Mathematical Structures for Computer Graphics

4.4 Affine Transformations

There is one glaring omission among all the transformations we have visited so far. We have not considered the simple translation that just moves all points in the same direction. A translation just adds a vector to every point: . This is certainly a useful, in fact essential, transformation for graphics work; objects are moved around all the time. Unfortunately, it is not technically a linear transformation because it does not preserve vector addition and scalar multiplication. (In particular, it does not fix the origin and is not, in general, equal to for constant .) This does not cripple its use in manipulating graphics objects because it certainly still preserves lines. Yet, since translation is not a linear transformation, it apparently cannot be represented by standard matrix multiplication, putting it outside the system we have developed for all the other linear transformations.

Nevertheless, the essential nature of translations means we immediately ...