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In-Class Exercise

Pb. 9.14 Translate the trapezoid of Example 9.1 by a vector of length 5 that
is making an angle of 30° with the x-axis.

9.2 Homogeneous Coordinates

As we have seen in Section 9.1, inversion about the origin, reflection about the
coordinate axes, rotation, and scaling are operations that can be represented by
a multiplicative matrix, and therefore the composite operation of acting succes-
sively on a figure by one or more of these operations can be described by a prod-
uct of matrices. The translation operation, on the other hand, is represented by
an addition, and thus cannot be incorporated, as yet, into the matrix multiplica-
tion scheme; and consequently, the expression for composite operations
becomes less tractable. We illustrate this situation with the following example:

Example 9.2
Find the new G that results from rotating the trapezoid of Example 9.1 by a
π/4 angle around the point Q (–5, 5).

Solution: Because we have thus far defined the rotation matrix only around
the origin, our task here is to generalize this result. We solve the problem by
reducing it to a combination of elementary operations thus far defined. The
strategy for solving the problem goes as follows:
1. Perform a translation to place Q at the origin of a new coordinate
system.
2. Perform a π/4 rotation around the new origin, using the above
form for rotation.
3. Translate back the origin to its initial location.

Written in matrix form, the above operations can be written sequentially as
follows:

1. G = G T ones+ * ( ,1 n + )1 (9.13)
1
 5 
where T=   (9.14)
 − 5 
and n = 4.

© 2001 by CRC Press LLC