Transformation - Mathematics for Computer Graphics - page 67

A common sequence for applying these rotations is roll, pitch, yaw , as seen
in the following transform:
⎡

⎤ ⎡ ⎤

x
y
z
1

x
y
z
1

⎣ ⎦ ⎣ ⎦

=[ yaw ] · [ pitch ] · [ roll ] · (7.64)

and if a translation is involved,
⎡ ⎤ ⎡ ⎤

x
y
z
1

x
y
z
1

⎣ ⎦ ⎣ ⎦

=[ translate ] · [ yaw ] · [ pitch ] · [ roll ] · (7.65)

When these rotation transforms are applied, the vertex is first rotated about
the z -axis (roll), followed by a rotation about the x -axis (pitch), followed by a
rotation about the y -axis (yaw). Euler rotations are relative to the fixed frame
of reference. This is not always easy to visualize, as one's attention is normally
with the rotating frame of reference. Let's consider a simple example where
an axial system is subjected to a pitch rotation followed by a yaw rotation
relative to fixed frame of reference.
We begin with two frames of reference XYZ and X Y Z mutually aligned.
Figure 7.10 shows the orientation of X Y Z after it is subjected to a pitch of
90 ◦ about the x -axis. Figure 7.11 shows the the final orientation after X Y Z
is subjected to a yaw of 90 ◦ about the y -axis.

7.4.4 Gimbal Lock

Let's take another example starting from the point where the two axial sys-
tems are mutually aligned. Figure 7.12 shows the orientation of X Y Z after
it is subjected to a roll of 45 ◦ about the z -axis, and Figure 7.13 shows the
orientation of X Y Z after it is subjected to a pitch of 90 ◦ about the x -axis.
Now the interesting thing about this orientation is that if we now performed

Y

pitch = 90

Y ′ X ′

Z X

Z ′

Fig. 7.10. The X Y Z axial system after a pitch of 90 ◦ .