Versor

This article is about the quaternions of norm one. For other uses, seeVersor (disambiguation).

Inmathematics,aversoris aquaternionofnormone (aunitquaternion). Each versor has the form

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

where ther²= −1 condition means thatris a unit-length vector quaternion (or that the first component ofris zero, and the last three components ofrare aunit vectorin 3 dimensions). The corresponding3-dimensionalrotation has the angle 2aabout the axisrinaxis–angle representation.In casea= π/2(aright angle), then$q=\mathbf {r}$,and the resulting unit vector is termed aright versor.

The collection of versors with quaternion multiplication forms agroup,and the set of versors is a3-spherein the 4-dimensional quaternion algebra.

Presentation on 3- and 2-spheres

arcAB+ arcBC= arcAC

Hamilton denoted theversorof a quaternionqby the symbolUq.He was then able to display the general quaternion inpolar coordinate form

q=TqUq,

whereTqis the norm ofq.The norm of a versor is always equal to one; hence they occupy the unit3-sphereinH.Examples of versors include the eight elements of thequaternion group.Of particular importance are theright versors,which haveangle π/2.These versors have zero scalar part, and so arevectorsof length one (unit vectors). The right versors form asphere of square roots of −1in the quaternion algebra. The generatorsi,j,andkare examples of right versors, as well as theiradditive inverses.Other versors include the twenty-fourHurwitz quaternionsthat have the norm 1 and form vertices of a24-cellpolychoron.

Hamilton defined a quaternion as the quotient of two vectors. A versor can be defined as the quotient of two unit vectors. For any fixedplaneΠ the quotient of two unit vectors lying in Π depends only on theangle(directed) between them, the sameaas in the unit vector–angle representation of a versor explained above. That's why it may be natural to understand corresponding versors as directedarcsthat connect pairs of unit vectors and lie on agreat circleformed by intersection of Π with theunit sphere,where the plane Π passes through the origin. Arcs of the same direction and length (or, the same,subtended angleinradians) areequipollentand correspond to the same versor.

Such an arc, although lying in thethree-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 vectorr,that isperpendicularto Π.

On three unit vectors, Hamilton writes

$q=\beta:\ Alpha =OB:OA\$and

$q'=\gamma:\beta =OC:OB$

imply

$q'q=\gamma:\ Alpha =OC:OA.$

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 twointersection points.Hence, one can always move the pointBand 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

$\exp(c\mathbf {r} )\exp(a\mathbf {s} )=\exp(b\mathbf {t} )\!$

implicitly specifies the unit vector–angle representation for the product of two versors. Its solution is an instance of the generalCampbell–Baker–Hausdorff formulainLie grouptheory. As the 3-sphere represented by versors in$\mathbb {H}$is a 3-parameter Lie group, practice with versor compositions is a step intoLie theory.Evidently versors are the image of theexponential mapapplied to a ball of radius π in the quaternion subspace of vectors.

Versors compose as aforementioned vector arcs, and Hamilton referred to thisgroup operationas "the sum of arcs", but as quaternions they simply multiply.

The geometry ofelliptic spacehas been described as the space of versors.

Representation of SO(3)

Theorthogonal groupin three dimensions,rotation group SO(3),is frequently interpreted with versors via theinner automorphism$q\mapsto u^{-1}qu$whereuis a versor. Indeed, if

$u=\exp(ar)$and vectorsis perpendicular tor,

then

$u^{-1}su=s\cos 2a+sr\sin 2a$

by calculation.The plane$\{x+yr:(x,y)\in \mathbb {R} ^{2}\}\subset H$is isomorphic to$\mathbb {C}$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 rotationactioncan also be viewed through thespecial unitary groupSU(2).

For a fixedr,versors of the form exp(ar) wherea∈(−π, π],form asubgroupisomorphic to thecircle group.Orbits of the left multiplication action of this subgroup are fibers of afiber bundleover the 2-sphere, known asHopf fibrationin the caser=i;other vectors give isomorphic, but not identical fibrations. In 2003 David W. Lyonswrote "the fibers of the Hopf map are circles in S³"(page 95). Lyons gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions.

