Graphics Reference
In-Depth Information
10.4 Composite Rotations
In Chap. 9 we went into some detail describing how point rotation transforms can be
combined into composite rotations about the three Cartesian axes. There are twelve
possible combinations that are listed in Appendix A. As any rotation transform can
be used to rotate a point in one direction, or a frame of reference in the opposite
direction, the previously computed composite transforms for rotating points, can be
used for rotating frames in the opposite direction.
For example, we previously computed
R
γ,z
R
β,y
R
α,x
:
⎡
⎤
c
γ
c
β
c
γ
s
β
s
α
−
s
γ
c
α
c
γ
s
β
c
α
+
s
γ
s
α
⎣
⎦
R
γ,z
R
β,y
R
α,x
=
s
γ
c
β
s
γ
s
β
s
α
+
c
γ
c
α
s
γ
s
β
c
α
−
c
γ
s
α
−
s
β
c
β
s
α
c
β
c
α
which rotates a point about a fixed frame of reference. But it can also be used to
rotate a frame of reference in the opposite directions:
⎡
⎤
c
γ
c
β
c
γ
s
β
s
α
−
s
γ
c
α
c
γ
s
β
c
α
+
s
γ
s
α
R
−
1
−
γ,z
R
−
1
β,y
R
−
1
⎣
⎦
.
α,x
=
s
γ
c
β
s
γ
s
β
s
α
+
c
γ
c
α
s
γ
s
β
c
α
−
c
γ
s
α
−
−
−
s
β
c
β
s
α
c
β
c
α
γ,z
R
−
1
In order to compute
R
−
1
β,y
R
−
1
α,x
we only have to reverse the sign of the sine terms
in the transform for
R
γ,z
R
β,y
R
α,x
:
⎡
⎤
c
γ
c
β
c
γ
s
β
s
α
+
s
γ
c
α
−
c
γ
s
β
c
α
+
s
γ
s
α
γ,z
R
−
1
R
−
1
β,y
R
−
1
⎣
⎦
.
(10.1)
−
−
s
γ
s
β
s
α
+
s
γ
s
β
c
α
+
α,x
=
s
γ
c
β
c
γ
c
α
c
γ
s
α
−
s
β
c
β
s
α
c
β
c
α
Let's test (
10.1
) by making
α
=
β
=
γ
=
90°:
⎡
⎤
001
0
R
−
1
90
°
,z
R
−
1
90
°
,y
R
−
1
⎣
⎦
.
90
°
,x
=
10
100
−
Figure
10.5
(a) shows the initial scenario, Fig.
10.5
(b) shows the frame rotated 90°
about the local
x
-axis, Fig.
10.5
(c) shows the frame rotated 90° about the local
y
-axis, and Fig.
10.5
(d) shows the frame rotated 90° about the local
z
-axis. If we
subject the coordinates of the unit cube to this composite transform we have
⎡
⎤
⎡
⎤
001
0
00001111
00110011
01010101
⎣
⎦
⎣
⎦
10
100
−
⎡
⎤
01 0
1 01 0
1
⎣
⎦
=
00
−
1
−
100
−
1
−
1
00 0
0 11 1
1
which are confirmed by Fig.
10.5
(d).
