Transformations and the Graphics Pipeline - Computer Graphics and Geometric Modeling - page 134

Figure 4.18. Parts of a homogeneous matrix.

1000
0700
0010
0000

1000
3100
0010
0001

Ê ˆ Ê ˆ

1 000
0100
0 010
1351

Ê ˆ

Á ˜ Á ˜

Á ˜

Á ˜

Á ˜ Á ˜

Á ˜ Á ˜ Á ˜

Á ˜ Á ˜ Á ˜

Á ˜

Á ˜ Á ˜

-

Ë ¯ Ë ¯

Ë ¯

(a) (b) (c)

1000
0700
0010
0005

1002
0103
0010
0001

Ê ˆ Ê ˆ

Á ˜ Á ˜

Á ˜ Á ˜

Á ˜ Á ˜

Á ˜ Á ˜

Á ˜ Á ˜

Ë ¯ Ë ¯

(d) (e)

Figure 4.19. Transformation examples.

transformation. Furthermore, computer hardware can be optimized to deal with 4 ¥
4 matrices to more than compensate for the inefficiency of computation issue men-
tioned above.

Let us look at the advantage of homogeneous coordinates in more detail. To see
the geometric power contained in a 4 ¥ 4 homogeneous matrix consider Figure 4.18.
The matrix can be divided into the four parts L, T, P, and S as shown, each of which
by itself has a simple geometric interpretation. The matrix corresponds to an affine
map if and only if P is zero and in that case we have a linear transformation defined
by L followed by a translation defined by T. If P is nonzero, then some plane will be
mapped to infinity. We illustrate this with the examples shown in Figure 4.19.
First, consider L. That matrix corresponds to a linear transformation of R 3 . If L
is a pure diagonal matrix, then we have a map that expands and/or contracts along