Versors have been used to represent rotations of theBloch spherewith quaternion multiplication.

Elliptic space

The facility of versors illustrateelliptic geometry,in particularelliptic space,a three-dimensional realm of rotations. The versors are the points of this elliptic space, though they refer torotations in 4-dimensional Euclidean space.Given two fixed versorsuandv,the mapping$q\mapsto uqv$is anelliptic motion.If one of the fixed versors is 1, then the motion is aClifford translationof the elliptic space, named afterWilliam Kingdon Cliffordwho was a proponent of the space. An elliptic line through versoruis$\{ue^{ar}:0\leq a<\pi \}.$Parallelism in the space is expressed byClifford parallels.One of the methods of viewing elliptic space uses theCayley transformto map the versors to$\mathbb {R} ^{3}$

Subgroups

The set of all versors, with their multiplication as quaternions, forms acontinuous groupG.For a fixed pair {r,−r} of right versors,$G_{1}=\{\exp ar:a\in \mathbb {R} \}$is aone-parameter subgroupthat is isomorphic to thecircle group.

Next consider the finite subgroups, beyond thequaternion groupQ₈:

As noted byAdolf Hurwitz,the 16 quaternions ( ±1 ±i ±j ±k)/2 all have norm one, so they are inG.Joined with Q₈,these unitHurwitz quaternionsform a groupG₂of order 24 called thebinary tetrahedral group.The group elements, taken as points on S³,form a24-cell.

By a process ofbitruncationof the 24-cell, the48-cellonGis obtained, and these versors multiply as thebinary octahedral group.

Another subgroup is formed by 120icosianswhich multiply in the manner of thebinary icosahedral group.

Hyperbolic versor

A hyperbolic versor is a generalization of quaternionic versors toindefinite orthogonal groups,such asLorentz group. It is defined as a quantity of the form

$\exp(a\mathbf {r} )=\cosh a+\mathbf {r} \sinh a$where$\mathbf {r} ^{2}=+1.$

Such elements arise insplit algebras,for examplesplit-complex numbersorsplit-quaternions.It was the algebra oftessarinesdiscovered byJames Cocklein 1848 that first provided hyperbolic versors. In fact, James Cockle wrote the above equation (withjin place ofr) when he found that the tessarines included the new type of imaginary element.

This versor was used byHomersham Cox(1882/83) in relation to quaternion multiplication.The primary exponent of hyperbolic versors wasAlexander Macfarlaneas he worked to shape quaternion theory to serve physical science.He saw the modelling power of hyperbolic versors operating on the split-complex number plane, and in 1891 he introducedhyperbolic quaternionsto extend the concept to 4-space. Problems in that algebra led to use ofbiquaternionsafter 1900. In a widely circulated review of 1899, Macfarlane said:

...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.

Today the concept of aone-parameter groupsubsumes the concepts of versor and hyperbolic versor as the terminology ofSophus Liehas replaced that of Hamilton and Macfarlane. In particular, for eachrsuch thatr r= +1orr r= −1,the mapping$a\mapsto \exp(a\,\mathbf {r} )$takes thereal lineto a group of hyperbolic or ordinary versors. In the ordinary case, whenrand−rareantipodeson a sphere, the one-parameter groups have the same points but are oppositely directed. In physics, this aspect ofrotational symmetryis termed adoublet.

In 1911Alfred Robbpublished hisOptical Geometry of Motionin which he identified the parameterrapiditywhich specifies a change inframe of reference.This rapidity parameter corresponds to the real variable in a one-parameter group of hyperbolic versors. With the further development ofspecial relativitythe action of a hyperbolic versor came to be called aLorentz boost.

