Graphics Reference
In-Depth Information
Y-a
xi
s
Note that Equation (9.1) is the same as
V
a
=(3,8)
V
a
T
(
t
x
,
t
y
)
S
(
s
x
,
s
y
)
T(
-1,-5)
or
V
ats
=(6,1.5)
V
at
=(2,3)
V
at
S
(
s
x
,
s
y
)=
V
a
T
(
t
x
,
t
y
)
S
(
s
x
,
s
y
)
.
X-axis
Origin
Figure 9.1 shows an example where
38
Figure 9.1.
The re-
sults of applying translate
and scale operators in suc-
cession.
=
,
V
a
with
t
x
=
−
1
T
,
t
y
=
−
5
with
s
x
=
3
S
.
s
y
=
0
.
5
If we apply the two operators separately, with translate first:
=
x
at
y
at
V
at
=
x
a
+
t
y
t
x
y
a
+
=
3
5
−
18
−
=
23
,
followed by the scale operator:
=
(
,
)
V
at s
V
at
S
s
x
s
y
=
x
at
s
y
×
s
x
y
at
×
=
2
5
,
×
×
.
33
0
we get
=
61
5
.
V
at s
.
We can verify this result with Equation (9.1):
=
(
y
s
V
at s
x
a
+
t
x
)
×
x
s
(
y
a
+
t
y
)
×
=
(
5
3
−
1
)
×
3
(
8
−
5
)
×
0
.
=
61
5
.
.
This result is applicable to all the transformation operators we have studied, where
given two or more transformation operators, it is possible to combine (or concate-
nate) these operators into a single operator. This concatenated operator has the
same net effect on vertices as applying the individual operators in the concate-
nation order. For the above given example, we can define a new operator
M
a
to
encode the translation followed by scaling operation:
M
a
=
TS
.
(9.2)
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