__Brief overview of the notion
of homogeneous coordinates.__

The translation of a point, vector, or object
defined by a set of points in the plane is performed by adding the same quantity
to
each x-coordinate and the same quantity
to each y-coordinate. (We emphasize that
and
are not required to be equal in magnitude.) We illustrate this in Figure 1 for a
point (x,y) in R^{2}, where the coordinates of the translated point are .

Figure 1.

It is easy to show that the operation of
translation given by
is not a linear transformation. (Verify.) Thus we can not perform a translation
in R^{2} using multiplication by a 2 by 2 matrix. In
order to have
rotations, contractions and expansions, shears, and projections
"play together nicely" with translations (that is, each can be
performed by matrix multiplication) we change the space in which we work. To
employ matrix multiplication to perform translations we adjoin another component
to vectors and border matrices (See Figure 2.) with another row and column. This
change is said to use** homogeneous coordinates**. To use homogeneous
coordinates we make the following identifications.

Each of the
matrices **M** associated with plane linear transformations is now identified
with a 3 × 3 matrix of the form

.

Figure 2.

For
example when using homogeneous coordinates for a reflection about the y-axis the
corresponding matrix is the 3 × 3 matrix
. Also when using homogeneous coordinates for a rotation by an angle
the corresponding matrix is the 3 × 3 matrix
. A translation
can be performed by matrix multiplication on data expressed in homogeneous
coordinates using the 3 × 3 matrix
.
We have
.