Transformations - Transformations and Projections in Computer Graphics - page 25

Using the ratio k , we can write a similarity (ignoring the translation part) as the
product

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

ac 0
−

k 00
0 k 0
001

a/k c/k 0

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ,

ca 0
001

− c/k a/k 0

0 0 1

which shows that a similarity is a combination of a scaling/reflection (by a factor k )
and a rotation. (The definition of k implies that ( a/k ) 2 +( c/k ) 2 = 1, so we can consider
c/k and a/k the sine and cosine of the rotation angle, respectively.)
1.2.6 A 180 ◦ Rotation
Another interesting example of combining transformations is a 180 ◦ rotation about a
fixed point P =( P x ,P y ). This combination is called a halfturn . It is performed, as
usual, by translating P to the origin, rotating about the origin, and translating back.
The transformation matrix is (notice that cos(180 ◦ )=

− 1)

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

1 0 0 − 100

0

100
010
P x

− 100

0

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = ⎝ ⎠ .

T = 0 1 0 10

001

− − 10

− P x − P y 1 P y 1 2 P x 2 P y 1

A general point ( x, y ) is therefore transformed by a halfturn to

⎛ ⎞

− 100

0

⎝ ⎠ =(

( x, y, 1) − 10 − x +2 P x , − y +2 P y , 1) (1.19)

2 P x 2 P y 1

(Figure 1.14a), but it's more interesting to explore the effect of two consecutive halfturns,
about points P and Q . The second halfturn transforms point ( − x +2 P x , − y +2 P y , 1)

to

⎛ ⎞

− 100

0

⎝ ⎠ =( x

( − x +2 P x , − y +2 P y , 1) − 10 − 2 P x +2 Q x ,y − 2 P y +2 Q y , 1) . (1.20)

2 Q x 2 Q y 1

If P = Q , then the result of the second halfturn is ( x, y ), showing how two identical
180 ◦ rotations return a point to its original location. If P and Q are different, the result
is a translation of the original point ( x, y )byfactors − 2 P x +2 Q x and − 2 P y +2 Q y

(Figure 1.14b).

Exercise 1.32: What is the result of three consecutive halfturns about the distinct
points P , Q ,and R ?

Things turn out best for the people who make the best out of the
way things turn out.

—Art Linkletter