a genus two shape en.wikipedia.org/wiki/Genus-2_surface. Note that an annulus does not have genus one, just two borders $\endgroup$ – rviertel Jan 18 '17 at 14:06
  • $\begingroup$ I also found this other post that addressed the same topic math.stackexchange.com/q/1541125/446004 $\endgroup$ – XavierStuvw Jun 12 '17 at 11:58

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