How can I calculate the mass and moment of inertia of a polyhedron?

Question

For use in a solid simulation, I want to calculate the mass and inertia tensor (moment of inertia), given the triangular grid representing the boundary of the (not necessarily convex) object and assuming a constant density in the inside.

Answer 1

Assuming your trim is closed (convex or not), there is a way!

As dmckee points out, the general approach is to build tetrahedra from each surface triangle, and then use the obvious math to sum the contributions of mass and moment from each tet. The trick comes when the surface of the body has concavities that create internal pockets when viewed from any reference point.

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Answer 2

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Answer 3

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Answer 4

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mass_polyhedron/mass_hypercube \approx points_internal/points_total.

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Answer 5

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Also note that the polyhedron must not be convex for the formulas to work, it must be simple .