The Order of Transformations
The order of transformations is critical.
A simple case of two transformations: a 45-degree
counterclockwise rotation about the origin around the z-axis, and a translation
down the x-axis.
If you rotate the object first and then
translate it, the rotated object appears on the x-axis. If you translate it
down the x-axis first, however, and then rotate about the origin, the object is
on the line y=x.