I introduce an important way to think about and construct a DCM: by implementing a yaw–pitch–roll sequence of rotations on a model aircraft. This does away with the widespread but rather involved method of describing the relative orientation of two axis sets by drawing them with a common origin. For this, we must distinguish the idea of a rotation in a sequence being about either a ‘space-fixed’ axis or a ‘carried-along’ axis. Users of these terms tend to fall into two groups, ‘active’ and ‘passive’. I state the ‘fundamental theorem of rotation sequences’, which does away with any need for the reader to stand in one group or the other. I also discuss the extraction of Euler angles from a DCM, and examine infinitesimal rotations. I discuss two methods of interpolating from an initial to a final orientation; one of these is used widely in computer graphics, but both methods must be discussed for the computer-graphics method to be understood. I end with a calculation of the position and attitude of a robot arm.

I introduce quaternions by recounting the story of how Hamilton discovered them, but in far more detail than other authors give. This detail is necessary for the reader to understand why Hamilton wrote his quaternion equations in the way that he did. I describe the role of quaternions in rotation, show how to convert between them and matrices, and discuss their role in modern computer graphics. I describe a modern problem in detail whereby Hamilton’s original definition has been ‘hijacked’ in a way that has now produced much confusion. I end by describing how quaternions play a role in topology and quantum mechanics.