Lie theory

Main article:Lie theory

Sophus Liewas 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 Robert Gilmore in his text on Lie theory.Sl(1,q) is thespecial linear groupof one dimension over quaternions, the "special" indicating that all elements are of norm one. The group is isomorphic to SU(2,c), aspecial unitary group,a frequently used designation since quaternions and versors are sometimes considered archaic for group theory. Thespecial orthogonal group SO(3,r) of rotations in three dimensionsis closely related: it is a 2:1 homomorphic image of SU(2,c).

The subspace$\{xi+yj+zk:x,y,z\in R\}\subset H$is called theLie algebraof the group of versors. The commutator product$[u,v]=uv-vu\,$just double thecross productof two vectors, forms the multiplication in the Lie algebra. The close relation to SU(1,c) and SO(3,r) is evident in the isomorphism of their Lie algebras.

Lie groups that contain hyperbolic versors include the group on theunit hyperbolaand the special unitary groupSU(1,1).

Etymology

The word is derived from Latinversari= "to turn" with the suffix-orforming a noun from the verb (i.e.versor= "the turner" ). It was introduced byWilliam Rowan Hamiltonin the 1840s in the context of hisquaterniontheory.

Versors in geometric algebra

Main article:Geometric algebra § Versor

The term "versor" is generalised ingeometric algebrato indicate a member$R$of the algebra that can be expressed as the product of invertible vectors,$R=v_{1}v_{2}\cdots v_{k}$.

Just as a quaternion versor$u$can be used to represent a rotation of a quaternion$q$,mapping$q\mapsto u^{-1}qu$,so a versor$R$in Geometric Algebra can be used to represent the result of$k$reflections on a member$A$of the algebra, mapping$A\mapsto (-1)^{k}RAR^{-1}$.

A rotation can be considered the result of two reflections, so it turns out a quaternion versor$u$can be identified as a 2-versor$R=v_{1}v_{2}$in the geometric algebra of three real dimensions${\mathcal {G}}(3,0)$.

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 asunit versorsin a geometric algebra if$R{\tilde {R}}=\pm 1$,where the tilde denotesreversionof the versor.

See also

• cis (mathematics)(cis(x) = cos(x) +isin(x))
• Quaternions and spatial rotation
• Rotations in 4-dimensional Euclidean space
• Turn (geometry)

References

• William Rowan Hamilton(1844 to 1850)On quaternions or a new system of imaginaries in algebra,Philosophical Magazine,link to David R. Wilkins collection atTrinity College, Dublin.
• William Rowan Hamilton (1899)Elements of Quaternions,2nd edition, edited by Charles Jasper Joly, Longmans Green & Company. See pp. 135–147.
• Arthur Sherburne Hardy(1887)Elements of Quaternions,pp. 71,2 "Representation of Versors by spherical arcs" and pp. 112–8 "Applications to Spherical Trigonometry".
• Arthur Stafford Hathaway(1896)A Primer on Quaternions,Chapter 2: Turns, Rotations, Arc Steps, fromProject Gutenberg
• Cibelle Celestino Silva, Roberto de Andrade Martins (2002) "Polar and Axial Vectors versus Quaternions",American Journal of Physics70:958. Section IV: Versors and unitary vectors in the system of quaternions. Section V: Versor and unitary vectors in vector algebra.
• Pieter Molenbroeck (1891)Theorie der Quaternionen,Seite 48, "Darstellung der Versoren mittelst Bogen auf der Einheitskugel", Leiden: Brill.
• Hestenes, David;Sobczyk, Garret (1984),Clifford Algebra to Geometric Calculus, a Unified Language for Mathematics and Physics,Springer Netherlands,ISBN978-90-277-1673-6
• Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007),Geometric algebra for computer science: an object-oriented approach to geometry,Elsevier,ISBN978-0-12-369465-2,OCLC132691969

External links

• VersoratEncyclopedia of Mathematics.
• Luis IbáñezQuaternion tutorialArchived2012-02-04 at theWayback MachinefromNational Library of Medicine