User:Jheald/sandbox/GA/Quaternions to Bivectors

Quaternions <--> Bivectors
Quaternion and Bivector multiplication differ, in that:
 * $$ \mathbf{i} \mathbf{j} = \mathbf {k} $$ (Quaternion)
 * $$ e_{12} e_{23} = e_1 e_2 e_2 e_3 = e_{13} = - e_{31} \;$$ (Bivector)

This leads to the identification
 * i = -e23; j = -e31; k = -e12

Rotations

 * clean up the different letters being used.

Quaternion:
 * $$\vec{v^\prime} = q \vec{v} q^{-1}$$

where

\begin{array}{lcl} q_0 &=& \cos(\theta/2)\\ q_1 &=& e_x\sin(\theta/2)\\ q_2 &=& e_y\sin(\theta/2)\\ q_3 &=& e_z\sin(\theta/2) \end{array} $$

Bivector:
 * $$a' = R a R^\dagger$$


 * $$R = \exp(- \hat u\theta/2)= \cos \theta/2 - \hat u \sin \theta/2$$

where u is a unit bivector, u = i v


 * Translation

In the quaternion calculation v is actually being stored in the non-scalar part of the quaternion; in GA therefore this is an equation that is mapping for
 * $$ R(-iv) = -iv^{\prime}$$

but this will fall out because in 3D the pseudoscalar commmutes with everything

q corresponds to the rotor:
 * $$ q = \cos(\theta) - iu \sin(\theta) \; $$

or, in terms of the bivector U = i u,
 * $$ q = \cos(\theta) - U \sin(\theta) \; $$