I am having difficulty understand how to get from a homogeneous transformation matrix:
$
M = \begin{pmatrix} R&t\\0&1 \end{pmatrix}
$
Where $R$ is a rotation matrix (or it could be any linear transformation, I just chose rotation for this example) and $t$ is a translation vector.
To it's inverse definition:
$
M^{-1} = \begin{pmatrix} R^{-1}&-R^{-1}t\\0&1 \end{pmatrix}
$
The part that I don't understand is the $-R^{-1}t$ term in the $(0,1)$ entry of $M^{-1}$.
My incorrect expectation was:
$
M^{-1} = \begin{pmatrix} R^{-1}&-t\\0&1 \end{pmatrix}
$
So where does the extra multiplication by $R^{-1}$ come from?
Thank you! (:
Best Answer
If $y=Rx+t$, then $x=R^{-1}(y-t)=R^{-1}y-R^{-1}t$.