Graphics Reference
In-Depth Information
⎡
⎤
cos
θ
cos
β
−
sin
θ
sin
β
−
cos
θ
sin
β
−
sin
θ
cos
β
0
⎣
⎦
=
sin
θ
cos
β
+
cos
θ
sin
β
−
sin
θ
sin
β
+
cos
θ
cos
β
0
0
0
1
⎡
⎤
cos
(θ
+
β)
−
sin
(θ
+
β)
0
⎣
⎦
=
sin
(θ
+
β)
cos
(θ
+
β)
0
0
0
1
which confirms that the composite rotation is equivalent to a single rotation of
θ
+
β
.
7.4 Inverse Transforms
Given a transform
A
, its inverse
A
−
1
is defined as such that
AA
−
1
A
−
1
A
I
where
I
is the identity matrix. So let's identify the inverse translation and rotation
transforms.
We know that the translation matrix is given by
=
=
⎡
⎤
10
t
x
01
t
y
00 1
⎣
⎦
T
t
x
,t
y
=
and we could reason that the inverse of
T
t
x
,t
y
must be a translation in the opposite
direction:
⎡
⎤
10
−
t
x
T
−
1
⎣
⎦
.
t
x
,t
y
=
01
−
t
y
00
1
We can confirm this by computing
T
−
1
t
x
,t
y
from the cofactor matrix of
T
t
x
,t
y
, trans-
posing it and dividing by its determinant:
⎡
⎤
1
0
0
⎣
⎦
cofactor matrix of
T
t
x
,t
y
=
0
1
0
−
t
x
−
t
y
1
⎡
⎤
10
−
t
x
T
t
x
,t
y
=
⎣
⎦
01
−
t
y
00
1
and as det
T
t
x
,t
y
=
1, we can write
⎡
⎤
10
−
t
x
T
−
1
⎣
⎦
.
t
x
,t
y
=
01
−
t
y
00
1
So our reasoning is correct. Furthermore,
T
t
x
,t
y
T
−
1
t
x
,t
y
=
I
:
⎡
⎤
⎡
⎤
⎡
⎤
10
t
x
01
t
y
00 1
10
−
t
x
100
010
001
⎣
⎦
⎣
⎦
=
⎣
⎦
.
T
t
x
,t
y
T
−
1
t
x
,t
y
=
01
−
t
y
00
1
