Where did the final equation on this slide come from? It seems like we substituted x'R = x-t, but the formula we are given (rigid motion) is the reverse: x' = R(x-t).
motoole2
The rotation matrix $R$ is unitary, meaning $R^T R = R R^T = I$. So we can re-write the left equation as
$R^T x' = R^T R (x - t) = (x-t)$
Transposing this result gives us
$x'^T R = (x-t)^T$
Finally, we simply have to plug in this result into the equation on the right hand side to get our desired result.
Where did the final equation on this slide come from? It seems like we substituted x'R = x-t, but the formula we are given (rigid motion) is the reverse: x' = R(x-t).
The rotation matrix $R$ is unitary, meaning $R^T R = R R^T = I$. So we can re-write the left equation as
$R^T x' = R^T R (x - t) = (x-t)$
Transposing this result gives us
$x'^T R = (x-t)^T$
Finally, we simply have to plug in this result into the equation on the right hand side to get our desired result.