# Have Inertia Tensor for Object. Find It for Object after Rotation

#### RoshanBBQ

1. Homework Statement
Let's say I have a coordinate system that has (0,0,0) at the CM of an object, and I know the object's inertia tensor for that coordinate, T. (T is a 3x3 inertia tensor where (1,1) is moment of inertia about x-axis, (2,2) is moment of inertia about y-axis, and (3,3) is moment of inertia about z-axis).

I then rotate the object about its CM by a degrees about x-axis, b degrees about y-axis, and c degrees about the z-axis. I am using the convention where a positive angle rotates the object clockwise if the positive portion of the axis about which rotation is happening is approaching you from your viewing spot.

Is there a simple matrix transform I can do to T to find the new inertia tensor, T'? This is for a coding project. As of now, I am recomputing T' for each orientation. I would like to speed things up if possible. I am searching online with little success. If one of you gentlemen or ladies knows the relation off the top of your head, it would save me much time!

EDIT: I found something about finding the inertia tensor for a new, rotated coordinate. Is it the case that rotating an object a b and c would be the tensor found about the coordinates rotated -a -b and -c?

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#### RoshanBBQ

Here is the answer for future people with this question:
Let us say you know the rotation matrix to rotate a set of x,y, and z coordinates like so:
$$\begin{pmatrix} x_{new}\\ y_{new}\\ z_{new} \end{pmatrix} = R \begin{pmatrix} x\\ y\\ z \end{pmatrix}$$
And you rotated your object using R. If you have T, the old inertia tensor before rotation, the new one is found by
$$T_{new} = RTR^T$$

However, and this is an annoying lack of complete resource from places I searched (I had to derive this myself), if you have (LIKE IN MATLAB) a rotation matrix used like so:
$$\begin{pmatrix} x_{new} & y_{new} & z_{new} \end{pmatrix} = \begin{pmatrix} x & y & z \end{pmatrix} R$$

Then, we have
$$T_{new} = R^T T R$$

R is a 3x3 rotation matrix. In MATLAB, your rotation matrix will be 4x4. Simply discard 4th row and column.

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