Graphics Reference
In-Depth Information
then
v
=
R
−
1
·
S
−
1
·
v
t
R
−
1
T
−
1
=
·
·
t
and
t
+
v
p
=
Note that
v
is not subjected to the translation transformation. Therefore, Eq. (6.24) becomes
2
x
v
y
v
−
x
v
x
t
+
y
v
y
t
±
r
2
x
v
+
y
v
−
x
t
y
t
=
(6.25)
x
v
+
y
v
Let's test Eq. (6.25) with the following conditions:
If the scale transform is
⎡
⎤
⎡
⎤
1
2
000
0
1
2
00
00
2
0
0001
2000
0200
0020
0001
⎣
⎦
⎣
⎦
then
S
−
1
S
=
=
If the rotate transform is
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
10 00
0 cos 90
1
0
0
0
sin 90
0
0 sin 90
cos 90
0
00 01
−
cos 90
sin 90
0
0
then
R
−
1
R
=
=
−
sin 90
cos 90
0
0
0
0
0
1
If the translate transform is
⎡
⎤
⎡
⎤
100 0
010 0
00110
000 1
100 0
010 0
001
⎣
⎦
⎣
⎦
then
T
−
1
T
=
=
10
000 1
−
If the ray is defined as
p
=
t
+
v
where
t
=
0
and
v
=
j
+
k
