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

Appendix C. 3D Quaternion Frames

We next outline the basic features for 3D orientation and quaternion frames, following the pattern now established in Appendix B for 2D orientation and complex numbers. A quaternion frame is a unit four-vector q = (q₀, q₁, q₂, q₃) = (q₀, q) with the following features.

Unit Norm

If we define the inner product of two quaternions as

Equation C.1. 

the components of a quaternion frame obey the constraint

Equation C.2. 

and therefore lie on S³, the three-sphere embedded in 4D Euclidean space .

Multiplication Rule

The quaternion ...