Transformations - Transformations and Projections in Computer Graphics - page 28

An improper rotation differs from a rotation in one important aspect. When we
rotate an object through a small angle and repeat this transformation, the object seems
to move smoothly along a circle. Each time we repeat an improper rotation, however,
the object “jumps” from one side of the coordinate plane to the other. The total effect
is very different from that of a smooth circular movement.
1.2.9 Decomposing Transformations
Sometimes, a certain transformation A may be equivalent to the combined effects of
several different transformations B , C ,and D .Wesaythat A can be decomposed into B ,
C ,and D . Mathematically, this is equivalent to saying that the original transformation
matrix T A equals the product T B T C T D . We have already seen that a rotation in two
dimensions can be decomposed into a scaling followed by a shearing; here are other
examples.
It may come as a surprise that the general two-dimensional transformation matrix,
Equation (1.7), can be written as a product of shearing, scaling, rotation, and translation
as follows:
⎡ ⎤

ab 0
cd 0
mn 1

⎣ ⎦ =

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

1 0 0 A 0 0 a/A b/A 0 100

010
mn 1

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ,

( ac + bd ) /A 2 10 0 ad − bc ) /A 0 − b/A a/A 0

0 0 1 0 0 1 0 0 1

(1.21)
where A = √ a 2 + b 2 . The third matrix produces rotation since ( a/A ) 2 +( b/A ) 2 =1.
Even something as simple as shearing in one direction can be written as the product
of a unit shearing and two scalings:
⎛

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

100
c 10
001

1 /c 00
010
001

100
110
001

c 00
010
001

⎝ ⎠ = ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ .

Even the simple transformation of a unit shearing can be decomposed into a product
that involves a scaling and two rotations. Note that the Golden Ratio φ is involved,
⎛

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

100
110
001

cos α − sin α 0 φ 00

0 /φ 0
001

cos β sin β 0

⎝ ⎠ = ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ,

sin α cos α 0 − sin β cos β 0

0 0 1 0 0 1

where α =tan − 1 φ 58 . 28 ◦ and β =tan − 1 (1 /φ )

31 . 72 ◦ .
(This is indeed a surprising result. It means that a clockwise rotation of 58 . 28 ◦ ,
followed by a scaling of φ in the x direction and 1 /φ in the y direction, followed by
a counterclockwise rotation of 31 . 72 ◦ , is equivalent to a unit shear in the x direction.
This is illustrated by Figure 1.17.)

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