[Bf-committers] Blender Center of Mass - Voxel data better algorithm?

[panjz]

> From: brechtvanlommel at pandora.be
> Date: Sun, 16 Jun 2013 05:38:58 +0200
> To: bf-committers at blender.org
> Subject: Re: [Bf-committers] Blender Center of Mass - Voxel data better	algorithm?
> 
> Indeed, you can compute the exact volume of a closed triangle mesh by
> summing signed volumes of tetrahedra. The tessellation doesn't even
> need to be any good, you can just use one tetrahedron for each
> triangle, constructed from the triangle vertices and one other fixed
> point (typically the origin).
> http://stackoverflow.com/questions/1406029/how-to-calculate-the-volume-of-a-3d-mesh-object-the-surface-of-which-is-made-up
> 
> By summing the centers of mass of these tetrahedra, weighted by the
> signed volume, then I guess you get the center of mass of the whole
> mesh too? I didn't check the math but intuitively it makes sense to
> me.

Yes, it's correct because the center of mass is $\frac{\int_M (x,y,z)dV}{\int_M dV}$,
after simplification it's weighted average of center of mass of individual tetrahedron.

> This algorithm does require the mesh to be closed. For non-closed
> meshes the fixed point could perhaps be the center of mass as computed
> now, that might give a reasonable approximation.
> 

IMO it's hard to define center of mass of non-closed mesh. We can only calculate center of mass of the hull in this case. Maybe add an option in UI to specify which type we want to calculate...