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I am building a game physics engine following Ian Millington's book Game Physics Engine Development and I am confused by one of his implementations when changing matrix bases.

In order to apply rotation to rigid bodies, I have stored the inverse of the inertia tensor in the RigidBody class. It is given in local space, however, so in order to use it, I need to transform it into world space.

This is the code he's using (I know it's a lot of code, but I didn't want to alter the original). The function takes in the inverse inertia tensor in local space (iitBody) and the rotation matrix of the rigid body (rotmat), and outputs the inverse inertia tensor in world space (iitWorld). 

static inline void _transformInertiaTensor(Matrix3 &iitWorld, const Quaternion &q, const Matrix3 &iitBody, const Matrix4 &rotmat)
{
    real t4  = rotmat.data[0]*iitBody.data[0] + rotmat.data[1]*iitBody.data[3] + rotmat.data[2]*iitBody.data[6];
    real t9  = rotmat.data[0]*iitBody.data[1] + rotmat.data[1]*iitBody.data[4] + rotmat.data[2]*iitBody.data[7];
    real t14 = rotmat.data[0]*iitBody.data[2] + rotmat.data[1]*iitBody.data[5] + rotmat.data[2]*iitBody.data[8];
    real t28 = rotmat.data[4]*iitBody.data[0] + rotmat.data[5]*iitBody.data[3] + rotmat.data[6]*iitBody.data[6];
    real t33 = rotmat.data[4]*iitBody.data[1] + rotmat.data[5]*iitBody.data[4] + rotmat.data[6]*iitBody.data[7];
    real t38 = rotmat.data[4]*iitBody.data[2] + rotmat.data[5]*iitBody.data[5] + rotmat.data[6]*iitBody.data[8];
    real t52 = rotmat.data[8]*iitBody.data[0] + rotmat.data[9]*iitBody.data[3] + rotmat.data[10]*iitBody.data[6];
    real t57 = rotmat.data[8]*iitBody.data[1] + rotmat.data[9]*iitBody.data[4] + rotmat.data[10]*iitBody.data[7];
    real t62 = rotmat.data[8]*iitBody.data[2] + rotmat.data[9]*iitBody.data[5] + rotmat.data[10]*iitBody.data[8];

    iitWorld.data[0] = t4*rotmat.data[0]  + t9*rotmat.data[1]  + t14*rotmat.data[2];
    iitWorld.data[1] = t4*rotmat.data[4]  + t9*rotmat.data[5]  + t14*rotmat.data[6];
    iitWorld.data[2] = t4*rotmat.data[8]  + t9*rotmat.data[9]  + t14*rotmat.data[10];
    iitWorld.data[3] = t28*rotmat.data[0] + t33*rotmat.data[1] + t38*rotmat.data[2];
    iitWorld.data[4] = t28*rotmat.data[4] + t33*rotmat.data[5] + t38*rotmat.data[6];
    iitWorld.data[5] = t28*rotmat.data[8] + t33*rotmat.data[9] + t38*rotmat.data[10];
    iitWorld.data[6] = t52*rotmat.data[0] + t57*rotmat.data[1] + t62*rotmat.data[2];
    iitWorld.data[7] = t52*rotmat.data[4] + t57*rotmat.data[5] + t62*rotmat.data[6];
    iitWorld.data[8] = t52*rotmat.data[8] + t57*rotmat.data[9] + t62*rotmat.data[10];
}

To my understanding, in order to change the matrix from local to world space, I would simply need to multiply the rotation matrix by the inverse inertia tensor in world space. However, Millington's implementation includes a second multiplication by the rotation matrix and I haven't been able to understand why.

1201ProgramAlarm
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Juan Diego Alzate Mejia
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  • I have found that https://stackoverflow.com/questions/18290798/calculating-rigid-body-inertia-tensor-world-coordinates?rq=1 answers the question and states that the right way is multiplying the rotation matrix by the inverse inertia tensor in local space. However, my question is more about _why_ the author did what he did. – Juan Diego Alzate Mejia Aug 24 '19 at 21:31
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    Maybe it's better to ask this question [here](https://math.stackexchange.com)! – csabinho Aug 24 '19 at 22:02
  • @csabinho Oh okay, thanks for the tip! I'll try that, too. – Juan Diego Alzate Mejia Aug 24 '19 at 22:05
  • I'm voting to close this question as off-topic because it is a math problem instead of a programming problem – L. F. Aug 25 '19 at 07:55

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