In mathematics, a versor is a quaternion of norm one (a unit quaternion). Each versor has the form
q=\exp(ar)=\cosa+r\sina, r2=-1, a\in[0,\pi],
q=r
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.
Hamilton denoted the versor of a quaternion q by the symbol U q. He was then able to display the general quaternion in polar coordinate form
q = T q U q,where T q is the norm of q. The norm of a versor is always equal to one; hence they occupy the unit 3-sphere in
H
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 the quotient of two unit vectors lying in depends only on the angle (directed) between them, the same 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.[1]
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, that is perpendicular to .
On three unit vectors, Hamilton writes
q=\beta:\alpha=OB:OA
q'=\gamma:\beta=OC:OB
q'q=\gamma:\alpha=OC:OA~.
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 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
\exp(c r) \exp(a s)=\exp(b t)
H
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.[2]
q\mapstou-1q u
u=\exp(a r)
then
u-1s u=s \cos(2a)+s r \sin(2a)
\{ x+y r : (x,y)\inR2 \}\subH
C
For a fixed, versors of the form
\exp(a r)
a\in\left(-\pi,\pi \right] ,
Versors have been used to represent rotations of the Bloch sphere with quaternion multiplication.[5]
The facility of versors illustrate elliptic geometry, in particular elliptic space, a three-dimensional realm of rotations. The versors are the points of this elliptic space, though they refer to rotations in 4-dimensional Euclidean space. Given two fixed versors and, the mapping
q\mapstou q v
\{ u \exp(a r) : 0\lea<\pi \}~.
R3~.
The set of all versors, with their multiplication as quaternions, forms a continuous group G. For a fixed pair
\{-r,+r \}
G1=\{ \exp(a r) : a\inR \}
Next consider the finite subgroups, beyond the quaternion group Q8:[6] [7]
As noted by Hurwitz, the 16 quaternions
\tfrac{ 1 }{2}\left(\pm1\pmi\pmj\pmk\right)
By a process of bitruncation of the 24-cell, the 48-cell on G is obtained, and these versors multiply as the binary octahedral group.
Another subgroup is formed by 120 icosians which multiply in the manner of the binary icosahedral group.
A hyperbolic versor is a generalization of quaternionic versors to indefinite orthogonal groups, such as Lorentz group.It is defined as a quantity of the form
\exp(ar)=\cosha+r\sinha~~
~~r2=+1~.
This versor was used by Homersham Cox (1882/1883) in relation to quaternion multiplication.[8] [9] The primary exponent of hyperbolic versors was Alexander Macfarlane, as he worked to shape quaternion theory to serve physical science.[10] He saw the modelling power of hyperbolic versors operating on the split-complex number plane, and in 1891 he introduced hyperbolic quaternions to extend the concept to 4-space. Problems in that algebra led to use of biquaternions after 1900. In a widely seen review, Macfarlane wrote:
... the root of a quadratic equation may be versor in nature or scalar in nature. If it is versor in nature, then the part affected by the radical involves the axis perpendicular to the plane of reference, and this is so, whether the radical involves the square root of minus one or not. In the former case the versor is circular, in the latter hyperbolic.[11]
Today the concept of a one-parameter group subsumes the concepts of versor and hyperbolic versor as the terminology of Sophus Lie has replaced that of Hamilton and Macfarlane.In particular, for each such that or, the mapping
a\mapsto\exp(ar)
defined the parameter rapidity, which specifies a change in frame of reference. This rapidity parameter corresponds to the real variable in a one-parameter group of hyperbolic versors. With the further development of special relativity the action of a hyperbolic versor came to be called a Lorentz boost.[12]
See main article: Lie theory. Sophus Lie was less than a year old when Hamilton first described quaternions, but Lie's name has become associated with all groups generated by exponentiation. The set of versors with their multiplication has been denoted Sl(1,q) by .[13] Sl(1,q) is the special linear group of one dimension over quaternions, the "special" indicating that all elements are of norm one. The group is isomorphic to SU(2,c), a special unitary group, a frequently used designation since quaternions and versors are sometimes considered archaic for group theory. The special orthogonal group SO(3,r) of rotations in three dimensions is closely related: it is a 2:1 homomorphic image of SU(2,c).
The subspace
\{ xi+yj+zk : x,y,z\inR \}\subsetH
[u,v]=uv-vu ,
Lie groups that contain hyperbolic versors include the group on the unit hyperbola and the special unitary group SU(1,1).
The word is derived from Latin versari = "to turn" with the suffix -or forming a noun from the verb (i.e. versor = "the turner"). It was introduced by William Rowan Hamilton in the 1840s in the context of his quaternion theory.
The term "versor" is generalised in geometric algebra to indicate a member
R
R=v1v2 … vk
Just as a quaternion versor
u
q
q\mapstou-1qu
R
k
A
A\mapsto(-1)kRAR-1
A rotation can be considered the result of two reflections, so it turns out a quaternion versor
u
R=v1v2
l{G}(3,0)
In a departure from Hamilton's definition, multivector versors are not required to have unit norm, just to be invertible. Normalisation can still be useful however, so it is convenient to designate versors as unit versors in a geometric algebra if
R\tilde{R}=\pm1
Book: Hardy, A.S. . Arthur Sherburne Hardy . 1887 . Elements of Quaternions . 112–118 . Applications to spherical trigonometry .
Section IV: Versors and unitary vectors in the system of quaternions.
Section V: Versor and unitary vectors in vector algebra.