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### #Actualmax343

Posted 02 December 2012 - 05:21 PM

I'm not sure whether I got you right, but why is it a bad thing to use the origin? You can choose x0 to be whatever you want, and as long as it's constant this formula will work.

I'll try to rephrase. In the vector field F you can choose x0 as any point in R^3. After some manipulations we can interpret the result as the sum of volumes of multiple tetrahedrons that are comprised of the faces of the mesh and have one vertex in common, that is x0.
Geometrically this makes a lot of sense for sphere-like (or more generally convex) meshes, just choose x0 to be somewhere within the object and the formula should come trivially to you without the divergence theorem. For arbitrary oriented manifolds (or when x0 is not within the enclosed volume) this is not so clear, but the divergence theorem takes care of that.

### #1max343

Posted 02 December 2012 - 03:28 PM

I'm not sure whether I got you right, but why is it a bad thing to use the origin? You can choose x0 to be whatever you want, and as long as it's constant this formula will work.

I'll try to rephrase. In the vector field F you can choose x0 as any point in R^3. After some manipulations we can interpret the result as the sum of volumes of multiple tetrahedrons that are comprised of the faces of the mesh and have one vertex in common, that is x0.
Geometrically this makes a lot of sense for sphere-like meshes, just choose x0 to be somewhere within the object and the formula should come trivially to you without the divergence theorem. For arbitrary oriented manifolds (or when x0 is not within the enclosed volume) this is not so clear, but the divergence theorem takes care of that.

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