Transformation - Mathematics for Computer Graphics - page 78

The same result is obtained simply by substituting - roll ,- pitch ,- yaw in the
original matrices. As described above, the virtual camera will normally be
translated from the origin by ( t x ,t y ,t z ), which implies that the transform
from the world space to the camera space must be evaluated as follows:
⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

x
y
z
1

x
y
z
1

=[ − roll ] · [ − pitch ] · [ − yaw ] · [ − translate ( − t x , − t y , − t z ) · (7.81)

which can be represented by a single homogeneous matrix:
⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦ ·

⎡
⎣

⎤
⎦

x
y
z
1

T 11 T 12 T 13 T 14 x

y
z
1

T 21 T 22 T 23 T 24

= (7.82)

T 31 T 32 T 33 T 34

T 41 T 42 T 43 T 44

where

T 11 =cos( yaw )cos( roll ) + sin( yaw ) sin( pitch ) sin( roll )
T 12 =cos( pitch ) sin( roll )
T 13 = sin( yaw )cos( roll )+cos( yaw ) sin( pitch ) sin( roll )

T 14 = − ( t x T 11 + t y T 12 + t z T 13 )
T 21 =

−

cos( yaw ) sin( roll ) + sin( yaw ) sin( pitch )cos( roll )
T 22 =cos( pitch )cos( roll )
T 23 =sin( yaw ) sin( roll )+cos( yaw ) sin( pitch )cos( roll )
T 24 =

−

( t x T 21 + t y T 22 + t z T 23 )
T 31 =sin( yaw )cos( pitch )
T 32 =

−

sin( pitch )
T 33 =cos( yaw )cos( pitch )
T 34 =

−

− ( t x T 31 + t y T 32 + t z T 33 )

T 41 =0
T 42 =0
T 43 =0
T 44 = 1 (7.83)

This, too, can be verified by a simple example. For instance, consider the
situation shown in Figure 7.22 where the following conditions prevail:

roll =0 ◦
pitch =0 ◦
yaw = 180 ◦
t x =10