Transformations - Transformations and Projections in Computer Graphics - page 6

y

4

3

2

1

Rectangle

x

1 2 3 4 5

(a) (b)

Figure 1.2: Scissors and Shearing.

x = R cos α and y = R sin α . From this, we get the expressions for x ∗ and y ∗

x ∗ = R cos( α

− θ )= R cos α cos θ + R sin α sin θ = x cos θ + y sin θ,

y ∗ = R sin( α

− θ )= − R cos α sin θ + R sin α cos θ = − x sin θ + y cos θ.

Hence, the clockwise rotation matrix in two dimensions is

cos θ , cos θ 0

0 s θ

1 .

which also
equals
the product

− sin θ − tan θ

(1.4)

sin θ cos θ tan θ 1

This shows that any rotation in two dimensions is a combination of scaling (and, per-
haps, reflection) by a factor of cos θ and shearing, an unexpected result (that's true for
all angles where tan θ is finite).

P

P *

θ

φ

α

x *

x

Figure 1.3: Clockwise Rotation.

Exercise 1.6: Show how a 45 ◦ rotation can be achieved by scaling followed by shearing.