# Flip rotation matrix

I'm doing character control with kinect. I need to mirror the joint orientation because the character faces the player. Somehow by Googling through internet I've done it and everything works very well. But I have little idea about how the math works, here's my code:

//-------------------------------------------------------------------------------------
Ogre::Quaternion JointOrientationCalculator::buildQuaternion(Ogre::Vector3 xAxis, Ogre::Vector3 yAxis, Ogre::Vector3 zAxis)
{
Ogre::Matrix3 mat;
if(isMirror)
{
mat = Ogre::Matrix3(xAxis.x, yAxis.x, zAxis.x,
xAxis.y, yAxis.y, zAxis.y,
xAxis.z, yAxis.z, zAxis.z);
Ogre::Matrix3 flipMat(1, 0,  0,
0, 1,  0,
0, 0, -1);
mat = flipMat * mat * flipMat;
}
else
{
mat = Ogre::Matrix3(xAxis.x, -yAxis.x, zAxis.x,
-xAxis.y, yAxis.y, -zAxis.y,
xAxis.z,  -yAxis.z, zAxis.z);
}

Ogre::Quaternion q;
q.FromRotationMatrix(mat);

return q;
}


When I need to mirror/flip it by axes z I calculate mat = flipMat * mat * flipMat; but I don't understand how this equation works.

Conceptually, it's similiar to what you'd do to rotate around a specified pivot point: you'd translate the pivot to the origin, then do the rotation, then undo the translation to return the pivot to its original location. In general you'd have a formula like inverse(matPivot) * matRotate * matPivot. However, in the case of your flip, flipMat is its own inverse, so the inverse() call is unnecessary. (By the way, a transform that is its own inverse is called an involution.)