Fast computation of inertia through affinely extended Euler tensor

We introduce an affine extension of the Euler tensor which encompasses all of the inertia properties of interest in a convenient linear format, and we show how it transforms under affine maps. This result generalizes the standard theorems on the action of rigid transformations (translations and rotations) on inertia properties, allowing for stretch and shear components of the transformation. More importantly, it provides extremely simple and highly efficient computational tools. By these means, a very fast computation of the inertia properties of polyhedral bodies and surfaces may be obtained. The paper contains some mathematical background, a discussion of the state of the art, and a detailed algorithmic description of the new method, which computes and transforms the Euler tensor (strictly related to the inertia) under general affine maps, through addition and multiplication of 4x4 matrices. Evidence is given that the transformation-based computational technique that is introduced is much faster than conventional domain integration.