In mathematics, a versor is a quaternion of norm one (a unit quaternion). Each versor has the form
where the r2 = −1 condition means that r is a unit-length vector quaternion (or that the first component of r is zero, and the last three components of r are a unit vector in 3 dimensions). The corresponding 3-dimensional rotation has the angle 2a about the axis r in axis–angle representation. In case (a right angle), then , and the resulting unit vector is termed a right versor.
The collection of versors with quaternion multiplication forms a group, and the set of versors is a 3-sphere in the 4-dimensional quaternion algebra.
Presentation on 3- and 2-spheres
Hamilton denoted the versor of a quaternion q by the symbol Uq. He was then able to display the general quaternion in polar coordinate form q = Tq Uq, where Tq is the norm of q. The norm of a versor is always equal to one; hence they occupy the unit 3-sphere in H. Examples of versors include the eight elements of the quaternion group. Of particular importance are the right versors, which have angle π/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, its subtended angle in radians) are equivalent, i.e. define 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
and
imply
Multiplication of quaternions of norm one corresponds to the (non-commutative) "addition" of great circle arcs on the unit sphere. Any pair of great circles either is the same circle or has two intersection points. Hence, one can always move the point B and the corresponding vector to one of these points such that the beginning of the second arc will be the same as the end of the first arc.
An equation
implicitly specifies the unit vector–angle representation for the product of two versors. Its solution is an instance of the general Campbell–Baker–Hausdorff formula in Lie group theory. As the 3-sphere represented by versors in is a 3-parameter Lie group, practice with versor compositions is a step into Lie theory. Evidently versors are the image of the exponential map applied to a ball of radius π in the quaternion subspace of vectors.
Versors compose as aforementioned vector arcs, and Hamilton referred to this group operation as "the sum of arcs", but as quaternions they simply multiply.
The geometry of elliptic space has been described as the space of versors.
Representation of SO(3) The orthogonal group in three dimensions, rotation group SO(3), is frequently interpreted with versors via the inner automorphism where u is a versor. Indeed, if
and vector s is perpendicular to r,
then
by calculation. The plane is isomorphic to and the inner automorphism, by commutativity, reduces to the identity mapping there. Since quaternions can be interpreted as an algebra of two complex dimensions, the rotation action can also be viewed through the special unitary group SU(2).