Graphics Reference
In-Depth Information
x x ym
¢=
+
+
y x yn
¢=
+
+
(2.25a)
where
ab
cd
π 0.
(2.25b)
2.4.3. Theorem.
The set of transformations defined by equations (2.25) form a
group under composition.
Proof.
This is straightforward. The main observation is that since the determinant
in (2.25b) is nonzero, the transformations have inverses. It is also easy to show that
the inverses are defined by equations of the same form.
The transformations defined by equations (2.25) clearly include the motions and
similarities. It is worth noting that they are simply the composition of a linear trans-
formation of the plane followed by a translation. There are two other interesting
special cases.
Definition.
The linear transformation of the plane defined by the equations
¢x x
y
¢=
dy
,
ad
π
0
(2.26)
is called a (
local
)
scaling transformation
. It is a
global scaling transformation
if a = d.
Note that the scaling transformation defined by equations (2.26) is orientation
reversing if ad < 0. It will be a similarity if a = d > 0. It is easy to check that the inverse
of the scaling transformation above is the scaling transformation
¢=
()
x
1a
x
(
)
y
¢=
d 1.
Definition.
A linear transformation of the plane defined by equations
¢xx
y xy
¢=
+
(2.27a)
is called a
shear in the x-direction
. A linear transformation defined by equations
xx y
¢=
+
(2.27a)
y
¢=
y
is called a
shear in the y-direction
.
It is easy to show that the inverse of a shear is a shear. See Exercise 2.4.1.
