Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors

We present the three main mathematical constructs used to represent the attitude of a rigid body in threedimensional space. These are (1) the rotation matrix, (2) a triple of Euler angles, and (3) the unit quaternion. To these we add a fourth, the rotation vector, which has many of the benefits of both Euler angles and quaternions, but neither the singularities of the former, nor the quadratic constraint of the latter. There are several other subsidiary representations, such as Cayley-Klein parameters and the axis-angle representation, whose relations to the three main representations are also described. Our exposition is catered to those who seek a thorough and unified reference on the whole subject; detailed derivations of some results are not presented. Keywords–Euler angles, quaternion, Euler-Rodrigues parameters, Cayley-Klein parameters, rotation matrix, direction cosine matrix, transformation matrix, Cardan angles, Tait-Bryan angles, nautical angles, rotation vector, orientation, attitude, roll, pitch, yaw, bank, heading, spin, nutation, precession, Slerp