Graphics Reference
In-Depth Information
Y
X'
Y'
Z'
X'
Z
X
Fig. 7.21.
The
X
Y
Z
axial system after a
roll
of 90
◦
.
consisting of a translation back to the origin, followed by a change of axial
systems. This can be expressed as
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
·
⎡
⎣
⎤
⎦
·
⎡
⎣
⎤
⎦
x
y
z
1
r
11
r
12
r
13
0
r
21
r
22
r
23
0
r
31
r
32
r
33
0
0001
100
−
t
x
x
y
z
1
010
−
t
y
=
(7.77)
001
−
t
z
000
1
As an example, consider the situation shown in Figure 7.22. The values of
(
t
x
,t
y
,t
z
) are (10, 1, 1), and the direction cosines are as shown in the following
matrix operation:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
·
⎡
⎣
⎤
⎦
·
⎡
⎣
⎤
⎦
x
y
z
1
−
10 00
01 00
00
100
−
10
x
y
z
1
010
−
1
=
10
00 01
−
001
−
1
000
1
which concatenates to
⎡
⎤
⎡
⎤
⎡
⎤
x
y
z
1
−
10 0
0
x
y
z
1
⎣
⎦
⎣
⎦
·
⎣
⎦
01 0
−
1
=
00
−
1
1
00 0
1
Substituting (0, 0, 0) for (
x
,
y
,
z
) in the above transform produces (10
,
1
,
1)
for (
x
,y
,z
), which can be confirmed from Figure 7.22. Similarly, substituting
(0, 1, 1) for (
x
,
y
,
z
) produces (10,0,0) for (
x
,y
,z
), which is also correct.
−
7.6.3 Euler Angles
Another approach for locating the virtual camera involves Euler angles, but
we must remember that they suffer from gimbal lock (see page 70). However, if
