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

Appendix B. 2D Complex Frames

The basic 2D rotation matrix changing the axes by an angle θ and its interpolation in terms of a local frame on a curve, with tangent vector T and normal vector N, may be defined as

Equation B.1. 

A double-valued quaternion-like parameterization of the 2D frame may be written in the form

Equation B.2. 

We can easily verify that if (a, b) is a point on S¹ embedded in (i.e., a² + b² = 1) this is an orthonormal parameterization ...