Hilariously Fast Volume Computation with the Divergence Theorem
This technical article explains a method for calculating the volume of a 3D mesh using the divergence theorem. It demonstrates how to simplify the triple integral of a volume into a surface integral over the mesh's triangular faces.
Why it matters
Efficient volume computation is a fundamental requirement in computer graphics, CAD software, and physical simulation engines.
The following presents a fast algorithm for volume computation of a simple, closed, triangulated 3D mesh. This assumption is a consequence of the divergence theorem. Further extensions may generalise to other meshes as well, although that is presently out of scope.
We begin with the definition of volume as the triple integral over a region of the constant one:
Let 𝐅 \mathbf{F} be a function in ℝ 3 \mathbb{R}^3 such that its divergence is equal to one. For the purposes of this paper, we choose:
𝐅 ( x , y , z ) = < x , 0 , 0 > \mathbf{F}(x, y, z) = <x, 0, 0>
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