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| title = Generalization of parallel axis theorem for rotational inertia
| volume = 85| doi-access = free
| bibcode = 2017AmJPh..85..791A
}}</ref> Let {{math|''I<sub>ij</sub>''}} denote the inertia tensor of a body as calculated at the center of mass. Then the inertia tensor {{math|''J<sub>ij</sub>''}} as calculated relative to a new point is
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where '''E'''<sub>3</sub> is the {{nobr|3 × 3}} [[identity matrix]] and <math>\otimes</math> is the [[outer product]].
Further generalization of the parallel axis theorem gives the inertia tensor about any set of orthogonal axes parallel to the reference set of axes x, y and z, associated with the reference inertia tensor, whether or not they pass through the center of mass.<ref name="Abdulghany"/> In this generalization, the inertia tensor can be moved from being reckoned about any reference point <math>\mathbf{R}_{ref}</math> to some final reference point <math>\mathbf{R}_F</math> via the relational matrix <math>M</math> as:
:<math> I_{F} = I_\mathrm{ref} + m(M[\mathbf{R},\mathbf{R}] - 2M[\mathbf{R},\mathbf{C}])</math>
where <math>\mathbf{C}</math> is the vector from the initial reference point to the object's center of mass and <math>\mathbf{R}</math> is the vector from the initial reference point to the final reference point (<math>\mathbf{R}_F = \mathbf{R}_{ref} + \mathbf{R}</math>). The relational matrix is given by
:<math> M[\mathbf{r},\mathbf{c}] = \left[\begin{array}{rrr}(r_y c_y + r_z c_z) & -1/2(r_x c_y + r_y c_x) & -1/2(r_x c_z + r_z c_x) \\
-1/2(r_x c_y + r_y c_x) & (r_x c_x + r_z c_z) & -1/2(r_y c_z + r_z c_y) \\
-1/2(r_x c_z + r_z c_x) & -1/2(r_y c_z + r_y c_z) & (r_x c_x + r_y c_y) \end{array}\right] </math>
==Second moment of area==
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y^2+z^2 & -xy & -xz \\ -y x & x^2+z^2 & -yz \\ -zx & -zy & x^2+y^2 \end{bmatrix}.</math>
This product can be computed using the matrix formed by the outer product ['''R''' '''R'''<sup>T</sup>] using the
:<math> -[R]^2 = |\mathbf{R}|^2[E_3] -[\mathbf{R}\mathbf{R}^T]=
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[[Category:Mechanics]]
[[Category:Physics theorems]]
[[Category:Moment (physics)]]
[[fr:Moment d'inertie#Théorème de transport (ou théorème d'Huygens
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