Graphics Reference
In-Depth Information
2.4.4. Theorem.
Every transformation of the plane defined by equations (2.25) is a
composition of translations, rotations, shears, and/or scaling transformations. Con-
versely, every composition of such maps can be described by equations of the form
(2.25).
Proof.
Let M be defined by equations (2.25) and set M
0
= TM, where T is the trans-
lation with translation vector (-m,-n). Then M
0
(
0
) =
0
and M
0
is a nonsingular linear
transformation. Let r = |M
0
(
e
1
)|, let R be the rotation about the origin that rotates
the unit vector (1/r)M
0
(
e
1
) into
e
1
, and let M
1
= RM
0
. It follows that M
1
is defined by
equations
()
==
M
ev e
r
11
1
1
()
== +.
M
ev ee
s
t
12
2
1
2
Define a linear transformation S by
()
=
S
vv
1
1
()
=
(
)
S
v
v
•
e
e
==
we
t
.
2
2
2
2
2
2
Since a linear transformation is completely defined once it is defined on a basis, S is
well defined. In fact, it is easy to show that S is a shear in the x-direction defined by
equations
x
¢=
x
s
t
xy.
y
¢=-
+
Figure 2.17 shows the effect of the maps R and S. The map M
2
= SM
1
is now the
scaling transformation defined by
¢x x
y
¢=
sy.
To summarize, M = T
-1
R
-1
S
-1
M
2
and the first part of the theorem is proved. Since the
converse of the theorem is obvious, Theorem 2.4.4 is proved.
M
0
(e
2
)
v
2
= M
1
(e
2
)
w
2
M
0
(e
1
)
e
2
v
1
= M
1
(e
1
)
q
v
1
e
1
shear
S
rotation
R
Figure 2.17.
The rotation and shear in the proof of Theorem 2.4.4.
