ABSTRACT
This chapter examines the difficult problem of describing the orientation of an object in 3D. It discusses the closely related concepts of rotation and angular displacement and describes the subtle differences between terms like “orientation,” “direction,” and “angular displacement.” The basic idea behind Euler angles is to define an angular displacement as a sequence of three rotations about three mutually perpendicular axes. The fixed-axis system is very closely related to the Euler angle system. In an Euler angle system, the rotation occurs about the body axes, which change after each rotation. The correspondence between quaternion multiplication and 3D vector rotations is more of a theoretical interest than a practical one. The chapter also discusses three most important methods—matrices, Euler angles, and quaternions—as well as two lesser known forms—axis-angle and exponential map.
