This Wikipedia article on distance from a point to a plane says the following:
If what is desired is the distance from a point not at the origin to the nearest point on a plane, this can be found by a change of variables that moves the origin to coincide with the given point.
Suppose we wish to find the nearest point on a plane to the point $(X_0, Y_0, Z_0)$, where the plane is given by $aX + bY + cZ = D$. We define $x = X - x_0$, $y = Y - Y_0$, $z = Z - Z_0$, and $d = D - aX_0 - bY_0 - cZ_0$, to obtain $ax + by + cz = d$ as the plane expressed in terms of the transformed variables. Now the problem has become one of finding the nearest point on this plane to the origin, and its distance from the origin.
So what's being described here is a translation, right?