Page 293 -

1 0 
P =   (9.3)
x
 0 −1 

−  1  0
P =   (9.4)
y
 0 1 

In-Class Exercise

Pb. 9.1 Using the trapezoid of Example 9.1, obtain all the transformed G’s
as a result of the action of each of the three transformations defined in Eqs.
(9.2) through (9.4), and plot the transformed figures on the same graph.
Pb. 9.2 In drawing the original trapezoid, we followed the counterclock-
wise direction in the sequencing of the different vertices. What is the sequenc-
ing of the respective points in each of the transformed G’s?
2
2
Pb. 9.3 Show that the quantity (x + y ) is invariant under separately the
action of P , P , or P.
y
x

9.1.3 Rotation around the Origin
The new coordinates of a point in the x-y plane rotated by an angle θ around
the z-axis can be directly derived through some elementary trigonometry.
Here, instead, we derive the new coordinates using results from the complex
numbers chapter (Chapter 6). Recall that every point in a 2-D plane repre-
sents a complex number, and multiplication by a complex number of modu-
lus 1 and argument θ results in a rotation of angle θ of the original point.
Therefore:

z ′ = ze jθ

x ′ + jy ′ = x ( + jy)(cos( θ) + j sin( θ)) (9.5)
= x ( cos( θ) − y sin( θ)) + j x ( sin( θ) + y cos( θ))

Equating separately the real parts and the imaginary parts, we deduce the
action of rotation on the coordinates of a point:

x ′ = xcos( )θ − y sin( )θ
(9.6)
y ′ = xsin( )θ + y cos( )θ

© 2001 by CRC Press LLC