Rotation in Three Dimensions - 3D Math Primer for Graphics and Game Development - page 236

system in which the first and last rotations are performed around the same
axis. These methods are more convenient in certain situations, such as
describing the motion of a top, where the three angles correspond to pre-
cession, nutation, and spin. You may encounter some purists who object
to the name “Euler angles” being attached to an asymmetric system, but
this usage is widespread in many fields, so rest assured that you outnum-
ber them. To distinguish between the two systems, the symmetric Euler
angles are sometimes called “proper” Euler angles, with the more com-
mon conventions being called Tait-Bryan angles, first documented by the
aerospace forefathers we mentioned [10]. O'Reilly [51] discusses proper Eu-
ler angles, even more methods of describing rotation, such as the Rodrigues
vector, Cayley-Klein parameters, and interesting historical remarks. James
Diebel's summary [13] compares different Euler angle conventions and the
other major methods for describing rotation, much as this chapter does,
but assumes a higher level of mathematical sophistication.
If you have to deal with Euler angles that use a different convention
from the one you prefer, we offer two pieces of advice:

• First, make sure you understand exactly how the other Euler angle

system works. Little details such as the definition of positive rotation
and order of rotations make a big difference.

• Second, the easiest way to convert the Euler angles to your format

is to convert them to matrix form and then convert the matrix back
to your style of Euler angles. We will learn how to perform these
conversions in Section 8.7. Fiddling with the angles directly is much
more di cult than it would seem. See [63] for more information.

8.3.3 Advantages of Euler Angles

Euler angles parameterize orientation using only three numbers, and these
numbers are angles. These two characteristics of Euler angles provide cer-
tain advantages over other forms of representing orientation.

• Euler angles are easy for humans to use —considerably easier than

matrices or quaternions. Perhaps this is because the numbers in an
Euler angle triple are angles, which is naturally how people think
about orientation. If the conventions most appropriate for the sit-
uation are chosen, then the most practical angles can be expressed
directly. For example, the angle of declination is expressed directly
by the heading-pitch-bank system. This ease of use is a serious ad-
vantage. When an orientation needs to be displayed numerically or
entered at the keyboard, Euler angles are really the only choice.