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New linear-time algorithm computes volume of closed triangulated meshes via divergence theorem
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Illustration only Photo by Redd Francisco on Unsplash
The paper presents a linear-time method that evaluates the volume of a simple, closed, triangulated 3D mesh by converting a triple integral into a surface integral using a vector field with unit divergence.
Volume calculation is a common sub-task in simulation, geometry processing, and manufacturing pipelines, and a method that avoids costly rendering and sampling can cut runtime dramatically. The algorithm’s O(n) complexity and minimal arithmetic per triangle make it attractive for large-scale meshes and real-time applications.
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The divergence theorem rewrites the volume integral as a sum of surface integrals over each triangle, using the field F(x,y,z)=<x,0,0> whose divergence equals one.
Each triangle contributes (Δ₁×Δ₂)_x · (T₀x+T₁x+T₂x)/6, requiring only seven additions and three multiplications per triangle, for a total of about 11 n floating-point operations.
The approach runs in O(n) time for n triangles and outperforms naive render-and-sample techniques, but it is limited to simple closed triangulated meshes.
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