Parallel Axis

Consider the inertia tensor $I_{x}$ defined by

I_{x} ≜ \int x \cdot x 1 - x \otimes x d α

The transformed tensor $I_{x + ρ}$ corresponding to a translation by $ρ$ is obtained by replacing $x$ with $x + ρ$ , furnishing

I_{x + ρ} = I_{x} + 2 A ρ \cdot \bar{ρ} 1 - A (\bar{ρ} \otimes ρ + ρ \otimes \bar{ρ}) + A (ρ \cdot ρ 1 - ρ \otimes ρ)

where the centroid $\bar{ρ}$ and area $A$ are defined as

\bar{ρ} ≜ \frac{1}{A} \int x d α and A ≜ \int d α

In the plane one often defines

J ≜ \int \hat{r} \otimes \hat{r} d α

In terms of $I$ this is

J = - i^{\times} I i^{\times}

and

I = I_{0} i \otimes i - \int {\hat{r}}^{\times} (i \otimes i) {\hat{r}}^{\times}

I = I_{0} i \otimes i - i^{\times} \int \hat{r} \otimes \hat{r} d α i^{\times}

so that

I = I_{0} i \otimes i - i^{\times} J i^{\times}

It is convenient to note that with coordinates $r = [x, y, z]$

[r^{\times}] [r^{\times}]^{t} = - {[\begin{array}{ccc} 0 & - z & y \\ z & 0 & - x \\ - y & x & 0 \end{array}]}^{2} = [\begin{array}{ccc} y^{2} + z^{2} & - x y & - x z \\ - y x & x^{2} + z^{2} & - y z \\ - z x & - z y & x^{2} + y^{2} \end{array}]