answer: This equation can be validated by evaluating both sides using a matrix representation

from either **Section 5-2 or Section 5-11**.

ans: Using the trigonometric identities for the sine and cosine of the sum of two angles,

we can express the elements of the product matrix for two successive rotations in the xy

plane about the coordinate origin as

Since the addition of angles is commutative, this composite transformation is commutative.

The commutative property of two successive rotations can also be shown by comparing

the elements of the composite matrix * R1 R2* with the elements of the composite

matrix

The composite matrix for two successive translations is given by **Eq. 5-28 (page 241) **. And

this matrix product is commutative because the addition of the translation parameters

is commutative.

The composite matrix for two successive scaling transformations is given by **Eq. 5-33 (page242) **.

This is also a commutative matrix product because the product of the scaling parameters

is commutative.

ans: The matrix representation for a two-dimensional transformation sequence consisting of

scaling followed by rotation, relative to the coordinate origin, is

And the matrix representation for a rotation followed by scaling, relative to the coordinate

origin, is

These two matrices are equivalent only when sx = sy.

ans: Two successive reflections about a single axis yields the identity matrix; i.e., the object is

returned to its original position. A reflection about one axis followed by a reflection about

the other axis is equivalent to a rotation of 180°, assuming that the reflection parameters

are either 1 or -1.

ans: A transformation sequence for obtaining the composite matrix for reflection about a given

line is:

(1) Translate the line so that it passes through the coordinate origin. Since parameter

*b* is the value for the *y-axis* intersection, the translation parameters are *tx = 0* and

*ty = −b*.

(2) Rotate the line onto the *x axis*. A horizontal line is already on the *x axis*, and

a vertical line is perpendicular to the *x axis (a 90° angle)*. Otherwise the angle

between the reflection line and the positive* x axis* is calculated from the slope of the

line: *Ø = tan ^{-1 }m*

If this angle is positive (*m > 0*), perform a clockwise rotation with respect to the

coordinate origin. If the angle is negative (*m < 0*), perform a counterclockwise

rotation with respect to the coordinate origin. Thus, in either case, set *θ = −θ* and

use the transformation matrix for a counterclockwise rotation.

(3) Perform the reflection about the *x axis* using transformation matrix *5-52*.

(4) Carry out the inverse of steps (2) and (1) to return the line to its original position.

Multiplying these five transformation matrices and simplifying the trigonometric expressions

using double-angle formulas, we obtain the transformation matrix for reflection

about the line ** y = mx+ b** as

In this matrix, angle* θ *is the negative of the line angle with respect to the *x axis*, as

calculated in step (2).

ans: Given a set of character definitions and a required shearing amount, we can apply matrix

**5-59 **on **page 256** to each character to shear it relative to its baseline before displaying the character.