Graphics Reference
In-Depth Information
Using the trace operation, we can write
Tr ( R 90 ° ,x R 90 ° ,y R 90 ° ,z )
=−
1
therefore,
β
=
arccos ((
1
1 )/ 2 )
=
180° .
As promised, let's explore another way of identifying the fixed axis of rotation,
which is an eigenvector. Consider the following argument where A is a simple rota-
tion transform:
If v is a fixed axis of rotation and A a rotation transform, then v suffers no rota-
tion:
Av
=
v
(9.9)
similarly,
A T v
=
v .
(9.10)
Subtracting ( 9.10 ) from ( 9.9 ), we have
A T v
Av
=
0
(9.11)
A
A T v
=
0
(9.12)
where 0 is a null vector.
In Chap. 4 we defined an antisymmetric matrix Q as
2 A
A T
1
Q
=
(9.13)
therefore,
A
A T =
2 Q .
(9.14)
Substituting ( 9.14 )in( 9.12 )wehave
2 Qv
=
0
Qv
=
0
which permits us to write
0
q 3
q 2
v 1
v 2
v 3
0
0
0
=
q 3
0
q 1
(9.15)
q 2
q 1
0
where
q 1 =
a 32
q 2 = a 31 a 13
q 3 =
a 23
a 12
a 21 .
Expanding ( 9.15 )wehave
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