Versor

In mathematics, a versor is a quaternion whose norm is one, also known as a unit quaternion. Each versor has the form

${\displaystyle \ u=\exp(a\mathbf {r} )=\cos a+\mathbf {r} \sin a,\qquad \mathbf {r} ^{2}=-1,\qquad a\in [0,\pi ]\ ,}$

where the condition ${\displaystyle \ \mathbf {r} ^{2}=-1\ }$ means that ${\displaystyle \ \mathbf {r} \ }$ 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 ${\displaystyle \ \mathbf {r} ~.}$ If ${\displaystyle \ a={\tfrac {\pi }{2}}\ }$ (when ${\displaystyle \ a\ }$ is a right angle), then ${\displaystyle \ u=\mathbf {r} \ ,}$ and it is called a right versor.

The mapping ${\displaystyle \ q\ \longmapsto \ u^{-1}q\ u\ }$ corresponds to 3-dimensional rotation, and has the angle ${\displaystyle \ 2\ a\ }$ about the axis ${\displaystyle \ \mathbf {r} \ }$ 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.