Graphics Reference
In-Depth Information
Appendix A
Composite Point Rotation Sequences
A.1 Euler Rotations
In Chap. 9 we considered composite Euler rotations comprising individual rotations
about the
x
,
y
and
z
-axes such as
R
γ,x
R
β,y
R
α,z
and
R
γ,z
R
β,y
R
α,x
. However, there
is nothing preventing us from creating other combinations such as
R
γ,x
R
β,y
R
α,x
or
R
γ,z
R
β,y
R
α,z
that do not include two consecutive rotations about the same axis. In
all, there are twelve possible combinations:
R
γ,x
R
β,y
R
α,x
,
R
γ,x
R
β,y
R
α,z
,
R
γ,x
R
β,z
R
α,x
,
R
γ,x
R
β,z
R
α,y
R
γ,y
R
β,x
R
α,y
,
R
γ,y
R
β,x
R
α,z
,
R
γ,y
R
β,z
R
α,x
,
R
γ,y
R
β,z
R
α,y
R
γ,z
R
β,x
R
α,y
,
R
γ,z
R
β,x
R
α,z
,
R
γ,z
R
β,y
R
α,x
,
R
γ,z
R
β,y
R
α,z
which we now cover in detail.
For each combination there are three Euler rotation matrices, the resulting com-
posite matrix, a matrix where the three angles equal 90°, the coordinates of the
rotated unit cube, the axis and angle of rotation and a figure illustrating the stages
of rotation. To compute the axis of rotation
T
[
v
1
v
2
v
3
]
we use
v
1
=
(a
22
−
1
)(a
33
−
1
)
−
a
23
a
32
v
2
=
(a
33
−
1
)(a
11
−
1
)
−
a
31
a
13
v
3
=
(a
11
−
1
)(a
22
−
1
)
−
a
12
a
21
where
⎡
⎤
a
11
a
12
a
13
⎣
⎦
R
=
a
21
a
22
a
23
a
31
a
32
a
33
and for the angle of rotation
δ
we use
2
Tr
(
R
)
1
.
1
cos
δ
=
−
We begin by defining the three principal Euler rotations:
