Game Development Reference
Local Coordinate Axes
Earlier, we defined the Cartesian coordinate system to use for your fixed global refer‐
ence, or world coordinates. This world coordinate system is all that's required when
treating particles; however, for rigid bodies you'll also use a set of local coordinates fixed
to the body. Specifically, this local coordinate system will be fixed at the body's center-
of-mass location. You'll use this coordinate system to track the orientation of the body
as it rotates.
For plane motion, we require only one scalar quantity to describe the body's orientation.
This is illustrated in Figure 2-8 .
Figure 2-8. Local coordinate axes
Here the orientation, Ω, is defined as the angular difference between the two sets of
coordinate axes: the fixed world axes and the local body axes. This is the so-called Euler
angle. In general 3D motion there is a total of three Euler angles, which are usually called
yaw , pitch , and roll in aerodynamic and hydrodynamic jargon. While these angular
representations are easy to visualize in terms of their physical meaning, they aren't so
nice from a numerical point of view, so you'll have to look for alternative representations
when writing your 3D real-time simulator. These issues are addressed in Chapter 9 .
Angular Velocity and Acceleration
In two-dimensional plane motion, as the body rotates, Ω will change, and the rate at
which it changes is the angular velocity, ω. Likewise, the rate at which ω changes is the
angular acceleration, α. These angular properties are analogous to the linear properties
of displacement, velocity, and acceleration. The units for angular displacement, velocity,