Graphics Reference
In-Depth Information
M a and M b with our example of V a =(
3
,
8
)
and T
(
1
,−
5
)
, S
(
3
,
0
.
5
)
.Wehave
already seen that
= 38 T
V a M a
(
1
,−
5
)
S
(
3
,
0
.
5
)
= 61
5 .
.
As for M b , where the order of the two transform operators are switched,
= 38 S
V a M b
(
3
,
0
.
5
)
T
(
1
,−
5
)
=
5
(
3
×
3
)
1
(
8
×
0
.
5
)
= 8
1 .
The results are very different!
9.2
Inverse Transformation
Y-a xi s
Figure 9.2 shows that in order to undo the transformation M a , we would:
Step 2:
T -1 (-1,-5)
V a
(
,
.
)
1. Undo the scaling by performing the inverse transform of of S
3
0
5
:
Step 1:
S -1 (3,0.5)
1
3 ,
S 1
(
3
,
0
.
5
)=
S
(
2
) .
V at
V ats
X-axis
Origin
M 1
Figure 9.2.
: undo-
a
2. Undo the translation by performing the inverse transform of T
(
1
,−
5
)
:
ing M a .
T 1
(
1
,−
5
)=
T
(
1
,
5
) .
From this example, we see that the operator to undo M a ,orthe inverse transform
of M a ,is
Order of operation. Remem-
ber that the operator on the
leftmost of a concatenation ex-
pression will operate on input
vertices first .
M 1
a
S 1 T 1
=
.
(9.3)
Recall from Equation (9.2) that M a is defined as
For example,
when applying
M a =
TS
.
M = TS
We s e e t h a t M a is applying the inverse transform of each of the operators in M a
in the reverse order. Instead of translate followed by scale, we apply the inverse
scale first, followed by the inverse translate.
In general, for any concatenated affine transform operator, the inverse of the
transform is always the inverse of each operator applied in reverse order.
to a point V ,
VM
=
VTS
,
the translation operator will
operate on the point V first.
For
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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