The Morgan Kaufmann Series in Interactive 3D Technology: Visualizing Quaternions

Chapter 29. The Relationship of 4D Rotations to Quaternions

What Happened in Three Dimensions

In three dimensions, there were many ways to deduce the quadratic mapping from quaternions to the 3 × 3 rotation matrix belonging to the group SO(3) and implementing a rotation on ordinary 3D frames. The one most directly derived from the quaternion algebra conjugates “pure” quaternion three-vectors v_i = (0, V_i) and pulls out the elements of the rotation matrix in the following way:

We easily find that the quadratic relationship between R₃(q) and q = (q₀, q₁, q₂, q₃) is

Equation 29.1. 

Quaternions and Four Dimensions

In the 4D case, which we should really regard ...