Tetrahedron Inertia Tensor - Maxima's use for the subject and a demand for the extension on the general Polyhedral 3D Objects - Messages
#1 Posted: 2/15/2026 12:59:53 PM
Hello Everybody,
I found a very interesting paper about the explicit formulas for the inertia tensor of a tetrahedron, defined with the 3 coordinates of its 4 vertices:
https://thescipub.com/abstract/jmssp.2005.8.11
The symbolic integration process for the obtention of the inertia tensor’s terms was conducted in the “F. Tonon’s” paper by using the “Mathematica” program.
I used the “SMath’ Maxima plugin” for the symbolic integrations.
My results gave identical numerical results as that of the “F. Tonon’s” example.
My formulas are available in an appended SMath file:
### Tetrahedron's Inertia Tensor - Ioan 2026.sm
I included some additional results obtained by using “Maxima” plugin, for “Eigenvalues and Eigenvectors”. Even if the “Maxima” plugin allows the obtention of symbolic results, the complexity of these formulas lets me prefer the numerical application.
I’m interested in having opinions of those familiar with this category of problems by looking in two directions:
Best Regards,
Ioan
I found a very interesting paper about the explicit formulas for the inertia tensor of a tetrahedron, defined with the 3 coordinates of its 4 vertices:
https://thescipub.com/abstract/jmssp.2005.8.11
The symbolic integration process for the obtention of the inertia tensor’s terms was conducted in the “F. Tonon’s” paper by using the “Mathematica” program.
I used the “SMath’ Maxima plugin” for the symbolic integrations.
My results gave identical numerical results as that of the “F. Tonon’s” example.
My formulas are available in an appended SMath file:
### Tetrahedron's Inertia Tensor - Ioan 2026.sm
I included some additional results obtained by using “Maxima” plugin, for “Eigenvalues and Eigenvectors”. Even if the “Maxima” plugin allows the obtention of symbolic results, the complexity of these formulas lets me prefer the numerical application.
I’m interested in having opinions of those familiar with this category of problems by looking in two directions:
a) the connection between the “Eigenvalues and Eigenvectors” results and the rotation classical procedure for the Inertia Moments Tensor.
b) Could it be formulated a procedure to find the Inertia Moments Tensor for the general case of Polyhedral Solid, similar to the classical procedure for 2D polygons, very nicely developed by Davide Carpi in the “Properties of generic Polygons” included in the “SMath Examples”. If the polygon problems are enough available in the literature, I do not know similar procedures for 3D polyhedral objects, defined by their vertex’s coordinates.
b) Could it be formulated a procedure to find the Inertia Moments Tensor for the general case of Polyhedral Solid, similar to the classical procedure for 2D polygons, very nicely developed by Davide Carpi in the “Properties of generic Polygons” included in the “SMath Examples”. If the polygon problems are enough available in the literature, I do not know similar procedures for 3D polyhedral objects, defined by their vertex’s coordinates.
Best Regards,
Ioan
Edited 2/15/2026 1:13:56 PM
Do to others as you would like them to do to you! Knowledge is of no value unless you put it into practice - Chekhov
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#2 Posted: 2/15/2026 3:26:38 PM
Once you have the inertia tensor of a tetrahedron in arbitrary coordinates, then you can compute the inertia of any body that has a surface triangulation (as available in the STL file format) using tetrahedra generated from a surface triangle and some reference point. The orientation of the triangles matters.
I assume that this approach is used in CAD software to compute the mass properties of 3D bodies.
Here is an example of the 2D procedure from my SMath book: https://smath.com/en-US/cloud/worksheet/LWrsp7uo
I assume that this approach is used in CAD software to compute the mass properties of 3D bodies.
Here is an example of the 2D procedure from my SMath book: https://smath.com/en-US/cloud/worksheet/LWrsp7uo
Edited 2/15/2026 3:46:59 PM
Martin KraskaPre-configured portable distribution of SMath Studio: https://en.smath.info/wiki/SMath%20with%20Plugins.ashx
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