edit
Quaternion and Bivector multiplication differ, in that:
i
j
=
k
{\displaystyle \mathbf {i} \mathbf {j} =\mathbf {k} }
e
12
e
23
=
e
1
e
2
e
2
e
3
=
e
13
=
−
e
31
{\displaystyle e_{12}e_{23}=e_{1}e_{2}e_{2}e_{3}=e_{13}=-e_{31}\;}
This leads to the identification
i = -e 23 ; j = -e 31 ; k = -e 12 clean up the different letters being used. Quaternion:
v
′
→
=
q
v
→
q
−
1
{\displaystyle {\vec {v^{\prime }}}=q{\vec {v}}q^{-1}}
where
q
0
=
cos
(
θ
/
2
)
q
1
=
e
x
sin
(
θ
/
2
)
q
2
=
e
y
sin
(
θ
/
2
)
q
3
=
e
z
sin
(
θ
/
2
)
{\displaystyle {\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
†
{\displaystyle a'=RaR^{\dagger }}
R
=
exp
(
−
u
^
θ
/
2
)
=
cos
θ
/
2
−
u
^
sin
θ
/
2
{\displaystyle 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
(
−
i
v
)
=
−
i
v
′
{\displaystyle R(-iv)=-iv^{\prime }}
but this will fall out because in 3D the pseudoscalar commmutes with everything
q corresponds to the rotor:
q
=
cos
(
θ
)
−
i
u
sin
(
θ
)
{\displaystyle q=\cos(\theta )-iu\sin(\theta )\;}
or, in terms of the bivector U = i u ,
q
=
cos
(
θ
)
−
U
sin
(
θ
)
{\displaystyle q=\cos(\theta )-U\sin(\theta )\;}
