Graphics Reference
In-Depth Information
Appendix B
Composite Frame Rotation Sequences
B.1 Euler Rotations
This appendix lists the twelve combinations of creating a composite frame rotation
sequence from
R
−
1
α,x
,
R
−
1
and
R
−
1
γ,z
, which are
β,y
R
−
1
γ,x
R
−
1
β,y
R
−
1
R
−
1
γ,x
R
−
1
β,y
R
−
1
R
−
1
γ,x
R
−
1
β,z
R
−
1
R
−
1
γ,x
R
−
1
β,z
R
−
1
α,x
,
α,z
,
α,x
,
α,y
γ,y
R
−
1
γ,y
R
−
1
γ,y
R
−
1
γ,y
R
−
1
R
−
1
β,x
R
−
1
R
−
1
β,x
R
−
1
R
−
1
β,z
R
−
1
R
−
1
β,z
R
−
1
α,y
,
α,z
,
α,x
,
α,y
R
−
1
γ,z
R
−
1
β,x
R
−
1
R
−
1
γ,z
R
−
1
β,x
R
−
1
R
−
1
γ,z
R
−
1
β,y
R
−
1
R
−
1
γ,z
R
−
1
β,y
R
−
1
α,z
.
For each combination there are three Euler frame rotation matrices, the resulting
composite matrix, a matrix where the three angles equal 90°, the coordinates of the
unit cube in the rotated frame, the axis and angle of rotation and a figure illustrating
the stages of rotation. To compute the axis of rotation
α,y
,
α,z
,
α,x
,
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 inverse Euler frame rotations:
⎡
⎤
1
0
0
⎣
⎦
R
−
1
α,x
=
0
c
α
s
α
rotate the frame
α
about the
x
-axis
0
−
s
α
c
α
