Game Development Reference
In-Depth Information
The numbers in a matrix aren't intuitive for humans to work with.
Not all matrices are valid for describing an orientation. Some matrices
contain mirroring or skew. We can end up with a ill-formed matrix
either by getting bad data from an external source or through matrix
creep.
8.3
Euler Angles
Another common method of representing orientation is known as Euler
angles. (Remember, Euler is pronounced “oiler,” not “yoolur.”) The tech-
nique is named after the famous mathematician who developed them, Leon-
hard Euler (1707-1783). Section 8.3.1 describes how Euler angles work and
discusses the most common conventions used for Euler angles. Section 8.3.2
discusses other conventions for Euler angles, including the important fixed
axis system. We consider the advantages and disadvantages of Euler an-
gles in Section 8.3.3 and Section 8.3.4. Section 8.3.5 summarizes the most
important concepts concerning of Euler angles.
This section utilizes many ideas, terms, and conventions from Section 7.3.2
concerning spherical coordinates.
8.3.1
What Are Euler Angles?
The basic idea behind Euler angles is to define an angular displacement
as a sequence of three rotations about three mutually perpendicular axes.
This sounds complicated, but actually it is quite intuitive. (In fact, its ease
of use by humans is one of its primary advantages.)
So Euler angles describe orientation as three rotations about three mu-
tually perpendicular axes. But which axes? And in what order? As it turns
out, any three axes in any order will work, but most people have found it
practical to use the cardinal axes in a particular order. The most common
convention, and the one we use in this topic, is the so-called “heading-pitch-
bank” convention for Euler angles. In this system, an orientation is defined
by a heading angle, a pitch angle, and a bank angle.
 
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