Physics for Game Developers, 2nd Edition

Chapter 11. Rotation in 3D Rigid-Body Simulators

A fundamental difference between particles and rigid bodies is that we cannot ignore rotation of rigid bodies. This applies to both 2D and 3D rigid bodies. In two dimensions, it’s quite easy to express the orientation of a rigid body; you need only a single scalar to represent the body’s rotation about a single axis. In three dimensions, however, there are three primary coordinate axes about each of which a rigid body may rotate. Moreover, a rigid body in three dimensions may rotate about any arbitrary axis, not necessarily one of the coordinate axes.

In two dimensions, we say that a rigid body has only one rotational degree of freedom, whereas in three dimensions we say that a rigid body has three rotational degrees of freedom. This may lead you to infer that in three dimensions, you must have three scalar quantities to represent a body’s rotation. Indeed, this is a minimum requirement, and you’re probably already familiar with a set of angles that represent the orientation of a rigid body in 3D—namely, the three Euler angles (roll, pitch, and yaw) that we’ll talk about in Chapter 15.

These three angles—roll, pitch, and yaw—are very intuitive and easy for us to visualize. For example, in an airplane the nose pitches up or down, the plane rolls (or banks) left or right, and the yaw (or heading) changes to the left or right. Unfortunately, there’s a problem with using these three Euler angles in rigid-body simulations. The problem is ...