Graphics Reference
In-Depth Information
Fig. 10.4
(
a
)and(
b
) The off-set axis and rotated frame. (
c
)and(
d
) The unit cube and rotated
frame
⎡
⎤
cos
α
sin
α
0
R
−
1
⎣
⎦
.
α,z
=
−
sin
α
cos
α
0
0
0
1
10.3.3 Rotated Frames About Off-Set Axes
In Chap. 9 we developed three transforms for rotating a point about an off-set axis
parallel with one of the three Cartesian axes. Let's develop three complementary
transforms for rotating a frame about the same off-set axes.
To ensure that we compute the correct transform, we must be very clear in our
own minds what we are attempting to do. The objective is to identify an off-set axis
parallel with the
z
-axis, for example, in the current
XYZ
frame of reference, about
which a frame is rotated. The first step, then, is to translate the frame, and then
rotate it.
Let's assume that the axis passes through the point
(t
x
,t
y
,
0
)
,asshownin
Fig.
10.4
(a). Therefore, given the following definitions for
T
−
1
t
x
,t
y
,
0
and
R
−
1
α,z
⎡
⎣
⎤
⎦
−
100
t
x
010
t
y
001 0
000 1
−
T
−
1
t
x
,t
y
,
0
=
