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 R2, where the coordinates of the translated point are .
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 R2 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
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