versor
In mathematics, a versor is a quaternion of norm one, also known as a unit quaternion. Each versor has the form
u
=
exp
(
a
r
)
=
cos
a
+
r
sin
a
,
r
2
=
−
1
,
a
∈
[
0
,
π
]
,
{\displaystyle u=\exp(a\mathbf {r} )=\cos a+\mathbf {r} \sin a,\quad \mathbf {r} ^{2}=-1,\quad a\in [0,\pi ],}
where the r2 = −1 condition means that r is an imaginary unit. There is a sphere of imaginary units in the quaternions. Note that the expression for a versor is just Euler's formula for the imaginary unit r. In case a = π/2 (a right angle), then
u
=
r
{\displaystyle u=\mathbf {r} }
, and it is called a right versor.
The mapping
q
→
u
−
1
q
u
{\displaystyle q\to u^{-1}qu}
corresponds to 3-dimensional rotation, and has the angle 2a about the axis r in axis–angle representation.
The collection of versors, with quaternion multiplication, forms a group, and appears as a 3-sphere in the 4-dimensional quaternion algebra.
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