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.
Presentation on 3- and 2-spheres
Sir William Rowan Hamilton wrote the versor of a quaternion q {\displaystyle \ q\ } as the symbol U q . {\displaystyle \ \mathbf {U} \ q~.} He was then able to display the general quaternion in polar coordinate form
q = T q U q , {\displaystyle \ q\ =\ \mathbf {T} \ \!q~~\mathbf {U} \ \!q\ ,}
where T q {\displaystyle \ \mathbf {T} \ \!q\ } is Hamilton's notation for the norm of q {\displaystyle \ q\ } (in modern notation T q = ‖ q ‖ {\displaystyle \ \mathbf {T} \ \!q=\|q\|} ). The norm of a versor is always equal to one; hence they occupy the unit 3-sphere in H {\displaystyle \ \mathbb {H} \ } . Examples of versors include the eight elements of the quaternion group. Of particular importance are the right versors, which have angle π 2 . {\displaystyle \ {\tfrac {\pi }{2}}~.} These versors have zero scalar part, and so are vectors of length one (unit vectors). The right versors form a sphere of square roots of −1 in the quaternion algebra. The generators i, j, and k are examples of right versors, as well as their additive inverses. Other versors include the twenty-four Hurwitz quaternions that have the norm 1 and form vertices of a 24-cell polychoron. Hamilton defined a quaternion as the quotient of two vectors. A versor can be defined as the quotient of two unit vectors. For any fixed plane Π the quotient of two unit vectors lying in Π depends only on the angle (directed) between them, the same a as in the unit vector–angle representation of a versor explained above. That's why it may be natural to understand corresponding versors as directed arcs that connect pairs of unit vectors and lie on a great circle formed by intersection of Π with the unit sphere, where the plane Π passes through the origin. Arcs of the same direction and length (or, the same, subtended angle in radians) are equipollent and correspond to the same versor. Such an arc, although lying in the three-dimensional space, does not represent a path of a point rotating as described with the sandwiched product with the versor. Indeed, it represents the left multiplication action of the versor on quaternions that preserves the plane Π and the corresponding great circle of 3-vectors. The 3-dimensional rotation defined by the versor has the angle two times the arc's subtended angle, and preserves the same plane. It is a rotation about the corresponding vector r, that is perpendicular to Π. On three unit vectors, Hamilton writes
q = β : α = O B : O A {\displaystyle q=\beta :\alpha =\mathrm {OB:OA} \qquad } and
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