Graphics Reference
In-Depth Information
Fig. 9.2
Rotating a point
about an axis parallel with the
z
-axis
where
T
creates a temporary origin
−
t
x
,
−
t
y
,
0
R
β,z
rotates
β
about the temporary
z
-axis
T
t
x
,t
y
,
0
returns to the original position
and the matrix transform is
⎡
⎣
⎤
⎦
−
−
+
cos
β
sin
β
0
t
x
(
1
cos
β)
t
y
sin
β
−
−
sin
β
cos
β
0
t
y
(
1
cos
β)
t
x
sin
β
T
t
x
,t
y
,
0
R
β,z
T
−
t
x
,
−
t
y
,
0
=
.
0
0
1
0
0
0
0
1
I hope you can see the similarity between rotating in 3D and 2D - the
x
- and
y
-coordinates are updated while the
z
-coordinate is held constant. We can now state
the other two matrices for rotating about an off-set axis parallel with the
x
-axis and
parallel with the
y
-axis:
⎡
⎣
⎤
⎦
1
0
0
0
0
cos
β
−
sin
β
y
(
1
−
cos
β)
+
t
z
sin
β
T
0
,t
y
,t
z
R
β,x
T
0
,
−
t
y
,
−
t
z
=
0 in
β
cos
β
z
(
1
−
cos
β)
−
t
y
sin
β
0
0
0
1
⎡
⎤
cos
β
0 in
β
x
(
1
−
cos
β)
−
t
z
sin
β
⎣
⎦
01
0
0
T
t
x
,
0
,t
z
R
β,y
T
t
z
=
.
−
t
x
,
0
,
−
−
sin
β
0
cos
β
z
(
1
−
cos
β)
+
t
x
sin
β
00
0
1
9.3 Composite Rotations
So far we have only considered single rotations about a Cartesian axis or a parallel
off-set axis, but there is nothing to stop us constructing a sequence of rotations
to create a composite rotation. For example, we could begin by rotating a point
α
about the
x
-axis followed by a rotation
β
about the
y
-axis, which in turn could be
