versor
In mathematics, a versor is a quaternion whose norm is 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,\qquad \mathbf {r} ^{2}=-1,\qquad a\in [0,\pi ]\ ,}
where the condition
r
2
=
−
1
{\displaystyle \ \mathbf {r} ^{2}=-1\ }
means that
r
{\displaystyle \ \mathbf {r} \ }
is an algebraic 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
.
{\displaystyle \ \mathbf {r} ~.}
If
a
=
π
2
{\displaystyle \ a={\tfrac {\pi }{2}}\ }
(when
a
{\displaystyle \ a\ }
is 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\ \longmapsto \ u^{-1}q\ u\ }
corresponds to 3-dimensional rotation, and has the angle
2
a
{\displaystyle \ 2\ a\ }
about the axis
r
{\displaystyle \ \mathbf {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.
blurred voices
- 2019-03-25T00:00:00.000000Z
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