Sequin's paper presents the extended equation as I - N_0 + N_1 - N_2 ...N_D = R_1 - R_2 + R_3...R_D, where "I" is # disjoint components, N_0 is vertices V, N_1 is edges E...R_1 is ring of edges, R_2 is ring of faces etc.
Now, how can I map it to V + F - E = 2 (S - g) + r, where S are Shells and g is genus, and r are inner loops?
Article Generalized Euler-Poincaré Theorem
Thanks Torsten. I agree with N0 N1 and N2...any suggestions for R1 R2 R3 , for both, manifold and non-manifold cases? Any papers and docs on this topics?
A very unusual approach to the problem is given by the paper.
The origin is algebraic topology. The number 2 in the Euler formula is the Euler characteristics of a manifold. Therefore the number of components (of disjoint components) modifies the Euler formula to
V+F-E = 2*(number of disjoint components) (maybe your shells S above)
Secondly, if your faces contain holes then one has a boundary and has 2-h (h the number of wholes on the faces) as Euler characteristics. Last, you mentioned the genus g (number of 1-handles) to get finally 2-h-2g for the most general case.
If I understand it correctly in the paper, one has:
I=S R3=2g R1=holes in the faces
R2=r (?)
The topological definition is so different from this view.
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